Fuzzy commutative algebra

Fuzzy Commutative Algebra

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Fuzzy Commutative Algebra

John N Mordeson D S Malik Creighton University

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

FUZZY COMMUTATIVE ALGEBRA Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-3628-X

This book is printed on acid-free paper.

Printed in Singapore by U t o P r i n t

To Our Parents

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vii

Contents FOREWORD

x

PREFACE

xii

LIST OF SYMBOLS

xv

1

1 1 6 10 15 20 29 32 34 36 39

L-SUBSETS A N D L - S U B G R O U P S 1.1 L-Subsets 1.2 L-Subgroups 1.3 Normal L-Subgroups 1.4 Homomorphisms and Isomorphisms 1.5 Complete and Weak Direct Products 1.6 Embedding of Fuzzy Power Sets 1.7 Representation of the Fuzzy Power Algebra 1.8 The Metatheorem 1.9 Applications and Unifications 1.10 EXERCISES

2 L - S U B G R O U P S OF A B E L I A N G R O U P S 2.1 Generators and Direct Sums of L-Subgroups 2.2 Independent Generators 2.3 Primary L-Subgroups 2.4 Divisible and Pure L-Subgroups 2.5 Invariants of L-Subgroups 2.6 Basic and p-Basic L-Subgroups 2.7 EXERCISES

41 41 47 48 50 56 60 68

3 L-SUBRINGS A N D L-IDEALS 3.1 Basic Concepts 3.2 Quotient L-Subrings 3.3 Direct Sums

70 70 83 90

VIII

3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 4

Maximal L-Ideals and Irreducible L-Ideals Prime L-Ideals and Semiprime L-Ideals Radicals of L-Ideals Primary L-Ideals ^-Radicals of L-Ideals 7£-Primary L-Ideals and 7^-Semiprimary L-Ideals Characterization of Artinian Rings L-Generators EXERCISES

L-SUBMODULES 4.1 Basic Concepts 4.2 L-Submodules of Quotient Modules 4.3 L-Submodules Generated by L-Subsets 4.4 Free L-Submodules 4.5 Residual Quotients 4.6 Primary L-Submodules 4.7 7r-Primary L-Submodules 4.8 Primary Decompositions 4.9 EXERCISES

96 100 104 108 Ill 115 119 121 129 131 131 136 139 140 145 149 154 156 160

5 L-SUBFIELDS 5.1 L-Subfields and L-Field Extensions 5.2 Separable and Inseparable Algebraic Extensions 5.3 Composites, Linear Disjointness, and Separability 5.4 Finite-Valued L-Field Extensions 5.5 Separability and Modularity 5.6 Neutrally Closed L-Subfields 5.7 Distinguished L-Subfields 5.8 Splitting 5.9 Purely Inseparable L-Field Extensions 5.10 EXERCISES

161 161 170 175 182 186 189 191 193 198 203

6

205 205 208 216 218 228 230 232 233

S T R U C T U R E OF L - S U B R I N G S A N D L-IDEALS 6.1 Comparison of Radicals 6.2 7£-Primary L-Representations 6.3 ^-Primary Representations 6.4 L-Prime Spectrum of a Ring 6.5 Quasi-local L-Subrings 6.6 Extension of L-Subsets 6.7 Extension of L-Subrings and L-Ideals 6.8 Extension of Prime L-Ideals

ix

6.9 6.10 6.11 6.12 6.13 6.14 6.15

L-Topological Spaces Complete L-Subrings ^Coefficient Fields Existence of L-Coefficient Fields Structure Results Completions EXERCISES

235 240 242 248 252 256 261

7

A L G E B R A I C L-VARIETIES A N D I N T E R S E C T I O N EQUA­ TIONS 266 7.1 Algebraic L-Varieties 267 7.2 Irreducible Algebraic L-Varieties 275 7.3 Localized L-Subrings 283 7.4 Local Examination 291 7.5 Fuzzy Intersection Equations 296 7.6 L-Intersection Equations 303 7.7 Union Equations 307 7.8 Applications and Examples 309 7.9 EXERCISES 311

8

L-SUBSPACES 8.1 Preliminary Results 8.2 L-Subspaces, //-Subgroups, and L-Subfields 8.3 Nonexistence of Bases 8.4 Existence of Bases 8.5 L-Bases 8.6 Dimension of L-Subspaces 8.7 Existence and Nonexistence of Bases 8.8 Examples 8.9 EXERCISES

315 315 321 329 332 334 337 343 347 353

9

GALOIS THEORY A N D G R O U P L-SUBALGEBRAS 9.1 Galois Theory 9.2 Dimension and Index 9.3 Infinite Fuzzy Galois Theory 9.4 The Galois Correspondence 9.5 Group L-Subalgebras 9.6 Construction of L-Field Extensions 9.7 EXERCISES

355 355 362 364 370 373 376 383

Bibliography

384

Index

401

X

FOREWORD In 1965 I read, with great delight, Lotfi Zadeh's groundbreaking paper "Fuzzy Sets" (Info. Control 8, 338-353). I especially enjoyed Lotfi's "natural" (and elegant!) generalizations of various basic mathematical concepts, starting with the algebra of sets (with <, V, A generalizing C, U, fl; but although (or be­ cause?) I was then busily engaged in writing a textbook on algebra (An In­ troduction to Algebraic Structures, Holden-Day, San Francisco, 1968), it didn't occur to me to try my hand at fuzzifying other kinds of mathematical struc­ tures. The idea of trying to fuzzify algebra finally dawned on me several years later, after Lotfi's student C. L. Chang had published his paper "Fuzzy topological spaces" (J. Math. Anal. Appl 24 (1968) 182-190). After somewhat belatedly coming across that paper, I said to myself "If Chang can do it for topological spaces, I can do it for algebraic structures". I took a copy of my own algebra book with me on a train ride from Washington to New York, with the intention of formulating natural fuzzifications of the basic concepts of algebra. Needless to say, I found a way to do this; in fact, by the end of the train ride I had written an essentially complete draft of my embarrassingly well-cited paper "Fuzzy groups" (J. Math. Anal. Appl 35 (1971) 512-517). As a former algebraist, I take special pleasure in having initiated the study of fuzzy algebraic structures. I regret that I have been unable to contribute to the subject myself since the appearance of my original paper; by the 1970's I had almost entirely abandoned algebra in favor of on assortment of other topics, most of them related to images and their geometric properties. My continued occasional involvement with fuzzy mathematics has expressed itself in non-algebraic directions, deahng primarily with fuzzy graph theory and geometry. Although I have been unable to participate personally in the vigorous growth of fuzzy algebra, I have derived great enjoyment from observing and encouraging its growth. It therefore gives me special pleasure to welcome the appearance of the first full-length book on the subject. I hope that this happy event is indicative of a new wave of activity in the field, leading both to further

XI

theoretical developments and to an increased interest in applications. Azriel Rosenfeld College Park, MD August 1, 1998

Xli

PREFACE The notion of a fuzzy subset of a set is due to Lotfi Zadeh. His seminal paper in 1965 has opened up new insights and applications in a wide range of scientific fields. Azriel Rosenfeld used the notion of a fuzzy subset to set down cornerstone papers in several areas of mathematics, among other disciplines. Rosenfeld is the father of fuzzy abstract algebra. This book is the first to be devoted entirely to fuzzy abstract algebra. The purpose of this book is to give an up to date version of fuzzy commutative algebra. We present a fuzzy ideal theory of commutative rings and apply our results to the solution of fuzzy intersection equations. This treatment includes all the work that has been done on algebraic L-varieties. The book includes all the important work that has been done on L-subspaces of a vector space and on L-subfields of a field. We focus our attention on the connection between L-subgroups of a group and L-subfields of a field. This connection is made in two ways. We present a Galois theory, both finite and infinite, which in the finite case gives a one-to-one inclusion reversing correspondence between finitevalued L-subgroups of the group of automorphisms of a normal field extension and its L-subfields. In the infinite case, we give a Galois correspondence be­ tween L-subgroups whose level sets are closed in the Krull topology of the group of automorphisms of a splitting field of a set of separable polynomials over a field. Our second focal point is the connection between L-subgroups of an Abelian group and L-subfields of a field constructed by means of a group al­ gebra. We do not give a complete treatise on L-subgroups. Rather we present results on L-subgroups which are of interest to the connections discussed above. In Chapter 1, we present the important work of Tom Head and Arthur Weinbereger concerning methods for deriving fuzzy theorems from crisp ver­ sions and embedding lattices of fuzzy algebras into lattices of crisp subalgebras. Other than that, we present only material that is needed for the development of the remainder of the book and that may be useful for the further develop­ ment of fuzzy Galois theory and fuzzy group subalgebras. Since the work of Weinberger is somewhat outside the confines of fuzzy commutative algebra, we only present a summary of it.

xiii

Essentially only those results concerning L-subgroups of abelian groups which are needed in the development of group L-subalgebras are presented in Chapter 2. In Chapter 3, we give some basic concepts and properties of L-ideals of a commutative ring with identity. These concepts include those of prime L-ideals, primary L-ideals, and various radicals of L-ideals. Many authors contributed to the development of L-ideal theory. Consequently, we will not attempt to mention them here. Our bibliography lists those who were respon­ sible for the development. The concepts of quotient L-submodules, free L-submodules, residual quo­ tients, and primary L-ideals are given Chapter 4. We do not give a complete treatment of L-submodules, but we do give an extensive bibliography on Lsubmodules. However we stop short of including papers which fall into the realm of category theory. Chapter 5 deals with L-subfields and L-field extensions. Concepts such as separability and inseparability for L-field extensions are developed. We introduce a concept for L-field extensions which is not present in ordinary field extensions, namely that of neutral L-field extensions. In Chapter 6, we delve deeper infco the realm of L-subrings. We present a primary representation theory for L-ideals which is used in the next chapter to examine fuzzy intersection equations. Other topics in this chapter include the L-prime spectrum of a ring, quasi-local L-subrings, and L-coefficient fields. The results of this chapter are mostly those of the authors. In Chapter 7, we give a complete account of algebraic L-varieties and fuzzy intersection equations. This material is entirely that of the authors. We also examine fuzzy intersection equations locally. In Chapter 8, we give an axiomatic approach to the concept of the gen­ eration of L-algebraic substructures. We determine the generating properties which yield the underpinnings for the existence of free generating sets and the uniqueness of their cardinalities for L-algebraic substructures. These Lsubstructures include concepts of L-subspaces of vector spaces, relative p-bases for L-subfields, transcendence bases for L-subfields, and L-subgroups in terms of p-bases and torsion-free bases. Many authors contributed to the develop­ ment of L-subspaces. They are included in our bibliography. We do not include in the bibliography papers dealing with topological vector spaces. In Chapter 9, we make a connection between L-subgroups and L-subfields in two ways. First, we make the connection by means of Galois theory and then by means of group algebras. Here also the work is entirely that of the authors. We assume that the reader is familiar with the basic results and definitions from abstract algebra and lattice theory. The authors are grateful to the staffs of World Scientific Publishers, es-

xiv pecially Sen Hu and S. H. Gan. We are indebted to Pr. Michael Proterra, Dean, Creighton College of Arts and Sciences and to Dr. and Mrs. George Haddix for their support of our work. We benefited from the work of Professor Yu Yandong during his visit. We also wish to thank Professor Paul Wang of Duke University for his support of fuzzy mathematics. The first author dedicates the book to his grandchildren, John, Mary Pat, Michael, Christine Marie, Matthew, Marc, and Jack. The second author dedicates the book to his daughter Shelly Malik, who is a joy and her smile is what counts. John N. Mordeson D. S. Malik

XV

LIST OF SYMBOLS A V

N Z Q R

0

I L

e

t

c c

D D

A\B Lx

[o,i] x

M(X)

Im(/i) M* Ma

P(S)

u n

foot(<S) a\b a/6 o

n

ft

i€/

/(*») /_1(")

minimum or infimum maximum or supremum the set of positive integers the set integers the set of rational numbers the set of real numbers the empty set an arbitrary nonempty index set complete Heyting algebra belongs to does not belong to subset proper subset contains properly contains set difference L-power set of X fuzzy power set of X fJ,(X) = {//(x) | x € X } , the image of /x Im(/x) = {fJ>(x) | x G X } , the image of /i fj,* = {x \ x £ X,ii(x) > 0}, the support of \x fj,a = {x | x € X,fj,(x) > a } , a-cut or a-level set of /x power set of the set S union of sets intersection of sets foot(<S) = {y | ya € 5 } a divides b a does not divide 6 composition product complete direct product image of \i under / the pre-image or inverse image of v under /

xvi /iOI/

M- 1 L(G) M. <M> NL(G)

NO*) \1
G/ii v/fi r^>

~ « =

the product of \x and 1/ the inverse of fi the set of all L-subgroups of G /x* = { i G G | JJL(X) = /x(e)}> where G is a group the L-subgroup of G generated by /i the set of all normal L-subgroups of G the normalizer of /i in G [i is a normal L-subgroup of the L-subgroup v the quotient group or factor group of G relative to the normal L-subgroup fi the quotient L-subgroup weakly homomorphic weakly isomorphic homomorphic isomorphic

n n

weak product

C(X) 9* £(F) L(p) ^(/*)

the crisp power set of X representation function the characteristic function of Y L(JJ) = {vG L{G) \vQfi «F(M) = {veLG \vCfi} the sum of /x and v the difference of \i and v the negative of fi the set of all L-subrings of R the set of all L-ideals of R /z* = {x £ R | fj.(x) = ^(0)}) where R is a ring

iei

Ji + V \X — V

-M

L(fl) L/(.R) ju

M1 n

/*

« /«W <M>

M

R/n

E*

^c

V? W F(J>) fj. : v M:C

L(F)

weak direct product

/i1 = / i /x n = / i 1 o / i n - 1

^>=/i

//"> = //v— 1 )

•the L-ideal of i? generated by // the quotient ring of R by /i weak direct sum the family of all prime L-ideals fj, of R such that £ C /z and £* C //* L-radical of £ 1l(/i)(x) = VneNM(zn)V i e i ? , the ^-radical of fi HP)

= {C € L* | c c P}

fi : v = U{77 | 77 G L H , 77 • 1/ C /i}, residual quotient /i : C = U{f | ^ L M , ( ^ C / i } , residual quotient the set of all L-subfields of F

xvii

v/n

L(u) ^ M# QM

Kdl(fi) X

v(x)

L-Spec(R) Fspec(-R) (R,M) R#

M{n) FS{V)

n») v?

VS(n)

°# L(F) L(F/K) AK C{F/K) K[G) [a:K] [F:<x]

H

[A: n] L(\)

L-field extension L{u) = {M € L(F) 1/xC./} neutral closure of an L-subfield \i fi# = {x€R\ fi(x) > /i(l)} Qn = {T)\T) is a £-prime L-ideals of i2 such that 77 D /x} fcd/(/x) = D r ^ Q ^ 3E = {77 | 77 is a prime L-ideal of R}

V(x) = iv e x | x c r,}

L-prime spectrum the fuzzy prime spectrum complete local ring R with unique maximal ideal M set of nonnegative real numbers algebraic L-variety of /x the set of all sets S of L-singletons of V such that if xa,Xb € <S, then a = b > 0 ^(/x) = { X G L v | X C /i} V^/T^a,!]) P5(/x) = {<S | <S e F S ( V ) , Vz a € 5 , x a C /x}. ( 7 # = { l G F | d(x) > (7(1)} the set of all L-subfields of F L(F/K) = {
L(\/K)

L(X/K)

s(0

5 ( 0 = { * « ! * € £ • , « * ) = <*} end of proof

■

= {4> € J"(F)

|/cC«/)CA}

1

Chapter 1

L-SUBSETS AND L-SUBGROUPS In this chapter, we present only material that is needed for the development of the remainder of the book and that may be useful for the further development of fuzzy Galois theory and fuzzy group subalgebras. A complete Heyting algebra L is a complete lattice such that for all A C L and for all b E L, V{a Ab \aeA} = (V{a | a G A}) Ab and A{aWb \ a e A} = (A{a | a€ A})Wb. Unless otherwise stated, L always denotes a complete Heyting algebra consisting of at least two elements. We at times assume that L is a chain or a dense chain. In these cases, L is always assumed to be complete. The meet, join, and partial ordering of L will be written as A, V, and <, respectively. We also write 1 and 0 for the maximal and minimal elements of L, respectively. L is said to be regular if Va, b € L such that a = ^ 0 ^ f c , a A 6 ^ 0. The closed interval [0,1] together with the operations min, max, and < form a complete Heyting lattice. We often write A for minimum or inftmum and V for maximum or supremum. It is assumed throughout that the least upper bound of the empty set is the 0 element of L and the greatest lower bound of the empty set is 1. We let X, Y, and Z denote nonempty sets. We let N denote the positive integers, Z the set of all integers, Q the set of rational numbers, R the set of real numbers, and 0 the empty set. Unless otherwise stated i" denotes an arbitrary nonempty index set. Many of the results of Sections 1.1-1.5 are taken from [250].

1.1

L-Subsets

Definition 1.1.1 By an L-subset of X, we mean a function from X into L. The set of all L-subsets of X is called the L-power set of X and is denoted

2

by Lx. In particular, when L is [0,1], the L-subsets of X are called fuzzy subsets and the set [0, l]x is referred to as the fuzzy power set of X. Definition 1.1.2 Let \x G Lx. Then the set {fi(x) \ x G X} is called the image of JJL, and is denoted by fi>(X) or Im(fj,). The set, {x | x 6 X, fi(x) > 0}, is called the support of \x and is denoted by fi*. In particular, \i is called a normal or unitary L-subset z/1 G fi>(X). [i is called a finite L-subset if (i* is a finite set, and an infinite L-subset otherwise. If n G Lx, then \i is said to have the sup property if every subset of [i(X) has a maximal element. Definition 1.1.3 When Y C X and a G L, we define ay G Lx , yW

a

N

as follows:

(a for x G Y = { 0 forxeX\Y.

In particular, if Y is a singleton, say, {y}, then a[yy is often referred to as an L-point (or L-singleton), and, sometimes, denoted by ya. We refer to l y as the characteristic function of Y. If S is a set of L-singletons, then we let foot(S) = {y | ya G S}. Definition 1.1.4 Let /z, v G Lx. If fi(x) < v(x) V x G l , then we say that fi is contained in v (or v contains fi) and we write / i C v (or v 3 /i). If ji C v and / I / I / , then fi is said to be properly contained in v (or v properly contains /x) and we write /J, C v (or v Z> fi). Clearly, the inclusion relation C is a partial ordering on

Lx.

Definition 1.1.5 Let fi, v G Lx. Define jiKJv and /J,C\V G Lx as follows: V x GX (fiUv)(x) = n(x) V v(x), (fiC\i/)(x) = /i(x) Av(x). Then fi U v and /x fl v are called the union and intersection respectively.

of \i and is,

Theorem 1.1.6 The L-power set Lx of X, together with the operations union and intersection, constitutes a complete Heyting algebra whose partial order­ ing is C . Its maximal and minimal elements are lx and Ox? respectively. Intersection is infinitely U-distributive on Lx. Moreover, the lattice L can be isomorphically embedded into the lattice Lx. ■

3

The proof of Theorem 1.1.6 is left to the reader. However, we would like to point out that for any family, {fii \ i G / } , of L-subsets of X, where / is a nonempty index set, the least upper bound U;€/Mz and the greatest lower bound DiziHi of the /Vs are given by Vx G X,

(uieIieIfJLi)(x) fii)(x) (U

== =

(n ( nieI * €fii)(x) //it)(a:)

===

V\zi ieI ^(x) G /m(x) A»Ai£i(j,i(x), //Zi(a?), e

respectively. Isomorphic embeddings of the lattice L into the lattice Lx, in general, are not unique, since for any nonempty subset 7 of X, the mapping a —► ay from L to Lx is clearly an isomorphic embedding. We write

r\iei^i rii nj^i/zi = MI fii n M jj,22 n., n... n ^n ..n^n €7/x* = nj^i/xi and Uieifii = UjLx/Xi = Mi U M2 U ... U iin when / = { l , 2 , . . . , n } . Furthermore, we have the following theorem. T h e o r e m 1.1.7 I7ie lattice Lx is an infinitely distributive (dense or Boolean) lattice if and only if L is infinitely distributive (dense or Boolean). I Definition 1.1.8 Let \x G Lx.

For a G L, define fia as follows:

fia = {X {x | x G Xjfi(x) X, ^(x) >> aa}. }. Ma /xa is ca/Zed #ie a-cut (or a-level set ) of p. It is easy to verify that t h a t for any /z> /z, v G G 7£Xx ,, fJLQl/ GL L => => nnaa CC i/i/ajaj C: v,, aa G ((1) i ) /i a < 6, a, 6 G L => M6 (2) a < 6, a, 6 G L => fibSCMa, Ma, (1\

<*-\ /# (3) /i = i/ < => /xa = 77i^a Y//7 Va £=G7".L. The next two theorems display some basic properties of cuts. Their proofs are straightforward. /;

—

1/

T h e o r e m 1.1.9 Suppose that {M* \ i G 1} C Lx.

Then for any a G L,

(1)UUiieIei{v>i)a (fJ,i)a C (Ui (Ut G j/Xi) 0 , W = ( n i e €/ //xt) / i i ) a 0, , (%> niGl(Hi)a #;n< G/(Mi)a = (n*

Moreover, when L is a finite chain, we have equality in (1). H T h e o r e m 1.1.10 Let M G Lx and {a; | i G 7} 6e a non-empty subset of L. Let b = Aieiai and c = V^/a;. Then, (1) U*€/ UieI [iai^ £ C /x&, w»

w (%) n» / / i ^

^iei€

0^i

■

= Mc ■

T h e o r e m 1.1.11 Let fi e Lx.

Then M = U0€i,a/za =

^ae^x)afxa-

4

Proof. Let x G X. Now (U^i/VaX^) = VaeLO>iia(x) = v { a G L | a < /i(rr)} = /i(x). Thus /i = UaeLaM«' Similarly, fi = ^aey.{X)a^a- " Definition 1.1.12 Let {Xi \ i G / } , where I is a nonempty index set, be a family of nonempty sets, and let X denote the Cartesian product of the Xi's, namely, }■ X-| Xi eXu i e / I}. X = Y[Xi = {{x%)iei {(*<)*£/ \xieX uie iei Let iii G LXi for all i G I. An L-subset \x G L x , given by Vx = (xi)iej

GX

x == AieAifJLi(Xi) fl(x) ieifJLi(Xi)

ri ) =

is called the complete direct product of the /x;'s and is denoted by

/* =

iei iei

Jil = { 1 , 2 , . . . , n } , then X

= = = =

UXi U^i iei iei

Xi x X 2 x .... . . x Xn , XfiJ 1 X^ \Z -^»-i) * — -L, ^ , • • . , n } , { ( x i , x 2 , . •. . -,^n) {(^1,^2,I ^ G Xi,z = 1,2,... , n } ,

and we write JJ/Xi |"JA**

= /Xi/i2§> n. /Xl§/i2§>.• .•. ®• ^®^n-

t€/

Clearly, if \ii, ^ G L X i with ^ C ^ for each i G / , then

Eh Definition 1.1.13 (Extension Principle) Let f be a mapping from X into Y", and let JJ, e Lx and v G LY. The L-subsets f(/i) G LY and / _ 1 ( ^ ) G L x , defined by Vy G Y, V{^(x) | xex, x € X, f(x) = y} y} t/ /-Hj/) 7^ 0, f(x) =-/ r „ w t A _ / V{/x(x) J/(/*)(
■{

and Vx € X, (/(*)), r »» (( xx )) === "K/(*)),

are called, respectively, the image of \x under f and the pre-image verse image) of v under f.

(or in-

5

Recall that in Definition 1.1.13, the least upper bound of the empty set is the zero element 0 of L. Theorem 1.1.14 Let f be a mapping from X into Y and g a mapping from Y into Z. Then the following assertions hold: x e /I,, /(Uie/Mi) /(Ui 6 /^i) = — ^i£if{^i) Uieif(fii) and hence G L,ix, i G (1) For any \ii eL x Mi Q Q H2 V2 => f(fi = > /ffai) 0 * I ) CC /(A * 22)) VV/ii,/i M I . ,^eL Ml 2 G.

Lx.

(2) For any Vj G LY, j G J, where J is a nonempty index set, /" _1(Uj€J^) /" "'(n.ej^)

= =

U

j e

j/'

"V;),

n j € j / " ">;),

ana7 hence v i C i ^ ^ / _ 1 (i>i) C / _ 1 ( ^ 2 ) VJ/I.^2 € L y . (3) / _ 1 ( / ( ^ ) ) 2/^V/iG L x . /n particular, if f is an injection, then

r

■\M) r1(f( fi))

-= =

xx ^v n^ij,eL fieL ,,

and hence, \i H-» /(/Z) is an injection from Lx into LY and v — i > f~x{y) Y x surjection from L onto L . (4) / ( / _ 1 ( ^ ) ) ^ z/ Vi/ G L y . in particular, if f is a surjection, then

is a

1 ( « ' ) ) ~==« uVue ' V i / € L yL, , / ( /r " ■\v)) Y

and hence, n *—► /(/u) is a surjection from Lx onto LY and u — i ► f~l(v) Y x injection from L into L . (5) /(/*) Qv&iiC f~\v) V/i € L x and Vi/ e £ v . f5j 9(/(M)) - (9 ° /)(/*) VM € L * and

z rr1(9-Hz)) (9°f)-Hz)vzeL = ( / ) - ■ (0 V£ € L .. V ■ (0) == x

1

Proof.

z

Clearly, assertions (1) and (2) hold. Next, for any /x G 1 //--1-(/(M))(^) (/(/*))(«)

= /(/*)(/(*)) /(**)(/(*)) -= D

V{/i(x') V { M ( x ' )| |x' x ' e X*,/(*') , / ( x ' ) == /(x)} = /(*)}

M(X)

Vx € X; and, in particular, / V{{MM (xM /-H/MX*) - V | X:'€X,f(x')-€ X , / ( X ' ) == //(*)} (X)} • ( / ( M ) ) ( X ) ^= f~ Oh x /i(x) = K) 1

V x G X when / is an injection. This shows that assertion (3) holds.

is an

6 Also for any v G L y , we have that

/(r»)G/) = v{f-1(")(x)\x£XJ(x) = y} =

<

vMf(x))\xeXJ(x) f v(y) for y € f(X) [ 0 otherwise v(v)

= y}

V y € K Thus f{f~l{v))(y) = ^(y) V y € Y when / is a surjection. This completes the proof of assertion (4). Assertion (5) is an immediate consequence of the four preceding assertions. Finally, for all n € Lx, g(f(n))(z)

= = =

V{f(p)(v)\veY,g(y) = z} V{V{n(x) \x€X, f{x) = y} | y € Y, g(y) = z} V{/x(z) I x € X, (go f)(x) = z}

=

(gof)M(z)

V z € Z. For any £ € L z ,

(G/o/rHOX*) = £(
- s-^x/o*)) = /-^(ox*)

V i G l . This proves assertion (6). ■

1.2

L-Subgroups

In the remainder of this chapter, G always denotes an arbitrary group with a multiphcative binary operation and identity e. In order to define the notion of L-subgroups and to examine their properties, we introduce some operations on L-subsets of a group G in terms of the group operation. Definition 1.2.1 We define the binary operation o on LG and the unary op­ eration _ 1 on LG as follows: V/x, v G L G , Vx G C?, (fi o v)(x) = W{fi(y) A v{z) \y,zeG,yz ^~x{x) We call /J, o v the product

= x},

=fi(x~1).

of fj, and v, and /x _1 the inverse

of /JL.

7

Theorem 1.2.2 Let /x, v, fi{ e LG, i G I. Let a = V{u(x) \ x eG}. (l){l*ov)(x)

Then

= Vyec(Ky) A v{y~xx)) =

VyeoiKxy-1)

Av(y))

V x eG; (2) (ay on)(x) = fiiy^x) \/x,y e G; (3) (/x o ay)(x) = n{xy~l) Vx, y e G;

«=>

/x" 1 C /i

<£> xx = / x - 1 <£> /x(x) < / x ( x - 1 ) V x G G

^ M ( ^ _ 1 ) < A*0*0 Vx G G <^> /x(x) = zx(x_1) Vx € G; f^j/iC i/^xx"1 Ci/"1; ( r 7;(Ui6/Mt)- 1 = u € / / x r 1 ; (s; ( n i € / / x i ) - 1 = n i 6 / xx- 1 ; (9; ( M O I / ) - 1 =

Z/-1O/X-1.B

Definition 1.2.3 An L-subset /x of G is called an L-subgroup of G if (Gl) /J,(xy) > fi(x) A ii(y) V x, y e G, and (GS) M * - 1 ) > /x(x) V x e G. If L = [0,1], £/ien an L-subgroup is called a fuzzy subgroup. Denote by L(G), the set of all L-subgroups of G. If /x e L{G), we let ix* = {x € G | /x(x) = fJ>(e)}. If [i e LG satisfies condition (Gl), then xx(xn) > /x(x) Vx e G, where n G N. Also, /x satisfies (Gl) and (G2) if and only if /x(x2/_1) > /x(x) A

Kv) Vx,y zG. Lemma 1.2.4 Let xx G L{G). Then V x G G, (1) xx(e) > xx(x); (2)/x(x) = /x(x- 1 ).

8

Proof. Let x G G. (1) //(e) = fi(xx~x) > fj,(x) A //(# - 1 ) > //(x) A fi(x) = //(x). (2) n(x) = //((aT-1) *) > /i(x _ 1 ) > fi(x). Hence //(x) = //(a: - 1 ). ■ Lemma 1.2.5 Let // G LG. Then fj, is an L-subgroup of G if and only if \xa is a subgroup of G V a G //(G) U {b G L \ b < //(e)}. Proof. Suppose /x is an L-subgroup of G. Let a G //(G). Since //(e) > //(#) V x G G, e G //a- Thus // a ^ 0. Let x,y G /i a . Then //(x) > a and //(?/) > a. Since /i is an L-subgroup, n{xy~l) > //(a;) A//(y) > aAa = a. Hence xy~l G // a and so // a is a subgroup of G . Similarly, if a < //(e), then \xa is a subgroup of G. Conversely, suppose // a is a subgroup of G V a G //(G) U {6 G L | 6 < //(e)}. Since e e fj,a Va € A^(G), //(e) > aV a e //(G). Let x,y e G and let /z(x) = a and /z(y) = 6. Let m = a Ab. Then x,y £ \Xm and ra < //(e). Thus /zm is a subgroup of G and hence xy~x G /z m . Hence fi{xy~l) >m = aAb = /Z(X)A/Z(T/). Thus // is an L-subgroup of G. I Corollary 1.2.6 / / / / G L(G), then //* zs a subgroup of G. M Theorem 1.2.7 / / / / G L(G) and L is regular, then //* zs a subgroup of G. I Theorem 1.2.8 Le£ // G L G . T/ien // G L(G) z'/ and on/y if // satisfies the following conditions: (Giy nonQn (G2)' fj,'1 C // ('or / i " 1 D //, or /z" 1 = / / ) . ■ Theorem 1.2.9 Let //, ^ G L(G). TTien // o z/ G L(G) Z/ and on/z/ if \iov — v o /x. Proof. Suppose that // o z/ G L(G). Then /ioi/ = / i _ 1 o z/ -1 = (z/ o /x) _1 = z/o /x. Conversely, suppose that // o z/ = z/ o /x. Then it follows that (// o v)~x = {y o / / ) _ 1 = // _ 1 o z/ -1 = jiov and (// o zy) o (/x o z/)

= = =

p o (is o fj.) o is fio (nov) ov (ji o ji) o {y o v)

C

p ZA

Consequently by Theorem 1.2.8, // o z/ G L(G). ■ Theorem 1.2.10 Le£ // G L(G) and H be a group. Suppose that f is an epimorphism of G onto H. Then /(//) G L{H).

9 Proof.

Let u,v G H. Then u = f(x) and v = f(y) for some x,y G G. Now

(/M)M

= V{MW | Z G G, /(*) = uv} > > = =

V{[M(xy) \x,yeG, f(x) = u, f(y) = v} W{fi(x) A fi(y) \x,yeG, f(x) = u, f(y) = v} (W{fi(x) | x G G, / ( * ) = u}) A (v{M(y) | y € G, /(
Clearly, (/(^))(u" 1 ) -

(/(M))(U).

Hence /(/x) G L(tf). ■

Theorem 1.2.11 Let H be a group and v G L(H). If f is a homomorphism ofG into H. Then f~l{v) G L(G). Proof. Let x,y e G. Now / - » ( x < / ) = v{f(xy)) = v{f(x)f{y)) > !/(/(*)) Ai/(/(y)) = / " » ( * ) A / - » ( < / ) • Also r 1 ^ ) ^ " 1 ) = ^ ( / ( x - 1 ) ) = v(fix)-1) = v{f{x)) = / - » ( * ) . Hence / - » € L(G). ■ Theorem 1.2.12 Let {/x; | i G 1} C L(G). TAera n^j/x; G L(G). ■ Definition 1.2.13 Let xx G LG. Lei (M) = n{i/1 / x c i / , i / € i ( G ) } . TTien (/i) is called the L-subgroup of G generated

by ji.

Clearly, (fi) is the smallest L-subgroup of G which contains /x. We shall present another procedure of constructing (/x). Define /i 1 = /x and fin = /x n - 1 o/i Vn G N, n > 1. T h e o r e m 1.2.14 Lei fi e LG and let a = A{n(e) | /x C 77, 77 G L(G)}. T/ien (xx) = a e U (U~=1(xi U fi-1)71) = U~ =1 (a e U /x U AX" 1 )". Proof.

Let i/ = a e U(U^ 0 = 1 (xxUxx- 1 r).

It is easily seen that xx C z/, a e C 1/, i/ _ 1 = z/, and 1/ o v — v. Thus v G L(G) by Theorem 1.2.8. Hence (/x) C 1/ since \i C z/. If £ G L(G) and /x C £, then a e C £ and (/x U /x" 1 ) 71 C f. Thus 1/ C £. Hence 1/ C (/x). Therefore, (/x) = ZA The other part of the theorem can be proved in a similar way. ■

10

1.3

Normal L-Subgroups

The notion of a normal subgroup is one of the central concepts of classical group theory. It serves as a powerful instrument for studying the general structure of groups. Just as a normal subgroup plays an important role in the classical group theory, a normal L-subgroup plays a similar role in the theory of L-subgroups. Theorem 1.3.1 Let fi G LG. Then (Nl) fi(yx) = fi{xy) V x, y G L-subset of G. (N2) fiixyx'1) = u(y) V x, y G (N3) nixyx-1) > u(y) V x, y G (N4) nixyx-1) < u(y) V x, y E (N5) fioi/ = uofi\f i/ e LG.

the following assertions are equivalent: G; in this case, fi is called an Abelian G. G. G.

Proof. (N1)=>(N2): Let x,y eG. Then ^(xyx'1) = ^ ( x - 1 • xy) = u(y). (N2)=>(N3): Immediate. (N3)=>(N4): nixyx-1) < ^ ( x " 1 • xyx" 1 • ( x " 1 ) " 1 ) = u(y) V x, y G G. (N4)=KN1): Let x,yeG. Then u{xy)

= < = <

//(x-i/x-x-1) l*(yx) u(y • xy • y'1) u{xy).

Hence fi{xy) = u(yx). (N1)=»(N5): Let x G G. Then (u o v){x)

=

V 2/€G {/x(xy- 1 ) A v(y)}

=

Vyeoiviy'1*)

A 1/(2/)}

Hence ^iov = uo^i. (N5)=»(N1): Now l { i r i } o/i = p l { y - 1 } \/y e G. Thus ( l { y - i } o u)(x) = (/i o l { 2 / -i } )(x) Vx,y G G. Hence fi(yx) = fi(xy) V x, y G G. ■ Definition 1.3.2 VFe say £/ia£ /x G L(G) zs a normal L-subgroup of G if it is an Abelian L-subset ofG. Also we denote by NL(G) the set of all normal Lsubgroups of G. If L — [0,1], we refer to normal L-subgroups of G as normal fuzzy subgroups.

11

If /x, v G L(G) and there exists u G G such that /x(#) = v(uxu~l) Vx G G, then ji and 1/ are called conjugate L-subgroups (with respect to u) and we write, fi = vu. Clearly, \Q and l e are normal L-subgroups as well. If G is a commutative group, every L-subgroup of G is normal. An L-subgroup zx of G is normal if and only if /x = \xz Vz G G. T h e o r e m 1.3.3 Le£ /i G L G . 77ien /x G NL(G) if and only if /x0 is a normal subgroup of G Va e fi(G) U {b G L | b < /x(e)}. Proof. Suppose that xx G NL(G). Let a G xx(G) U {b G L | 6 < //(e)}. Since /i G L(G), jia is a subgroup of G. When x e G and y G /ia> it follows from Theorem 1.3.1 that ^{xyx'1) = fi(y) > a. Thus xyx'1 G \xa. Hence \xa is a normal subgroup of G. Conversely, assume that /xa is a normal subgroup of G Va G /z(G) U {b G L I & < / i ( e )}- By Lemma 1.2.5, we know that fj, G L(G). Let x, y e G and a = /i(2/). Then 2/ G /xa and so xyx"1 G /i a . Hence fi(xyx~x) > a = /x(y). That is, /i satisfies condition (N3). Consequently, it follows from Theorem 1.3.1 that /x €NL(G).m T h e o r e m 1.3.4 Let \x G NL(G). Then /x* is a normal subgroup of G. More­ over, /x* is a normal subgroup of G when L is regular. Proof. Since /x G L(G), it follows from Lemma 1.2.5 that xx* is a subgroup of G, and, moreover, /x* is a subgroup of G when L is regular. Let x G G and y G /x*. Since zx satisfies condition (N2), we have ^(xyx-1) = fi(y) = /x(e) and thus xyx-1 G *x*. Hence /1* is a normal subgroup of G. Suppose that L is regular. Let x G G and y e fi*. Since /i satisfies condition (N2), it follows that ^(xyx-1) = fi(y) > 0 and so xyx~l G /x*. Therefore, /x* is a normal subgroup

ofG. ■ The proof of the following result appears in [250, Theorem 5.2.6, p. 122]. T h e o r e m 1.3.5 Let a G L,a ^ 0, and let NLa{G) = {/i G A/'L(G) | zx(e) = a}. The set NLa(G) together with C constitutes a complete lattice whose meet is L-subset intersection 0 and whose join is the product o. Moreover, NLa(G) is closed under intersection. I T h e o r e m 1.3.6 Let fi G L(G). Set iV(/x) = {x \ x G G, ii{xy) = /x(yx) V y G G}. T/ien iV(/x) is a subgroup of G and the restriction of /x £0 iV(/i), /i|jv(/u)> 25 a normal L-subgroup of N(ji).

12

Proof.

Clearly, e G N(fx). Let x, y G N(fi). For any z G G, we see that fi(xy~l • z)

= /x(x • t/" 1 *) = \i{3Txz • x) = fi{x~xz~x • y) = ^(yx-1^-1) = ^(z-xsT1).

Thus x y - 1 G iV(/x). Hence N(fi) is a subgroup of G. Also, it can be readily seen that ^\N(^) £ -^C^CAO) lAN{n)(yx) V x, 2/ G iV(/x). Therefore, ^| N ( / l ) G NL(N(fi)).

an

d ■

A*|AT(/X)0E2/)

=

The subgroup iV(^) of G defined in Theorem 1.3.6 is called the normalizer of fi in G. Theorem 1.3.7 Let v G L(G). Then the cardinal number of the set {vu \ u G G} is equal to the index [G : N(v)] of the normalizer N(v) in G. Proof. Let u, v G G. Then vu = vv <& v(uxu~l) = i/(vxv~l) V x G G <=> l v(uv~ • x) = v(x • uv~l) V x e G ^ m ; - 1 G AT(*/) «» w_1iV(i/) = v~lN(v). Thus i/u — i ► u~lN(v) is a bijection from {i/u | u € G} onto {liAT(^) | w G G}. I Theorem 1.3.8 Let v G £(G). Then nuec^u € NL(G) and is the largest normal L-subgroup of G that is contained in v. Proof. Since vu G L(G) V u G G, n nG G^u € £(G). For any x G G, it is clear that \yu \ u G G} = {J/UX | ZX G G}. Thus A-uGG^nC^^"1)

= = =

AUGGKUXJ/OXX) -1 ) ^ueGVux(y) AueGVu(y)

V x,y e G. Hence by Theorem 1.3.1, n u € c ^ n € NL(G). Next, let /z G NL(G) with /x C IA Clearly, /x = /xu C vu V u G G. Thus /x C fl u€ Gi/ w . Therefore, n u € G^u is the largest normal L-subgroup of G that is contained in v. ■ Let ft G £(G) and x G G. The L-subsets /x(e)^} o /i and xz O ^ ( e ) ^ } are referred to as the left coset and right coset of xz with respect to x, and written as xfi and fix, respectively. From Theorem 1.3.1, we know that if /x G NL(G), then the left coset x/x is just the right coset fix. So, in this case, we will call x/i a coset for short. Theorem 1.3.9 Let fi G L(G). Then for any x, y G G, (1) X/J, = yfi & xfi* = y\i*\ (2) fix = fiy O fi*x = fi*y.

13

Proof. Suppose that x\i = yji. Then fi{x~lz) = ji(y~1z) V z G G. Choosing z = y yields fj,(x~xy) = ji{y~1y) = /x(e) and thus x~ly G /x*. Hence xfi* = y[i*. Conversely, suppose that xfj,* = y/j,*. Then x~xy G /x* and y~x x G /x*. Hence /x(x _1 z) = /x(z_12/ -y~xz)

> ^(x^y) =

A^y^z)

e

_1

M )AM2/ 2)

= Kv~lz) V z G G. Similarly, fj,(y~1z) > fi(x~lz) Vz G G. Therefore, fi(x~lz) \iz G G. This proves (1). Similarly, (2) holds. ■ Theorem 1.3.10 Let \x G NL(G)

= ii{y~1z)

and x, y e G. If xfi = y^i, then /x(x) =

Ml/)Proof. Suppose that X/J, = y/i. By Theorem 1.3.9, x~ly G /x* and 2/ -1 x G xx*. Thus jx(x) = fi(y~lxy) > xx(2/-1x) A fi(y) = fi(e) A fi(y) = fj,(y). Similarly, /J>(y) > /x(x). Consequently, /x(x) = /x(y). H Theorem 1.3.11 Lei xx G NL(G). Set G//x = {x/x | x G G}. TAen

(X> (*/*) ° (2/^) = 0*2/)/* V x,y

eG.

(2) (G/fj,, o) 25 a group.

(3) G/n = G/M*. (X) Le£ //(*) € L G / / i 6e de/mecf fry fi^(x/j.)

= (i(x) V x € G. Then ^

6

NL(G/v). Proof.

(1) For any x, y G G, we have (x/x) o (y^)

=

(/i(e) { x } o xx) o (/x(e) { y } o fj) Ke){x}°(v0Ke){y})°V

= e

=

M ){x}°(P^)°M e ){y}

= = = = =

V>(e){x}°V°Ke){y} Ke){x}°(v°Ke){y}) fi(e){x}o(fi(e){y}ofi) (Ke){x}°Ke){y})ofi (xy)fi

by using Theorems 1.2.2 and 1.3.1. (2) By (1), G/fi is closed under the operation o. Also, o satisfies the asso­ ciative law. Now /i o (x/x) = (e/x) o (x/x) = (ex)/x = x/x Vx G G and (x -1 /x) o (x/x) = (x -1 x)/x = e\i = /x Vx G G.

14

Hence (G//x, o) is a group. (3) Clearly, x/x — i ► x/x* is an isomorphism from G//x onto G//x*. (4) Since 1 W / ^ (W( x( ^W x )_"11)) = M /xW(x-V) MC*" M(x) -= M MW( (*M) *W) M (* _ V) -= M ( * _ ) = MOO

V x G G and xx(*)((x/x)c 3

(w)) > =

/x<*)(xy/x) /x(xy) /x(x) A /x(y) /i(y) //x(*)(x,x)A/x(*)(y/x) x^(x/x)A/xW «

V x, y G G, it follows that //*) G L(G//x). Moreover, since /x(*)((xxx)c //*) ((x/x) o5 (y/x)) {vv)) = = = = =

/x(*)(xy/x) /x<*>(xy/x) /x(xy) /^y) /x(yx) /*(y*) fi^(yxfi) n{*\yxfi) /x(*)((^)o (x/x)) /x(*)((y/x)o(x/x))

V x, y G G. We conclude that /x<*) G A/X(G//x). ■ The group G//x defined in Theorem 1.3.11 is called the quotient group (or factor group) of G relative to the normal L-subgroup /x. Theorem 1.3.12 Let v G L(G) and let N be a normal subgroup of G. Define £ G L°fN as follows: £([*]) = V{i/(*) | z G [x]} Vx G G, w/iere [x] denotes the coset xN. Then £ G Proof.

Now

L(G/N).

1 1 aw]) ^[x]")1) = = saxsa*-1]) 11 = vl^lzefx= v{i/(*) | z e [x- ]} ]} _1 1 = = = = =

V{i/(u; ) | u r e [a; [x --11 ]} \/{v{w) e [x]} \/{i/(w) | «;\we[x}}

*(N) *(N)

VxeG;

4(MM) -=

= > > = -=

V{i/(s) v { ^ )| |z2Ge[xy]} N} V{^(m;) [x],v Ge[y}} [y]} \Z{u(uv) |\uzx G e[x],v V{i/(u) A i/(v) \ue\ x],vG[y]} V{v(u) A u(v) \u€[x],v€ [y]} (V{i/(u) (V{i/(«) |I u« G e [x]}) [ar]}) AA' (V{v(v) [»]}) ( v M v ) \1v ve ^Ml) *([*]) «(M) *([*]) A A *([y])

15

V x,y eG. Hence <£ G L(G/N). ■ The L-subgroup £ defined in Theorem 1.3.12 is called the quotient Lsubgroup (or factor L-subgroup) of the L-subgroup v of G relative to the normal subgroup N of G and is denoted by v/N. Theorem 1.3.13 Let /J, G NL(G) and H be a group. Suppose that f is an epimorphism of G onto H. Then f(fi) G NL{H). Proof. By Theorem 1.2.10, /(/i) G L(H). Now let x, y G H. Since / is a surjection, f(u) = x for some u G G. Thus f{li){xyx-1)

= = = =

V{n(w) | w G G,f(w) = xyx-1} W{ii(u~1wu) | w G G, f(u~1wu) = y} W{fi(w) | uwu~l G G, f(w) = y} V{/iiw)\weGJ(w)=y}

= /MMTherefore, it follows from Theorem 1.3.1 that /(/i) G NL(H). Theorem 1.3.14 £e£ H be a group and v G NL(H). from G into H, then / - 1 ( » G NL(G).

■

If f is a homomorphism

Proof. By Theorem 1.2.11, / - 1 ( » G £(G). Now for any x, y e G, we have f-\v){xy) = i/(/(x2/)) = v(f(x)f(y)) = u(f(y)f(x)) = u(f(yx)) = f-^^iyx). Hence Z " 1 ^ ) G iVL(G). ■

1.4

Homomorphisms and Isomorphisms

This section consists of two parts. The first part is mainly concerned with a generalization of the concept of normal L-subgroups. We will introduce the concept of normal L-subgroups of an L-subgroup and examine their basic prop­ erties. In the second part, we will introduce the concepts of homomorphisms and isomorphisms of L-subgroups, and use them to develop results concerning L-subgroups of an L-subgroup. Definition 1.4.1 Let ^, v G L(G) and ji C v. Then \x is called a normal L-subgroup of the L-subgroup v, written fi
> fi(y) A v(x) Vx,y G G.

The following statements are immediate from Definition 1.4.1. (1) If Gi and G2 are two subgroups of G, then G\ is a normal subgroup of G2 if and only if IGI is a normal L-subgroup of 1<32.

16

(2) If [i G NL(G), v G L(G), and / i O , then /x is a normal L-subgroup of v. (3) Every L-subgroup is a normal L-subgroup of itself. (4) \i G LG is a normal L-subgroup of G if and only if /x is a normal L-subgroup of the L-subgroup 1GTheorem 1.4.2 Let /x, v G L(G) and fi C v. Then the following assertions are equivalent: (1) fj, is a normal L-subgroup of v. (2) fi(yx) > fj,(xy) A u(y) V x, y G G. (3) v(e){x} o / i D ( / i o M(e){*}) n 1/ V x G G. ■

Theorem 1.4.3 Let xx, v G L(G). T/ien ^ is a normal L-subgroup of v if and only if Ha is a normal subgroup of va Va G \i{G) U {b G L | 6 < /x(e)}. ■ Theorem 1.4.4 Let fi, v G L(G) and \i be a normal L-subgroup of v. Then [i* is a normal subgroup of v*. If L is regular, then /x* is a normal subgroup of i>-.m

Theorem 1.4.5 / / fi G NL(G) subgroup of v. Proof.

and v G L(G), then /x n v is a normal L-

Clearly, /x fl v G L(G) and fiHiy C u. Now (/i fi i/)(x2/x _1 )

= =

^{xyx-1) fi(y) A

f\v{xyx~x) ^(xyx'1)

> =

n(y) A v(x) A ^(y) A z/(x _ 1 ) (Mni/)(y)Ai/(x)

V x, ?/ G G. Hence xx fl */ is a normal L-subgroup of v. ■ Theorem 1.4.6 Let /x, v, £ G L(G) 6e stzc/i £/ia£ /x and v are normal Lsubgroups of £. TTien fi Civ is a normal L-subgroup of £. Proof.

Clearly, fidu

e L(G) and /z n i/ C f. Now

(/i fl i/)(xyx - 1 )

= >

yi(xyx~l) A v{xyx~x) Wl/)A((x))AM!/)A$(x))

>

(xxH */)(
Therefore, /x fl v is a normal L-subgroup of £. ■ Theorem 1.4.7 Let /x, ^ G L{G) and xx be a normal L-subgroup of v. Let H be a group and f a homomorphism from G into H. Then /(/i) is a normal L-subgroup of f{v).

17

Proof.

Clearly, f(ji), f{y) € L(H) and f(ji) C f(v). Now

(/(/x))(xyx- 1 )

= > > = =

V{M(2)|26G,/(2) = x ^ - 1 } y{n{uvu~l) \u,v eG, f(u) - x, f(v) - y} \Z{fi(v) A u{u) \u,veG, f(u) = x, f(v) = y} (V{p(v) \veG, f(v) = y}) A (V{i/(u) \ u G G, f(u) = x}) (/(/i))(y) A (/(«/))(*)

V x, y G H. Hence f(fi) is a normal L-subgroup of f(v). M Theorem 1.4.8 Let H be a group. Let /x, v G L(H) and fi be a normal Lsubgroup of v. Let f be a homomorphism from G into H. Then / _ 1 ( / i ) is a normal L-subgroup o / / - 1 ( i / ) . Proof. Clearly, f-\v). Now

/_1(M),

f^Mixyx-1)

/ - 1 ( ^ ) € £(
/_1(A0

£

nif^xyx-1))

= KfixMvKfi*))-1) >

M/(l/))Ai/(/(x))

= r^x*) A (/->))(*)

V x, 7/ G G. Hence / _ 1 (/x) is a normal L-subgroup of f~1{u).

■

Definition 1.4.9 Let G and H be groups and let \i G L{G) and v G L(H). (1) A homomorphism f of G onto H is called a weak homomorphism of \i into v if f(fi) C-v. If f is a weak homomorphism of fi into v, then we say that \i is weakly homomorphic to v and we write [i ~ v, or simply \x ~ v. (2) An isomorphism f of G onto H is called a weak isomorphism of fi into v if /(/i) C v. If f is a weak isomorphism of n into v, then we say that /i is weakly isomorphic to v and we write \i — v, or simply \i~v. (3) A homomorphism f of G onto H is called a homomorphism of fi onto v if /(/x) = v. If f is a homomorphism of n onto v, then we say that fi is homomorphic to v and we write /i ~ v, or simply /x ~ v. (4) An isomorphism f of G onto H is called an isomorphism of /i onto v if f(n) = v. If f is an isomorphism of \i onto v, then we say that fj, is isomorphic

L

to v and we write \i — v, or simply fi = v.

Let /i, v G L(G). Suppose that ji is a normal L-subgroup of v and that L is regular. Then /i* is a normal subgroup of v*. Clearly, v\v* is an L-subgroup of z/\ Thus by Theorem 1.3.12, the factor L-subgroup of v\v* relative to /i* exists. For convenience sake, we denote this factor L-subgroup by v/ji and call it the quotient L-subgroup (or factor L-subgroup) of v relative to /i.

18

T h e o r e m 1.4.10 Let fi, v G L(G) and fi be a normal L-subgroup of v. Suppose that L is regular. Then v\v* w v/fi. Proof.

Let / be the natural homomorphism from v* onto i/*//x*. Then

f
V{i/(y) | y € [x]}

V x G ^ * , where [#] = x/x*. Therefore, v\v* w i///x. ■ T h e o r e m 1.4.11 Ze£ ^ G L(G). Suppose that H is a group and £ G £ ( # ) w such that v « £ and £/ia£ L is regular. Then there exists a normal L-subgroup \x of v such that vj\± = £|f* . Proof. Since v « £, there exists an epimorphism f oi G onto i J such that /(iz) = £. Define /x G L G as follows: V x e G, ^

, x _ f i/(x) if x G Ker / \ 0 otherwise.

Clearly, /z G £(G) and /J, C v. If x e Ker / , then yxy~x G Ker f V y e G, and so ^{yxy'1) = v(yxy~1) > v{x) A 1/(3/) = /x(x) A i/(y) Vy G G. If a; G G\Ker / , then /x(x) = 0 and so li(yxy~l)

> fi(x) A i/(y) My G G.

Hence /i is a normal //-subgroup of v. Next, since v « £, /(^*) = £*. Let p = /|^* . Then p is a homomorphism of v* onto £* and Ker g = /z*. Thus there exists an isomorphism h of v* j\i* onto £* such that h([x]) = g(x) = f(x) V x G i/*. For such an h, we have

h(u/n)(z)

= V{(v/n)([x})\xep*,h([x]) =

= z}

V{V{»/(») I y € [x]} | x € i/*, 0(1) = z}

= v{i/(j/)|yei/*,s(j/) = z} = vM»)|veG,/(!/) = z} V 2 G f . Therefore, u/fi = £ | e . ■ T h e o r e m 1.4.12 £e£ jx G NL(G) Suppose that L is regular. Then

and v G L(G) 6e such that /x(e) = i/(e).

v/(fj, fl ^) ~ (/x o i/)//i.

19

Proof. From Theorem 1.3.4, we have that /i* is a normal subgroup of G. By the Second Isomorphism Theorem for Groups, u*/(jf

n^*)=(/i* i / * ) / ^ -

Since L is regular, it is easy to verify that (/in ")* == / i * n i / ' ,

0"

(fj, oot))* v)* = = [i*i/*. ffu*.

Consequently, we have

"70* n ")*

/ = (/" °«/)7/*7

where / is given by f{x(fjLfM/)*) /(*(/* n !/)•) ==X/X** = Xfl

Vi6/. f(y/(n

Thus D is))(yn*)

= = = < < < < = =

(is/(ft (is/(fi (~) H is))(y(fi is))(y(fin H is)*) is)*)

(since (since/ /is isone-to-one) one-to-one)

v{v(z) v{i/(*) | zey(iinis)*} zeyfanv)*}

V{(MOI/)(Z) 1 z € y ( / i * n i / * ) }

V{(Moi/)(z)|z€y(/x*ni/*)} V{(/LX o */)(z) | 2 G y^*} (since /x(e) = is(e)) V{(/LX o is)(z) | z G y^*} (since /x(e) = K e ) ) ((fiois)//i)(y^) ((/^O^/MXW*)

V y e IS*. Hence f (is / (/J, f) is)) C (/iois)/fi. Therefore, is/(fif]is) ~ (fiois)ffi. Theorem 1.4.13 Le£ /LX, Z/, £ G L(G), \x be a normal L-subgroup of is, and fi and v be normal L-subgroups of £. Suppose that L is regular. Then

Wri/WrfH/v. Proof. By Theorem 1.4.4, /z* is a normal subgroup of i/*, and both /z* and ^* are normal subgroups of £*. By the Third Isomorphism Theorem for Groups,

(r/A**)/("7M*)=r/"*, 7/0 =r/"*, (r/^)/(" where / is given by

vx e r •

/(x/x* f(xfi* ■ (i/*/Ai*)) === £^1 / ** Va: € £*• • K/AO)

■

20

Thus /((£/ M )/(IVM))(^*)

= (Wfi)/(v/ri)W = ((€/Ai)/(^//i))(a:/i*-K/^)) • ("7M')) = V{(£//i)(wi*) 1 V € W* e€*M* */** • •KK//M A *' )*}) } = v{(£//*)(wi*) 12/ e T,T,J/M* = v{v{£(z) | z e 2/M*} 1 v e r , w* e ^ K//**)} = V{V{£(z) | z e J/M*} Itfe e,2/M* € * M * •• (i/*//i*)} V { ^ ) | z € r , 2 / J * e a ! / i * ( / V ) } = = V{£(z) I z e T , zf € x/** • (I/*//**)}

v{e(z) \zzxf-

(«/7/i*)}

= V{£(z) I z e xf ■ K//**)} = = v{e(z)|zer,/Wex^} = v{^)| 26 f,/We^} = U/i/)(x^)

V i e f . Hence

■ (tlti/iy/n) KM/W/O = = */"•■ 11*. f

Definition 1.4.14 Suppose that L is regular. Let /x be an L-subgroup of G. Then (1) fi is called Abelian if /J,* is Abelian; (2) \i is said to be solvable if there exist fuzzy subgroups ^ of G, i = 1 , . . . ,fc,such that \i{ < \ii-\ and /ii-i/'\x% is Abelian, i = 1 , . . . ,fc,where /J,Q = ix and [ik — e^e) • Theorem 1.4.15 Suppose that L is regular. Let /x, v be L-subgroups of G. (1) If fi is solvable, then /x* is solvable. (2) If \x C v and v is solvable, then /x is solvable. Proof. (1) Immediate. (2) Let Vi be L-subgroups of G such that v\ < Vi-\ and vi-i/vi is Abelian, i = 1 , . . . ,fc,where I/Q = v and Vk = e„(ey Then \x O Vi < /x n i/i_i and (/x n i/i_i)/(/x fl i/i) is Abelian, z = 1 , . . . ,fc.Hence it follows that /x is solvable. ■

1.5

Complete a n d Weak Direct P r o d u c t s

In order to define the complete direct product of L-subgroups, we first review the concept of the complete direct product of groups. Let {Gi | i G / } be a collection of groups and denote by e; the identity of d for each i G I. Multiplication on the Cartesian product Y\ieI Gi is defined by = (x (s (Xi)iel(yi)iel i)iel(yi)i€l == (x*2/*)t€/» iy%)i£li V(xi)i € j, (yi)iei G Yliei ^i% ^ s 1S w e ^ kn own > the Cartesian product Yliei ^ together with the multiplication just defined form a group with identity (e;)i € j. This group is called the complete direct product of the Gi's and is denoted

21

by Y[ Gi. In particular, when I = { 1 , 2 , . . . , n ) (where n G N \{1}), we regard iei the Cartesian product Yiiei @i as the set

Gi x G2 x ... x Gn = {(xi,x 2 ,...,x n ) | Xi e Gi, i el} ~ ~ ~ ~ and denote the complete direct product Yl Gi as G\ (g> G2 0 •.. ® G n . *G/

It follows that if G = Yl Gi and ^ G L{Gi) for all i G J, then fl Vi £ iei iei L(G), where \f(xi)ieI (Yl^i)((xi)iei) iei

=

hi€lHi(xi).

In this case, it is natural to name the L-subgroup Yl ^i °f G the complete iei direct product of the m's. Throughout the following theorems, it is assumed that Gi is a group with identity e; for each i G / , G = Yl Gi, and s o e = {^i)ieiiei Theorem 1.5.1 Let ^ G NL{Gi) V i € I. Then \i = fl \H € iVL(G). *€/

Proof. Now fi G L(G). For any two elements x = (xi)iei and 2/ = (yi)iei m G, we have /x(x?/) = AieiHi(xiyi) = Aiei/J>i(yiXi) = Kvx)- Hence /x G NL(G). ■ We need the following preparation in order to introduce the notion of a weak direct product of L-subgroups. Let Xi G G for i G / , where \I\ > 1. Suppose that the £;'s are pairwise commutative, and that there are at most finitely many Xi not equal to e. We define YiieiXi a s f°ll° w s : if xi ls equal to e for every i G / , then we interpret Yli^i xi 3s e:> a n d if Xi ^ e for some i G / , then YiieiXi *s m e r e l y the product of those Xi that are not equal to e. Note that if the x^s are not pairwise commutative or there are infinitely many x^s not equal to e, then we do not have a specific interpretation for Yli£ixii m other words, r L e / X i 1S undefined- For the sake of convenience, however, it is assumed that YiieiXi 1S me&ning£ul whenever it appears in the ensuing discussion. In the remainder of this section \I\ > 1 and we write Ylxi f° r Yiiei Xi' Clearly, every element x of G can be expressed in the form x = fj xi. Now let Gi be a submonoid of G for each i e i and assume that Xi e Gi and Xj G Gj for i j e l with i ^ j => x ^ = 2 ^ .

22

Then x=

IIXi

anc y =

*

IIyu w ^ ere Xi,Ui € Gi V i E I => xy = J"J(x»2/»).

For subsets A; (z E / ) of G, the set {[\xi I #i E Ai, i € / } is called the * weak product of the Ai's and written as Yl Ai. iei

If e E Ai V i E / , then Aj C f[ A» V j G / . For subgroups G^ of G (z E J), the group G is called the weak direct product of the Gi's, and written as G = J} Gi if the following conditions are iei

satisfied:

a) G =ftft, »G/

(2) Xi E Gi and x^ E Gj for z, j E I with z ^ j => x^Xj = XjXi, (3) f ] ^ i = Il2/*» where x^ y{ e Gi => Xi = y{V i e I. It can be shown that condition (2) is equivalent to the following condition: (2)' every Gi is a normal subgroup of G; and condition (3) is equivalent to either of the following two conditions: ( 3 / e = Yl xi, where X{ E Gi => Xi = e V i E / ; (3)"G,-n(

fl

G0 = { e } V i 6 J .

(When I \ {j} is a singleton {z'o}, we write

* Yl Gi = Gi0.) iei\{j}

If G is the weak direct product Yl Gi and / is finite, say J = { l , 2 , . . . , n } , iei

then we sometimes use the notation G\ ® G2 ® . . . ® Gn to denote the weak direct product of the Gi's, and call it the direct product of the Gi's. Now let us define the weak product of L-subsets as follows. Definition 1.5.2 For each i E / , let fii E LG. Define /x E LG as follows: ji(x) = V{(Atej/Zi(z*)) I Xi E G,z E 7,]^[xi = x} \f x E G. Then /1 is called the weak product *

of the m 's and is denoted by

23

Let I = { l , 2 , . . . , n } , where n G N \{1}. If fa(e) > fij(x) and /i* C /x V z , j G / and Vx G G, then ** J J fa C fa o /i2 /i 2 o ..,.. ..Oof J ^/, L n .n. iei

However, if the fa's also satisfy the following condition: Xi G /i* and Zj G /x* for i j e l with i ^ j => x{Xj — XjXi, then

*

Y[ Hi = Ml O fJL2 ° •..■ o/x n . iei *€/

Therefore, the product of L-subsets of a commutative group can be treated as a special case of the weak product. * Theorem 1.5.3 For each i G J, let fa G LG, and let fi = Yl Hi- Then the iei following assertions hold:

(i)t**2 no**)*. ^

7/V(( U i € / ^ ( G ) ) \{1}) < 1, tten fa = f[ (/*)*• ■ *€/ * Theorem 1.5.4 For each i G I, let fa G LG, and let fi = Yl fa. If L is regular, iei * t/ien /x* = f| ii*. ■ *€/ Let fi G L G . Then /i is called an L-subsemigroup of G if [i(xy) > fi(x) A fi(y) Va:,2/ G G and an L-submonoid of G if in addition to being an Lsubsemigroup of G, /i(e) > /x(x) Vx G G. * Theorem 1.5.5 For each i G / , Ze£ /z; G L G , and let /J, = Yl Hi- Suppose that G is a commutative group. Then the following assertions hold: (1) If every fa is an L-sub semigroup of G, then so is ji(2) If every fa is an L-submonoid of G and fa(e) = fa(e) Vz, j G / , then [i is the smallest L-submonoid of G that contains all fa's. (3) If every fa is an L-subgroup of G and fa{e) = fa(e) Vz, j G 7, then fi is the smallest L-subgroup of G that contains all fa 's, that is, JJL = (Ui^jfa).

24

Proof.

If every fa is an L-subsemigroup of G, then

fjb(xy) = > > >

V{Ai€lfJLi(zi) \ ZieG,ie I,l\zi = xy} V{AieiiJLi(xiyi) \ xit yi G G, i G 7, fl*» = z>II Vi = v) V{AieI(fii(xi) A fH(y%)) I a:*, Vi € G,i e I,Y[xi = x,Y[yi = y} V{(AieIfa(xi)) A (AieIfa(yi)) I a?i,2/» G G,i G 7,

n^i =a:,Il2/i=2/} =

(V{Ai€//i»(xi) | Xi G G, z G 7, [ ] xi = x}) A(y{AieIfii(yi) \yieG,ieI,Y[yi =y})

= M*) A My) V x, y € G. Hence /z is an L-subsemigroup of G. When every fa is an L-submonoid of G, >LA is also an L-subsemigroup G, and in addition we have fi(e) = AieIfa{e). Thus /J, is an L-submonoid of G. Now since /ii C 11 Vi G 7 and for any L-submonoid v of G that contains all /Vs, x = J T x ; , where x,Xi G G for z G 7 => Ai^jfa(xi) < Ai£iv(xi)

< i/(a;),

it follows that n Q v. Therefore, /x is the smallest L-submonoid of G that contains all fa's. When every fa is an L-subgroup of G, // is an L-submonoid of G. Also ^(rr- 1 )

\ yi G G,z G 7 , 1 1 ^ = ^ - 1 }

=

y{AieIfj,i(yi)

=

v{A i G /^(^ r l ) I z/i e G,z G 7,n
=

Kx)

V x e G. Thus // is also an L-subgroup of G. It has been shown that //, being an L-submonoid of G, is the smallest L-submonoid of G that contains all fa's. Naturally, fi is also the smallest L-subgroup of G that contains all fa's, that is, /i = (UieIfa). ■ In the preceding theorem, G is assumed to be a commutative group. For an arbitrary group, we have the following theorem. * Theorem 1.5.6 For each i G 7, let fa G L G and /J, = Yl fa. Suppose that the fa's satisfy the following condition: Xi G /i* and Xj G ^ for i ^ j => x ^ - = a ; ^ , and £/m£ L zs regular. If every fa is an L-subsemigroup (L-submonoid, Lsubgroup, normal L-subgroup) of G with fa(e) — fij(e) Vzjj G 7, then so is fa

25 Proof. The proofs of the first three parts of the theorem are similar to those of Theorem 1.5.5 and thus are omitted. Let us now prove the last part. Assume that every \ii is a normal L-subgroup of G. By the third part, we conclude that ^ is an //-subgroup of G. Hence ^i{xyx~l)

= = =

V{AieJfa(zi) | Zi E G, i E / , fl z% = xyx"1} 1 ViAieifiiixyiX- ) \ y{ G G,i G I^xyiX'1 = V{/\ieim(yi) \Vi£G,ieI,]\yi = y}

xyx~1}

= riv) V x, y G G. Thus /i is a normal L-subgroup of G. ■ Theorem 1.5.7 For each i G I, let fa G I/(G) 6e such that fa(e) = fij(e) * Vz, j G J. Lei /x = fl A^i- Suppose that H and Hi {i G I) are subgroups of G iei

such that fi* C Hi V i G I and LT = JJ Hi, i.e., H is the weak direct product %ei of the Hi's. Then the following assertions hold. (1) fx e L(G). (2) Every fa is a normal L-subgroup of the L-subgroup /i.

(3) H n (

ft

Mi) = M(e){e} V j e / .

i€/\{j'}

(When I\{j}

* f] fa = fa0).

is a singleton {io}> we write

iei\{j}

(4) Iffa\H

e NL(H)

V i e / , then p\H G

NL(H).

Proof. Let x,y G G and consider fi{xy). For either x (£ H or y £ H, clearly nixy) > 0 = ^(x) A ^0/)- When x, y G J7, we have xy E H and the proof that (1) holds follows as in the proof of Theorem 1.5.5. Now we show assertion (2) holds. Let j G I. Since fij G L(G) and /ij C ^Z, in view of part (2) of Theorem 1.4.2, it needs only to be shown that Pj(yx) > lij{xy) A fj.(y) Vx,?/ G G. When either xy £ Hj or y <£ H, it is clear that lij(yx) >0 = iij{xy) A /x(y). Now assume that xy G ii/j and y E H. Since iij C H, we have x G H. Also, from the fact that Hj is a normal subgroup of H, it follows that yx = x~x -xy-x G ^i/^-. Thus using the fact that xy, yx E Hj, we see that fi(xy) = fj.j(xy) and fi(yx) = fij(yx).

26

Hence Uj(yx)

= = >

u(yx) uiy-xy-y'1) u(xy) A u(y)

Therefore, Uj is a normal L-subgroup of //, that is, assertion (2) holds. Clearly, assertion (3) holds. Finally, suppose that U{\H £ NL(H) V i G / . It is easy to see that U\H £ L(H). Next, for any x, y G H, x and y can only be uniquely expressed as x = ]Jxi and y — X\yi, where Xi, yi G #< V i G / . So, xy = n(x*2/*) a n d Vx = Y[(yiXi). Thus u(xy) = /\ieiUi(xiyi)

= AiGiUi(yiXi) = u(yx).

Therefore, a\H G NL(H). That is, assertion (4) holds. ■ Note that it is easy to verify that U\H in Theorem 1.5.7 is isomorphic to some L-subgroup of group f ] Hi. iei

Definition 1.5.8 Let u G L{G) and Ui G L{G) for each i G / . Suppose that fj>i(e) = Uj(e) *ii,j G / . Then a is called the weak direct product of the L-subgroups Ui, written a = Yl fa > if the following conditions are satisfied. iei

(V a=

*

U faiei

(2) Every Ui is a normal L-subgroup of a. (3)ujn( ft Ui) = u(e){e}Vj el. iei\{j}

In addition, when / = { l , 2 , , . . , n } ( n G N\{1}), we sometimes write fii ® /i2 ® • ■ • ® Mn f°r El fa

anc

^

cau<

^ the direct product of the /^'s.

»€/

Theorem 1.5.9 Let u G L(G) and, for each i G / , Ui G L(G). Suppose that Ui(e) = /ij(e) Vi,.7 G / and £/ia£ L is regular. Then a necessary and sufficient condition for u = Yl fa ^s that

/z* = Ylti iei

* arid u = Y[fa iei

27

Proof. The sufficiency is a direct consequence of Theorem 1.5.7 with Hi = \i\ and H = fi*. Let us now establish the necessity as follows: Suppose that /x = Yl /^. iei * By Theorem 1.5.4, we first note that /x* = Yl lA- Then by Theorem 1.4.4, it iei follows that for each j G / , /x!- is a normal subgroup of /x*. Finally, by Theorem 1.5.4, for all j G J,

xG/x*n( n

M?) <* *e^*n( n MO*

<e/\{i}

*e/\0"}

* <£> /XJ(X) > 0 and ( Yl

/^)(z) > 0

i€/\{i} *

^

(«n( n

/**))(*) >o

*€/\{j}

<£> x = e. Therefore, the desired result holds. ■

Theorem 1.5.10 Let /x G £(G) and {xx; | z G / } be a collection of L-subgroups of G such that fii C /x V i G I. Suppose that either (1) L is a chain and Uigj \/ii(G)\ is finite or (2) L is regular and \x* = Yl /x*. iei * * Then /x = I~[ /^ */ an^ on^y if Pa = Yl (pi)a *e/\{i} ie/\{j} xx(e)}.

VaG{6GL|6<

Proof.

We first show that /Xi(e) = xx(e) V z G / in either direction of the * proof. Suppose that /i = Yl Pi- Then /x(e) = V{A{/ii(x;) | i G / } | e = YlieiXi) iei < V{A{^(e) \ i e 1} \ e = Ylieixi} ( s i n c e ^ ( ^ ) ^ M e )) = A{xxi(e) | i G 7} * < /x(e) (since /i; C /x). Thus xx(e) = xx;(e) Vz G 7. Suppose xxa = YliPi)* V i€/

* a G {6 G L | b < /x(e)}. Now e G /x^(e) and so e G I1(M»)/X(C)- Thus (/x;)M(e) *€/

^ 0. Hence /Xi(e) > xx(e) Vz G 7.

28

Suppose that \i — Yl Hi- Then iei * *

xe(Y\vi)a

<=>

(Ylni)(x)>a

<£> V{A{vi(xi) \iel}\x = Uiei xi) ^ a <= ^i{xi) > a for some representation of x = Yl xi *

an<

^ Vz G /

& xe n(^)aiei

If condition (2) holds, then x has a unique representation. Clearly then, either condition (1) or condition (2) implies that the <= becomes <4> in the above * sequence of implications. Conversely, suppose that \xa = n (Mi)a V a G {b € L iei I b < fi(e)}. Let x G G and ji{x) = a. Then x G fia and so there exist yi G (fii)a such that x = YlVi- Thus iei *

( H ^ X ^ ) = V{A{^(xi) | i G 1} | x = Y[xi} > a = Kx)iei

iei

* Hence it follows that \x = Yl iei

^.M

Theorem 1.5.11 Suppose that L is regular. Let fi G L(G) and {^ | i G / } be a collection of L-subgroups of G such that fii C \i V i G I. Suppose that /x;(e) = fij(e) \/i,j G / . Then

fi = Yl^i ^ and

0nl

V ft Pa = H(^i)o

iei \/ae{beL\

iei

b
Proof.

Suppose that \x — Y\v>i- Then \x* = Yl M* by Theorem 1.5.9. Then iei iei b V a G { 6 G L | 6 < /x(e)}\{0}, /i a = f l (/*<)<* Y Theorem 1.5.10 and iei (Vi)a H (^j)a C / 4 n /x* = {e} if i ^ j . Thus

^a = 1 1 ^ ) ° *€/

29 Vae{beL\b<

y(e)}\{0}

in view of Theorem 1.4.3. Conversely, suppose

t h a t \ia = n ( / i i ) a V a G { & G L | 6 < /x(e)}\{0}. T h e n by Theorem 1.5.10, /x iei * = n Mi- Let x G /x* D/x*. T h e n for a = ^{x)

A ^ - ( x ) , we have t h a t a > 0 a n d

£ 6 (^i)a n (fjLj)a = {e} for i / j . Hence x = e a n d so /x* fi /xj = {e} for i ^ j . T h u s M* = I I Mi m v i e w of Theorem 1.4.4. Hence \x = [ ] /x* by Theorem iG/ i€/ 1.5.9. ■ T h e o r e m 1 . 5 . 1 2 Let \i G L(G) and {fii \ i e 1} be a collection of L-subgroups of G such that /x; C /x V i G I. Suppose that y = Y[ f^i and JA* = n ( M i ) * iG/

iG/

Then y* = I I (Mi)*Proof. By Corollary 1.2.6, /x* is a subgroup of G. Since by definition, every \ii is a normal L-subgroup of /x, it follows from Theorem 1.4.4 t h a t every (/x;)* * is a normal subgroup of /x*. Let j € I a n d x G (/^)* H f| (Mi)*- T h e n x G *G/\{i}

* Mj^

11 iei\{j}

lA

an(

i

so

*

(Mi ^ (

El Mi))( x ) > 0- Hence x = e. Consequently, iei\{j}

M* = I I (Mi)*-■

*€/ T h e following two theorems are direct consequences of Theorems 1.4.5 and 1.4.6, respectively.

T h e o r e m 1 . 5 . 1 3 Let y G L(G),

{fii \ i G / } be a collection

of G such that /x; C /x V i G I. Let vi G NL(G).

of

L-subgroups

Suppose that /^ — II Mi iei

ano

^

*

fe* £ = n (M* n ^ ) - T / i e n £ = n (Mi n u^. m iei iei T h e o r e m 1 . 5 . 1 4 Let /J, G L{G) and {/x* \ i E 1} be a collection of L-subgroups of G such that Hi C /j, V i G I. Suppose that for each i E I, Vi is a normal * L-subgroup of y and /x = n Mi- ^ £ — El (Mi n vi)- Then f = n (Mi n vi). ■ iei iei iei

1.6

Embedding of Fuzzy Power Sets

T h e results of t h e remaining sections of this chapter are mainly from [61]. In this section, we let L = [0,1]. Let X be a non-empty set and let F(X) = Lx.

30

Let C(X) denote {0,1} X and P(X) the power set of X. We caU C(X) the crisp power set of X. Then P(X) and C(X) are in a natural one-to-one correspondence and C(X) C F(X). Many results concerning relationships between classes of crisp subsets can be carried over to similar relationships between classes of fuzzy subsets. The purpose of the next few sections is to provide a simple conceptual framework for transferring crisp results into fuzzy results, when such transference is possible. This is accomplished by the presentation of a suitable metatheorem. The focus of the work was the paper [109] dealing with fuzzy subsemigroups and the paper [8] dealing with the lattice of fuzzy subgroups. In Section 1.8, we present a metatheorem and two closely associated subdirect product representation theorems. These results arise from first estab­ lishing a representation function SH : F(X) —> C(X)J, where J = [0,1). The concept behind £H is very closely related to the concepts of level sets and strong cuts as widely used in the fuzzy literature. The results here are generated by exploiting the exact details of the definition of fH in a thoroughly systematic way. We develop the concept and fundamental properties of £H, the first subdirect product theorem, and the metatheorem. The use of the metatheorem in [61] is illustrated by showing how to obtain results of [109], and a few other papers, in a unified way from the metatheorem. The second subdirect product theorem is given and is applied to extend the modularity results of [9]. The function C : P{X) -> C{X) denned by VY G P(X), £(Y)(x) = 1 if and only if x G Y, establishes a bijection of P{X) with C(X). For each such Y, <£(Y) is the characteristic function of Y Each crisp subset is also a fuzzy subset. Recall that F(X) has the pointwise partial order C defined by V^, v G F(X), H C v if and only if fi(x) < v(x) Vx G X. This makes F(X) a complete lattice with C(X) a complete sublattice. P(X) is a complete lattice with respect to the partial order of containment and the function C : P(X) —> C{X) is an isomorphism of complete lattices. The function (5 : C(X) —> P(X) defined by &(fi) = {x 6 X | fj,(x) = 1} is the isomorphism that is the inverse of €. Via the functions € and €>, every assertion about crisp subsets is also an assertion about (ordinary) subsets and conversely. We exploit the representation function 9t presented in the following defini­ tion. Recall that J denotes the half open unit interval J = [0,1). Definition 1.6.1 Let X be a non-empty set. Let Dl : F(X) defined by V> G F(X), Vr G J, and \/x G X,

*(/i)(r)(a:) = | J0

if fi(x) < r otherwise.

Proposition 1.6.2 91 is an injective function.

-> C(X)J

be

31

Proof. Suppose SH(/z) = 9S(i/) and let x £ X. Since £H(/x)(i/(x))(a;) = 9t(z/)(i/(x))(x) = 0, it follows from the definition of SH that /x(x) < i/(x). Similarly, since 9^(i/)(/x(x))(x) = 9\(fj,)(fi(x))(x) = 0, 1/(2;) < p.{x). Thus /x(a:) = i/(x) for all x G X and so [i = i/. ■ The partial order C of G(X) provides the following pointwise partial order < for C(X)J : For G,H e C(X)J, G < H if and only if Vr G J, G(r) C # ( r ) . Note that this latter inequality, G(r) C H(r), holds if and only if Vx G X, G(r)(a;) = 0 whenever H(r)(x) = 0. Equivalently, G(r) C ff(r) holds if and only if H(r)(x) = 1 whenever G(r)(x) = 1. P r o p o s i t i o n 1.6.3 Jra(SK) = {G : J ^ g>r}}.

C(X)

| Vr G J, G(r) = U{G(g) |

Proof. Suppose that G = 9t(/x) for some ^ G 1*X-X). Let r G J and x £ X. Then G(r)(x) = SH(^)(r)(x) = 0 if and only if fi(x) < r. However, /x(x) < r if and only if fi(x) < q for all q > r. Consequently, G(r)(x) = 0 if and only if G{q){x) = 0 for all q > r. Thus G(r)(x) = V{G(q)(x) \ q > r } . Since this equation holds V x G X, G(r) = U{G(g) | g > r } as required. Suppose now that G : J —> C(X) has the property that Vr G J, G(r) = U{G(g) | q > r}. Let /x : X —> 7 be the function defined Vx G X by /x(x) = A{p G J | G(p)(x) = 0}. We confirm that 9 % ) = G by observing that Vr £ J and V x G l , the following five assertions are equivalent: (l)5l(Ax)(r)(x)=0. (2) M (x) < r. (3) For all r, fi(x) < q. (4) For a l l g > r , G{q)(x) = 0. (5) G(r)(x) = 0 . B Note that, V/z G -F(-X), 9*(A0 i s a n order reversing function from J into G(X), i.e., Vx G X, lH(/x)(g)(x) < SK(/i)(r)(a;) whenever q > r. This information is implicit in the statement of Proposition 1.6.3. Proposition 1.6.4 SH commutes with infima of finite sets of fuzzy subsets and with suprema of arbitrary sets of fuzzy subsets. Proof. Let 5 be a finite set of fuzzy subsets of X. The assertion that Dt commutes with inf means that SH(n{^ G S}) = A{SK(/x) | fi G S}. This equality is demonstrated by observing that Vr £ J and Vx G X, the following five statements are equivalent:

(1) m(n{fi e 5})(r)(x) = 0. (2) n{ju € S}(r)(x) < r. (3) There exists \x € 5 for which /x(a;) < r. (4) There exists \i e 5 for which 9t(/i)(r)(a:) = 0.

32

(5) A{W(/i) | M € S}(r)Or) = 0. Let S be a set of fuzzy subsets of X. The assertion that SH commutes with sup means that £K(U{^ G S}) = V{*R(/i) | /z e S}. This equality is demon­ strated by observing that Vr e J and Vx G X, the following five statements are equivalent: (6) *(U{/z € S})(r)(x) = 0. (7) U{p 6 5}(r)(a;) < r. (8) For all fj, 6 5, /x(a:) < r. (9) For all p 6 5, !*(/*) (r) (a:) = 0. (10) V{9%) | M 6 5}(r)(x) - 0. ■ In the proof above, (2)=>(3) fails when 5 is infinite. It is easily verified that 9\ does not, in general, commute with infima of descending sequences of fuzzy subsets even when X is a singleton. Let I(X) be the image of the function 9 t : F(X) -> C(X)J. By Proposition 1.6.4, I(X) is a sublattice and also a complete upper subsemilattice of C(X)J. By Propositions 1.6.2 and 1.6.4, 9t is an order isomorphism of F(X) onto I(X).

1.7

Representation of the Fuzzy Power Algebra

Definition 1.7.1 Let X have an n-ary operation * : Xn —> X, where n is a positive integer. We enlarge the domain of * to provide an n-ary operation * : F(X)n —» F(X) on F(X) by the standard convolution method: * ( ^ i , . . . , ^ n ) ( x ) = V{^i(xi) A . . . A/J,n(xn)

| X = *(a?i,...,x n )}.

The operation * on X also provides an operation * on P(X) defined by for all subsets X\,..., Xn of X, * ( X i , . . . , X n ) = {*(ari,...,x n ) | x. E l i , z = 1,2, . . . , n } . We summarize related information that is easily confirmed, but absolutely fundamental. Proposition 1.7.2 For each n-ary operation * on X, with n > 1, C(X) is closed with respect to the convolution extension of * to F(X). The bisection £ : P(X) —► C(X) commutes with the * operation on P(X) and C(X). I A convolution extension of * from X to F(X) induces an operation * in C{X) which then provides a pointwise operation * for C(X)J. Thus for Gi,...,GnGC(X)J, *(Gi,.--,Gn)(r)=*(Gi(r),...,Gn(r)). Proposition 1.7.3 For each n-ary operation * on X, with n > 1, the repres­ entation function Dl : F(X) —► G ( X ) J commutes with the convolution exten­ sion of*.

33

Proof. F(X),

The assertion that 9* commutes with * means that V/xi,..., \xn G *R(*(/il, . • ■ ,Mn)) = *(J»(A*l), ■ • • ,«(Mn)).

This equality is demonstrated by observing that V/*i,... ,fin £ F(X), r £ J, and x € X the following eight assertions are equivalent: ( l ) W ( * ( M l , . . . , M n ) ) ( r ) ( x ) = l. (2) *(/i!,...,/x„)(x) > r . (3) V{/xi(a;i) A . . . A^ n {x n ) \ x = *{x1,...,xn)} > r. (4) There exists X\,... ,xn € X for which x = * ( x l t . . . ,xn) and fi\(xi) > r,..., iin(xn) >r. (5) There exists xi,...,xn € X for which x = * ( # i , . . . , x „ ) and fR( Ml )(r)(:n) = 1 , . . . , m{fxn)(r)(xn)

= 1.

(6) V{9t(/ii)(r)(a:i) A . . . A 9t(/in)(r)(a: n ) | a: = * ( ! ! , . . . , xn)} = 1. (7)*(9l(/i 1 )(r),...,!R( A t n )(r))( a :) = l. (8)*(fH( A t l ),...,«( M n ))(r)(x) = l . B Now let X be an algebra of arbitrary structure. This means that X is a non-empty set provided with n-ary operations * i , . . . , *fc for various values of n > 1. (X may also have O-ary operations, but these will play no role in our work here.) Example 1.7.4 (1) X may be a semigroup having one binary operation, usu­ ally denoted by •. (2) X may be a group having one unary operation, usually denoted by - 1 , and one binary operation, usually denoted by •. (3) X may be a ring having one unary operation, usually denoted by —, and two binary operations usually denoted by + and •. (4) X may be a lattice with two binary operations, V and A. We have seen that the n-ary operations * i , . . . , *&, n > 1, extends to op­ erations on P(X),C(X), and F(X). RecaU that P(X),C(X), and F(X) have two additional binary operations, fl and U, as discussed in Section 1.6. Thus P ( X ) , C ( X ) , and F(X) all become algebras having the corresponding opera­ tions n, u, *i,...,*fc. Propositions 1.6.2, 1.6.4 and 1.7.3 tell us that EH is an injective homomorphism of the algebra F(X) into the product algebra C(X)J that establishes an algebraic and order-theoretic isomorphism of F(X) with its image I(X). For all r € J, the projection of F(X) into the r t h coordinate space of C(X)J is a surjection. In fact, Vr € J, Dl(fi)(r) maps the subset C(X) of F(X) bijectively onto the r t h coordinate space C(X) : Let ji e C(X). Then Vr G J, x e X, 9%)(r)(x) = M (x), i.e., M(/i)(r) = ft.

34

Recall that an algebra A is said to be a subdirect product of a family of algebras {Ab \ b E B}, where B is an arbitrary index set, if A is isomorphic to a subalgebra of the product algebra U{Ab \ b e B} with the property that its projection into each coordinate space Ab is a surjection. Since ^H defines an isomorphism of F(X) onto I(X) and the projection of I(X) into each coordinate space of C(X)J is a surjection, we have established: The Subdirect Product Theorem. Let X be an algebra having n-ary operations *i,...,*& for various values of n > 1. Then £H : F(X) —> C(X)^0,1^ is a representation of the (n, U, * i , . . . , * n )-algebra F(X) as a subdirect product of copies of the (n, U, * i , . . . , *fc)-algebra C{X).

1.8

The Metatheorem

To establish our notation, we review the recursive definition of an expression over a set Var of variables, a set Op of operations, and the three auxiliary symbols left paren, right paren, and comma. Each operation symbol * in Op has an integer n(*) > 1 as its arity. The following two sentences constitute the definition of an expression over Var and Op : For each v G Var, v is an expression. For each * G Op and each list of expressions E\,..., En^, where n(*) is the arity of *, *(E\,..., En^) is an expression. Note that each expression is a finite string of symbols having a syntax rigidly specified by the recursive definition. Let E be an expression. Then the number of distinct symbols in Var that occur in E is a finite positive integer and the number of distinct symbols in Op that occur in E is a finite non-negative integer. Each of the symbols occurring in E may occur numerous times. When no variable other than those in the set {^i,..., v m } occurs in E, E may be denoted conveniently by writing E = E(vi,..., v m ). Note that the order in which the variable symbols are hsted is immaterial and that a variable symbol may occur in the list even though it does not occur in the expression E. The convenience of this notation is indicated in the next paragraph. Let X be a non-empty set provided with n-ary operations * i , . . . , *& for various values of n > 1. Then P(X), C{X), and F{X) are algebras with oper­ ations fl, U, * i , . . . , *fc. Let E = E{v\,..., Vm) be an expression over the set of variables Var = {v\,..., v m } and the set of operations Op — {fl, U, * i , . . . , *&}. Then for any m elements / z 1 ? . . . , / i m G F(X), E([ii,..., /xm) is the element of F(X) which results when, in E, each occurrence of each V{ is replaced by fii and the result is evaluated in F(X). We say that a class C of fuzzy subsets of X is closed under projections if, V/x G C,\/r e J, the crisp set £H(/z)(r) is also in C. Theorem 1.8.1 (The Metatheorem) LetX be an algebra having n-ary op­ erations * i , . . . , *fc for various values of n> 1. Let F{X) be provided with the

35

operations n, U, * i , . . . , **.. Let Diy^,..., vm) and E{yx,..., vm) be expressions over the variable set {vi,..., vm} and the operation set {fl, U, * i , . . . , *&}. Let C\,..., Cm be classes of fuzzy subsets of X that are closed under projections. The inequality £>Qzi,...,/i m ) REL E(fjL1,...,fim) holds for all fuzzy sets fii G C;, i = 1, 2 , . . . , m if and only if it holds for all crisp sets \ii e Ci, i = 1,2,..., ra, w/iere REL is any one of the three relations

Proof. We consider only the case in which REL is C, since the case of 2 follows by symmetry and then the case of = follows by combining C and D . If the inequality C fails, then 3x G X for which £>(/zi,...,//m)(a:) >

Let r =

E(JJLI,

E(fjLU...,fjLm)(x).

. . . , /i m )(x). Then

^^(/i!)^),...,^^)^))^)

= = =

JD(M(^ 1 ),...,lH( A x m ))(r)(x) W(Z?(/ii J ...,M n 0)(r))(x) 1,

E(fH( Ml )(r),...,£H(/x m )(r))(x)

= = =

£(*(Mi), •.. , % m ) ) M ( x )
but

Let A be an (A, V, * i , . . . , *fc)-algebra. For expressions D(vi,..., vm) and i 5 ( v i , . . . , v m ) over Var = { v i , . . . , v m } and Op = {A, V , * i , . . . ,**.}, we say that D(vi,..., Vm) = E(vi,..., vm) is an (A, V, * i , . . . , *fc)-identity. The alge­ bra A satisfies the identity D(v\,..., vm) = E(v\,..., vm) if V/xi,..., jim £ A JD(/XI, . . . , ^xm) = 2S(^i,..., /x m )- When either the metatheorem or the subdirect product theorem is combined with the isomorphism of P(X) with C(X), we obtain the following result. Corollary 1.8.2 For each algebra X, having n-ary operations * i , . . . , *&, for various n > 1, tfie tfiree (fl, U, * i , . . . , *k)-algebras P(X), C(X), and F(X) all satisfy precisely the same (fl, U, * i , . . . , *k)-identities. ■ Let X be an algebra as in Corollary 1.8.2. Note that v\ V v[ = 1, where the prime denotes the appropriate complementation, may be regarded as an identity that is satisfied by P{X) and C(X), but not by F(X). However, viV v[ = 1 is not an ( 0 , 0 , * ! , . . . , *fc)-identity: Since 1 does not contain a

36

variable, 1 is not an expression over Var = {vi} and Op = {fl, U, * i , . . . , *&} and, for this reason alone, v\ V v[ = 1 does not satisfy the definition of an (fl, U, * i , . . . , *fc)-identity. However, a close analysis will also show that, even when X = P(Y), X = C(Y), or X = F(Y), for some set Y, the prime required in the identity is not the convolution extension of the complement appropriate to such an X. Corollary 1.8.2 provides two elementary known facts that are fundamental to the theory of fuzzy algebraic structures. The first refers only to the in­ formation concerning fl and U [254]: For each non-empty set X, the lattice of fuzzy sets, F(X), is distributive. The second refers to the operations * i , . . . , *& and the identities they may satisfy. In some cases, identities satisfied by the (*i, • • •, *fc)-algebra X are also satisfied by P{X). This is true, for example, for the associativity identity and for the commutativity identity. Recall that a set with an associative binary operation is called a semigroup. Thus the metatheorem gives: When (X, *) is a semigroup, so is (F(X), *). When (X, *) is commutative, so is (F(X), *).

1.9

Applications and Unifications

For the next group of applications, the classes C; in the metatheorem are required to satisfy restrictions that are closely related to the structure of the ( * i , . . . , *fc)-algebra X. That these conditions are easily dealt with in the present exposition is a consequence of the following property of 9t which follows immediately from its definition. Proposition 1.9.1 For each x,y e X and V^x G F(X), /i(x) REL fi(y) holds if and only if Mr G J, £R(/z)(r)(a;) REL DK(/j,)(r)(y) holds, where REL is any one of the three relations <, = , or > . M Several fundamental results in the literature can be observed simultaneously by noting that they are consequences of Proposition 1.9.1. Generic patterns for defining fuzzy versions of crisp concepts will allow many results to be treated simultaneously. For each class C of crisp subsets of X, a fuzzy subset /i G F(X) will be called a fuzzy C subset if and only if Vr G J, D^(/x)(r) G C. This generic definition is consistent with many definitions from semigroup theory [110] as pointed out in [61]. We do not discuss them here since they do not fall within the framework of commutative algebra. However, we will illustrate the ideas presented at various locations in the book. We note that Zadeh's classic definition of a fuzzy convex set [254] and Weiss' definition of a fuzzy open set for the topology induced by a crisp topology [243] are also consistent with the proposed generic pattern. The convenience of this general concept of a fuzzy C class is illustrated by deriving generalizations of results in a unified

37

manner by reference to the following principle which is immediate from the definition of a fuzzy C class: For classes C and D of crisp subsets, the class of fuzzy C subsets is contained in the class of fuzzy D subsets if and only if C is contained in D. Prom this principle and the observations made above, we may state numerous propositions. For example, see [61]. We now give an application to lattice modularity. Let (X~l , *) be a group. Let NL(X) be the set of all fuzzy normal sub­ groups in F(X). Recall that /i G F(X) is a fuzzy normal subgroup if it is a fuzzy subsemigroup and has the two additional properties: For all x G X, /i(x - 1 ) = /i(x) and, Vx, y G X, ji{x * y) — fi(y * x). With respect to the partial order, C, of F ( X ) , NL(X) is a lattice for which the INF coincides with the inf of the lattice F(X), but for which the SUP does not. Thanks to normality, this SUP is very well behaved: SUP(/i, v) = /J, * v V/i, v G NL(X). In the present exposition, the fact that the SUP of {//, v} is /z*z/, is an elementary consequence of Proposition 1.9.1 and the definition of a fuzzy normal subgroup: Since fi and v are fuzzy subgroups, 9t(//)(r) and *R(v)(r) are crisp normal subgroups Vr G J. Since £R commutes with *, 9l(/i. * v)(r) = fH(/x)(r) * 9t(i/)(r) holds Vr G J and is the SUP of its two displayed crisp factors. Since 9t(^ * v)(r) is normal Vr, fi * v is normal. Since D\(/J, * v)(r) = 5C/P{9l(/i)(r),9t(z/)(r)} Vr, ii * v — SUP{n, v\ if and only if \i{e) — z/(e). It is shown in [62] that D\ commutes with SUP. Corollary 1.9.2 Let (X," 1 ,*) be a group. Let NL(X) be the lattice of fuzzy normal subgroups in F(X). Let LV\ be the restriction o/SH : F(X) —> C(X)J to NL(X). Let LI(X) be the image of LDK and let LC consist of the crisp subsets in NL(X). Then L9^ is an isomorphism of the lattice NL(X) onto LI(X) that represents NL(X) as a subdirect product in the lattice LCJ. Proof. This result is a corollary of the subdirect product theorem since from that theorem we know that SH is an injection that commutes with PI and *. Since NL(X) is closed under fl and *, L9K is also an injection that commutes with fl and *. Since * provides the SUP operation of iVX(X),the result follows. ■ From Corollary 1.9.2, we obtain immediately the following result. Corollary 1.9.3 For each group ( X , - 1 , * ) the lattice of fuzzy normal sub­ groups and the lattice of crisp normal subgroups satisfy precisely the same (fl = INF, * = SUP)-identities. ■ A lattice is modular if and only if satisfies the identity: vi V (^2 A (vi V v3)) = (vi V v2) A (vi V v3)

38

[232, Theorem 30, p. 87]. Since the lattice of normal subgroups of each group is known to be modular, we have from Corollary 1.9.3: For each group (X~l, *) the lattice of fuzzy normal subgroups is modular. Let LF consist of the fuzzy normal subgroups fj, G F(X) for which the set {fi(x) | x G X} is finite. Let the identity element of the group X be e and let a G [0,1]. Let LFa = {fj, G LF \ 11(e) = a}. Both A and * preserve the property of having a finite image set and the property of assuming a given value at e. Consequently LF and each of the LFa are sublattices of the lattice of fuzzy normal, subgroups in F(X). This observation gives the following result [8, Theorem 3.10]: For each group ( X , - 1 , *) the lattice LF is modular and so is each LFa. When the concept of a fuzzy characteristic subgroup and a fuzzy fully in­ variant subgroup are defined generically, we have immediately that for each group ( X , - 1 , * ) , the fuzzy characteristic subgroups (respectively, the fuzzy fully invariant subgroups) form a sublattice of the lattice on fuzzy normal subgroups which must also be modular. The metatheorem and the subdirect product theorems given here apply widely to the study of fuzzy algebraic struc­ tures. This work should be applicable in various branches of fuzzy algebra and especially wherever fuzzy analogs of known crisp results are desired. As a final illustration, we provide the analogous fuzzy ring theoretic modularity result that follows from the fact that the lattice of fuzzy left (respectively, right, two-sided) ideals of a ring (X, —,+,-•) is a sublattice of the lattice of normal subgroups of the additive group (X, —, +) : The lattice of fuzzy left (respec­ tively, right, two-sided) ideals of a ring is modular. The interested reader may desire to refer to [61] for suggestions for further research. We also recommend that the reader examine a paper by JJ3. Golan [53]. We would like to point out that Weinberger [239] has given a different approach to obtain the results of [8] and [110]. Let A be a universal algebra. In [239], Weinberger shows not only that the lattice of crisp subalgebras of A embeds into the lattice of fuzzy subalgebras of A, but that sometimes the lattice of fuzzy subalgebras of A can be embedded into the lattice of crisp subalgebras of a possible different subalgebra A!. The algebras A and A! generate the same variety. Also, if A is a group, then the fuzzy normal subgroups of A are mapped to the crisp normal subgroups of A!. If A is airing, then the fuzzy ideals of A are mapped to the crisp ideals of A!. As a corollary to the embedding theorem in [239], it is shown that the lattice of fuzzy normal subgroups of a group is modular and that the lattice of fuzzy ideals of a ring is modular. The interested reader is encouraged to examine [239].

39

1.10

EXERCISES

1. (191) Suppose that G is a finite group. Let /x be a fuzzy subgroup of G. Let T\ = {xfi | x G G} and .?> = {/xx|x G G}. Prove the following statements: (a) | ^ i | = |J>| (call |^/| the index of \i in G). (b) The index of /z in G divides the order of G. 2. (13, 208, 245) Suppose that G is a finite group and let \i and v be fuzzy subgroups of G. Let r M (G) = {^a | a G [0,1]}. Define ~ on L(G) as follows: ji ~ v <^ (Vx,y G G, /x(x) > /x(y) ^ v{x) > fi(y)). Prove the following statements: (a) ~ is an equivalence relation on L(G).

(b)/jL~v&rfl(G)=r„(G). (c) There exist fuzzy subgroups /z, z/ (normal), and i/> (level normal) of G such that r ^ is a maximal chain of subgroups of G, TU(G) is a maximal chain of normal subgroups of G, and r ^ ( G ) is a composition series of G. (d) Let d be subgroups of G, i = 1,

, n, such that G = G\ <S> Gi 0

. . . (g) G n and o(Gi) and o(Gj) are relatively prime Vi,j such that 2 ^ j . Then ^ is a direct product of fuzzy subgroups. (e) If G is as in (d) and is impotent, then \i is a direct product of fuzzy subgroups of Sylow subgroups of G. (f) If G is as in (d) and is Abelian, then \x is a direct product of fuzzy subgroups. (g) If G is as in (d) and is cyclic, then \i is a direct product of fuzzy subgroups of Sylow subgroups of G. (h) If G is the symmetric of degree 3, determine the cardinality of L{G)/ ~ . 3. (19) Suppose that G is a finite group. Prove the following statements: (a) G is Dedekind if and only if every fuzzy subgroup of G is normal. (b) [35] Suppose that G is of order p\p2 • • -Pn? the p^s being primes, but not necessarily distinct. Then G is solvable if and only if there exists a fuzzy subgroup [i of G such that /iao, \iai,..., /xan are the only level subgroups of fi, /x(G) = { a o , a i , . . . , a n } , ao < a\ < . . . < a n , and the level subgroups form a composition series. 4. (61) Let (X, *) be a semigroup. Use the techniques of Section 1.9 to prove that every fuzzy left ideal is a fuzzy two-sided ideal if and only if every crisp left ideal is a crisp two-sided ideal.

40

5. (61) Let (A*,*) be a semigroup. Use the techniques of Section 1.9 to prove that the equality \x * v = fi n v holds for all fuzzy right ideals \x of X and for all fuzzy left ideals v of X if and only if it holds for all crisp right ideals of X and for all crisp left ideals of X. 6. (61) Let (X,*) be a semigroup. Use the techniques of Section 1.9 to prove that the equality \x * v — v * fi holds for all fuzzy biideals \i, v of X if and only if it holds for all crisp biideals of X. Prove also that the equality fi * /x = fi holds for all fuzzy biideals [i of X if and only if it holds for all crisp biideals of X. 7. (61) Prove that a semigroup (X, *) is a semilattice of groups if and only if the set of all fuzzy biideals of X is a semilattice under multiplication of fuzzy subsets.

41

Chapter 2

L-SUBGROUPS OF ABELIAN GROUPS Commutative group theory includes some of the best examples of algebraic structure theory. It is also a principal motive of research in module theory. In this chapter, we give structure results for fuzzy subgroups of Abelian groups. Our ultimate goal is to make a connection between such fuzzy subgroups and certain fuzzy field extensions. Throughout this chapter L is a chain and G is an (additive) Abelian group. The results from this chapter come mainly from [147, 171, 180].

2.1

Generators and Direct Sums of L-Subgroups

Let fi G L(G). We introduce the notion of a minimal generating set for // when the support of fi is a direct sum of cyclic groups. We show that if the support of fi is cyclic, then \i has a minimal generating set. We show that this result can break down when the support of fi is a direct sum of two or more cyclic groups. These examples also show that although a subgroup of a group G which is a direct sum of cyclic groups is a again a direct sum of cyclic groups, an ^-subgroup of G need not have the corresponding property. By the summation x = ^ a € ^ x«> w e m e a n ^ but a finite number of the xa are zero. If {fia \ a G ft} is a collection of L-subgroups of G, then we define the L-subset ^2aeCt /j,a of G as follows: Vx G G, (^2 A4*)0*0 = V{A{^ a (x a )|a G Q}\x = ] T

xa}.

We see that this definition corresponds to the definition of weak product given

42

in Definition 1.5.2. Hence all the results derived in Chapter 1 for the weak product hold here in a corresponding manner. In a similar manner, we speak of the weak direct sum rather than the weak direct product. We make the obvious change in notation, i.e., we replace Yl

an

d <8> with 0 .

Definition 2.1.1 Let fi G L(G). Let L(fi) = {ue L(G)

\vCfi}.

Let x be an L-subset of G such that X £ M- Define (x) to be the intersection of all 7 6 L(/J,) such that x ^ 7- Then x is called a generating set for (x) in fi. Clearly, (x) is an L-subgroup of G. In fact, (x) is the smallest L-subgroup of G in L(/J>) containing xDefinition 2.1.2 Let /x G L(G). Let S denote a set of L-points such that if G S, then a = b > 0 and x ^ 0. Define the L-subset x(^) as follows:

xGS)(*) = j

0

other otherwise.

Set (S) — (x(«S)) • $ is called a minimal generating 0^(0) and n D (S\{xa}) U 0M(0) for any xa G S.

set for /J, if ji = (S) U

Theorem 2.1.3 Let fi G L(G). Ifxi, x fi(xn), then p(xi + X2+ . . . + xn) = n(xn). Proof. Let x = X\ -f x A ^ V C ^ i ) > fi(xn). Hence /x(xn) = //(—a; + x + x n ) > /x(x) A fi(x + x n ) > ^0*0 A M^n) = Kxn)> Thus /i(x) A ^(ar + xn) = fi(xn). Since /z(x) > fi(xn), /i(x + x n ) = fjb(xn). ■ Example 2.1.4 Let L = {1, a, 6, c, 0}, where 1 > a > c > 0 and 1 > 6 > c > 0, a A 6 = c, and a and 6 are noncomparable. Then L is regular. Let /J, E Lz be defined by /x(x) = 1 if x G (6), ^(x) = a z/ x G (2) \ (6) , /z(x) = b if x G (3) \ (6), ^(x) = c otherwise. Then /zi = (6), /xa = (2), fn, = (3), and /i c = Z. 77i^ /J, is an L-subgroup of Z. Le£ x = 2 ana7 2/ = 1. TTien //(x) = a > c = fi(y). But, /z(x + y) = /x(3) = b ^ /^(y). Hence Theorem 2.1.3 zs no^ £nze z/L zs no£ a chain. Theorem 2.1.5 Let fi G L(G) and fi(G)\{0} = { a o , a i , . . . , a n } . Suppose that a;_i > ai, i e I\{0}, where I = { 0 , 1 , . . . , n } . 7/ £/iere exists a subgroup Hi of fiai such that [iai — /i ai _i ©-Hi for each i G A W ? then there exists \ii G L(G), i G 7, such tfm£ /i = (BieifJ>%, Mo — Ma0> a n a ? Mi = -Hi /or e a c ^ i € A W -

43

Proof. Since fiai = /j,ai_1 0 Hi for each i G 7\{0}, it follows that /i a n = fiao 0 Hi 0 . . . 0 Hn. Define the L-subset ^ of G as follows: /x;(0) = a 0 , ^0*0 = ai if x G i7;\{0}, and fii(x) = 0 otherwise, i G 7, where iJ 0 = AW Then ^ G L{G) and /J,* = Hi, i e I. Clearly, A*O = ^ on i7o and fa = JJ, on Hi since 7/^ C /x0. andflin/x a i _i = {0}, 2 G 7\{0}. Let x G /x*. Then x = x 0 + zi + .. . + x n , where x» G Hi, i G 7. Hence /i(x) = /io(ar) A ^i(x) A . . . A [JLn(x) by Theorem 2.1.3. Now (J2iei Vi)(x)

= =

V{A^ =0 /i(^)|x = y 0 + 2/i + • • • + 2/n,y» e jxj,i G 7} /x(x0) A /x(xi) A . . . A /i(x n )

since the expression x = XQ + x\ + . . . 4- xn is unique. Hence fi = ^2ieI HiThus /JL = 0 i G / ^ i by Theorem 1.5.9. ■ Theorem 2.1.6 Let /J, G L(G) and fi(G)\{0} = {a0,ai,... , a n } . Suppose that ai-i > ai, i G 7\{0}, w/iere 7 = { 0 , 1 , . . . , n } . 7/p/i* = {0} / o r p a prime, then pi is a weak direct sum of L-subgroups whose supports are cyclic. Proof. Since p/j,* = {0}, ^ a i _i is pure in /j,ai and [xai-x ls a direct summand of fiai , say fidi-x © Hi f° r some subgroup i7i of G, i G 7\{0}. Thus by Theorem 2.1.5, ^* = /xO0 0 i7i 0 . . . 0 i7 n and \x = (BieiVi for L-subgroups \ii of G such that fii* = Hi, fiQ = jiao = Ho. Since pji* = {0}, each Hi is a weak direct sum of cyclic groups, say Hi = (BjiCJiKjn ji G Ji (an index set), i = 0 , 1 , . . . ,n. Define the L-subsets ^ of G as follows: c.(~\-f ?jiW

A*iO*0 if are ^ | o otherwise

ji G Ji, i = 0 , 1 , . . . , n. Then ^ is an L-subgroup of G and £J. = 7C,. is cychc. Since each £j. is the same constant ai on 7^-^(0} and each \ii equals a^ on Hi\{0}, fa = £ i i G J i & = ®ji€J&i- Thus ^ = © i € / ( ®ji€j^jt). ■ Proposition 2.1.7 Let \i G L(G). Let x then x + V £ M-

and

V be L-subsets ofG. Ifx, V Q V->

Proof. Let z G G. Then (x + T/)(Z) = V{x(x) A 77(2/) | z = x + 3/} < V{/x(x) A n(y) \ z = x + y} < V{/z(x + y) \ z = x + y} = /i(z). ■ Proposition 2.1.8 Let /x G L(G). Let x be an L-subset of G. If x Q A*> then

44

Proof. By Definition 1.2.1, (-x)G*) = x ( - z ) < fi(-z) = fi(z). ■ For any L-singleton xa of G, it follows easily that —(xa) = {—x)a. T h e o r e m 2.1.9 Let fi G L(G). Let x be an L-subset of G such that x ^ / i . Let a be the L-subset of G defined by V y G G, o~{y) = V{(ei(xi) a]L + . . . + en(xn)an)(y) \ Xi E G, ai e L, x(xi) = a., et = ± 1, z = 1 , . . . , n; n G N}. 27ien cr = ( x ) . Proof. By Proposition 2.1.8, e;(z;) a . C (x) for i — 1 , . . . , n. Thus cr C (x) since e i ( x i ) a i + . . . + en(xn)an C (x) by Proposition 2.1.7. To show that (x) £ a, it suffices to show that a G L(/J,) and x Q ^ Let x G G and x(#) = a. Then clearly xa{x) < a(x) since x a = e(xa) with e = 1 and n = 1. Hence x £ 0\ Let x, y E G. Then <J(:C) and <7(y) are supremums of numbers of the form (ei(xi) a i + • •• + en(xn)an)(x) and (/i(2/1)61+ . . . + frn(ym)brn)(y), respectively. Suppose that cr(x) > 0 and a(y) > 0. Then 3 sequences aj = ai. A . . . A anj. and && = 6 ^ A . . . A bmk such that aj —► cr(x) and 6& —► cr(?/). If x G ( x i , . . . , x n ) a n d y G (3/1,... , y m ) , t h e n x + ?/ G ( x i , . . . , x n , 2 / i , . . . ,ym). T h u s a ( x + 7/) > W{ajAbk \ j , k = 1,2,...} = (V{a,- | j = 1,2,...}) A (V{bk \ k = 1,2,...}) = cr(x) A
\ e{ = ± 1 , i = 1 , . . . ,n; n G

Corollary 2.1.11 Let fi G L(G). Let S be a set of L-singletons such that VxaeS,0 0 & V{(ei(xi) a i + . . . + e fc (x fc ) a J(:r) | {xi)ai G 5 , ei = ± 1 , i = 1 , . . . ,/c, k G N} > 0 <=> x = ei^i + .. . + ekXk for some {xi)ai G 5 , i = l , . . . , t ^ x G (5*). ■ Recall that by a sum of the form Ylueimiu) m u ( n u # ) a u ) where rau, n u G N U {0}, we mean only a finite number of mu are different from 0. L e m m a 2.1.12 Let /i G G. Suppose that /J,* = (x) for some x G fx*. For all u G /i(G)\{0}, let nu denote the smallest positive integer such that fiu = (nux) if l^u 7^ {0} and let au = ji{nux). Then (1) u = au; (2) fi = ({(nux)au\u e fi(G)}).

45

Proof. (1) au — n(nux) > u. Let z G /i*\{0} be such that fi(z) = u. Then z G \iu and so 3m G Z such that z = m(nux). Thus u = /x(z) > fi{nux) = au. Hence u = au. (2) Clearly, ({(nux)au\u G /x(G)}) C ^. Let z G ^ . Now ({(nux)au\u G M(G)}) (z) = V{( J2uen(G) mu(nux)au)(z) | mu G Z}. Hence if ^x(z) = u, then 2: G jxn and s o 2 = m(nux) for some m G Z. Clearly, then

/*(*) < ( { K x ) a J w G Ai(G)}> (z). ■ Let % be an L-subset of G such that x ^ ^> where fi e G. Let («S(x)) denote the intersection of all L-subgroups of G in L(/x) such that x a C /x V x a G <S(x)Clearly, <S(X)> = (x) • Theorem 2.1.13 Let /1 G G. Suppose that /x* = (x) for some x G /x*. For a// ix G /J>(G), let nu denote the smallest positive integer such that fiu = (nux) . Then {(nux)u \ u G M ( G ) \ { 0 } } is a minimal generating set for fi. Proof. Let I — lm(^x)\{0}. By Lemma 2.1.12, {{nux)u \ u G / } is a set of generators of/x. Let v e I and v = ({(nux)u\u G I}\{(nvx)v}). Then v G L(/x). m Now v = n(nvx) and v(nvx) = V{ixi A . . . A ixn} | nvx = tluei\{v} u{nux), mu G Z, rau 7^ 0 for ix G {ixi,... ,ix n }, n G N} = V{ix| n v x = mu(nux), u < v, u G / } ( if 3 u G i such that u < v) < v, where = and < occur since / is well-ordered under < . Hence v C \i. Now suppose there does not exist u e I such that u < v. Suppose that v(nvx) > 0. Then nvx = J2i=irnui(nuix) for some nUi, mUi, (ui > v), i = 1 , . . . , k. Hence nvx G (nux) = ^iu for some u > v, a contradiction. Thus v(nvx) = 0. Hence v C \i. I Proposition 2.1.14 Let fj, G G. Let {/xa | a G 0 } C L(^x). Suppose that M* — ©a€n(Ma)*, w/iere V a G ft, (^a)* = (#a) /or some rr0 G G. Let «Sa — {((na)uaxa)Ua I ixa G yLxa(G), (n a ) U Q G N} 6e a minimal generating set for fia, a G ft. T/ien ^ Uaefic>a zs a minimal generating set for 0aefiMc*; (%) Uaefi*^a is a minimal generating set for ji if and only if \i = ©aeftM<*Proof. Let v — ® a 6 n / i a . Then z/(0) = /xQ(0) Va G ft. Let 2: G (M/?)* for some /? G ft. Then i/(z) = V{A{/xa(;za)|a G ft} | z = £ a G O j z a , 2:a G (/xa)* Va G ft} = A{/x/3(z) A/xa(0) I a G ft\{/3}} = jJLp(z) since the representation z = ^aen za is unique and i/(0) = /xa(0) Va G ft. That is, v = pp on (M/?)*. Let y a G 5^. Then (U a € f i 5 a ) > = 0 a e n ^ a and so (UaenSa\{ya}) (y) = ( 0a € n,/3^ t t (5 a ) © (5/3\{2/a}))(y) = (5/3\{ya}> (z/) < (5/j) (y) = i/(y). Thus U a € f i 5 a is a minimal generating set for v. ■ Proposition 2.1.14 shows that when fj,* is a direct sum of cyclic groups, the question of when /1 has a minimal set of generators is equivalent to that of fj, being a direct sum of L-subgroups whose supports are cyclic.

46

E x a m p l e 2.1.15 Let L = [0,1]. Let G = Z(p)@Z(p2) and let H = ((l,p)>. Define the fuzzy subsets /x, z/, and 7 of G by /x(x) = 1 if x G H, /x(x) = £ otherwise; i/((0,0)) = 1, i/(x) = £ z/ x € Z(p) 0 {0}\{(0,0)}, i/(a?) = 0 otherwise; 7(0,0) = 1, 7(2;) = | i / i G {O}0Z(p 2 )\{(O,O)}, 7(0:) = 0 otherwise. Then /x, z/, 7 are /zxzz?/ subgroups G such that v, 7 C \x and fi* = z/* 0 7*. However, \x ^ v + j since /x((l, p)) = 1, w/iile (j/ + 7)((l, p)) = 3. Now ira(z/) = {1, 1,0} = 7m( 7 ), 1* - {(0,0)}, I/J = Z(p) 0 {0}, 71 = {(0,0)}, 71 = {0} ©Z(p 2 ). Tnzxs { m ( l , 0 ) i } a^irf { m i ( 0 , l ) i } are minimal generating sets for v and 7 respectively, where m = 1 = m i . xVow { ( l , 0 ) i } U {(0, l ) i } is a minimal generating set for v + 7, &IA£ noi a generating set for /x since /x ^ z/ + 7. However, [i = 1// 0 7 and 50 /i has a minimal generating set. Theorem 2.1.16 Suppose that G = ©2aGi, where Gi is a cyclic subgroup of G for i = 1 , . . . , m. Suppose that 3 a cyclic subgroup H of G such that H is not contained in any cyclic direct summand of G. Then 3 an L-subgroup fi of G such that \x is not a direct sum of L-subgroups of G whose supports are cyclic. Proof. Define the L-subset /x of G by /i(x) = 1 if x G H and ji(x) = \ otherwise. Then /x is an L-subgroup of G and /x* = G. By hypothesis 3 jii G L(xx) for i = 1 , . . . , m such that /x* = 0 2 i M i a n d Mi = (Ci) f° r some Q G G, Z = 1 , . . . , m. We show that ji ^ YMLI Mi s o t n a t M 7^ 02iMi- Let 2: G G be such that H = (z). Then (X)™! Mi)0*) = V{A{Mi(zi)K = 1, . . . ,m} | z = zx+ . . . + £ m } . Thus (5^2=1 A^X*) = 0 if ^ ^ /x* for some z, while /x(z) = 1. Suppose that (Y^tLi MiX2) — 1- Then 3 z i , . . . , z m G G such that z = z\+ . . . + z m and Hi{zi) = 1 for i = 1 , . . . , m. Since /Xi C /x, 2^ G i J Pi /x*, z = 1 , . . . , m. Suppose that I if I = r. Then z has order r and so Z{ has order r for some z. Since Zi G if D /i*, H = (zi) C /x*. Thus i l is contained in a direct summand of G, a contradiction. Suppose that \H\ — 00. Then zi = qiz for some integers q^ i = 1 , . . . ,m. If qi ± 0 / gj for z / j , then fi* H fi* D (&*) H (^2) D {0}, a contradiction. Hence 3 z such that V j,j ^ z, g^ = 0 , qi = ± 1 . Thus HC.fi*, a contradiction. Therefore, ( ] £ 2 i A^X2) 7^ 1 a n ( i s o ]CHi Mi C /x. ■ E x a m p l e 2.1.17 Let L = [0,1]. Let G = Gi 0 G 2 , where d is a cyclic sub­ group of G for 2 = 1,2. Suppose that \G\ \ = 00 and IG2I = pr for some prime p. Then we show that 3 a fuzzy subgroup fi of G such that fi is not a direct sum of fuzzy subgroups ofG whose supports are cyclic. Let Gi = (xi), z = 1,2. Let H — ((pxi,x2)). Let K be a subgroup of G such that H D K = {(0,0)}. If (si^i, s2x2) G K for si, s2 G Z, t/ien (p r + 1 5iXi,0) G H D K and so s\ = 0. Thus if H is contained in a direct summand of G, say G = H' 0 K with H C W', t/ien K C. G2. Hence K = G2 since G2 is a torsion subgroup of G. J4£SO i J ' is q/c/z'c and of infinite order. Let H' = ((tiXi,t2x2)) . Then 3 q G Z 5zxc/i tfiat (p^i, x 2 ) = ^(tia:!, t2x2). Hence qti = p a n d p / gt 2 . Tte5 g = ± 1

47

and so t\ = =f P- Hence H = H'. But then (xi, 0) ^ H' 0 K, a contradiction. That is, H is not contained in a direct summand of G. Hence the desired result follow from Theorem 2.1.16. Example 2.1.18 Let L = [0,1]. Let G = G\ 0 G2, where Gi is a cyclic sub­ group o/G, i = 1,2 and |Gi| = p r , IG2I = ps for s — r > 2. T/ien we show that 3 a fuzzy subgroup fi of G such that fi is not a direct sum of fuzzy subgroups of G whose supports are cyclic. Let G{ = (x{), i = 1,2. Let H = ((xi,pkX2)) for 0 < k < s — r. Let K and J be any cyclic subgroups of G such that \K\ — Pr> \J\ — Vs\ and G — K 0 J. Now J = ((c,d)) for some c G G\ and d G G 2 . Now |iJ| = ps~k and s - k > r. Thus H %. K. Suppose that H C J. Then 3 g G Z suc/i £/ia£ g(c, d) = (#i,p fe £2)- A^oit; o(gc) = pr. Thus p)(q. Hence o(qd) = o(d) = ps, a contradiction. Thus H is not contained in a direct summand of G. The result now follows from Theorem 2.1.16.

2.2

Independent Generators

Let n G N and xa be a fuzzy singleton in G. Then n x a denotes x a + . . . + xa= N

v

'

n times

Mfl. Definition 2.2.1 Let fi G L(G). A system of L-singletons { ( x i ) a i , . . . , (#fc)afc}> where 0 < ai < fi(xi) for i = 1 , . . . , k is said to be linearly independent in \i */ni(Ei)ai+ • • • + nk(xk)ak = 0 a implies n&i = ... = nkxk = 0 where Ui G Z /or z = 1 , . . . , k and a G £\{0}. A system of L-singletons is called linearly dependent if it is not independent. An arbitrary system S of L-singletons is independent in fj, if every finite subsystem of S is independent. We let S denote a system of L-singletons such that V xa G 5 , 0 < a < n(x). Let <S* = {x\ xa G S} and Sa = p.a n 5* V a G {a G L\{0} | a < /x(0)}. Proposition 2.2.2 Le£ fj, G L(G). The following conditions are equivalent on S. (1) S is independent in ji\ (2) 5* is independent in /x*; (3) Sa is independent in \xa V a G {a G £\{0} | a < MO)}. Proof. (1) =>• (2). Let rix^i + . . . + nkxk = 0, where xi, ...,xk G 5*. Then n i ( x i ) a i + . . . + nk(xk)afc = 0 a , where (xi)ai G 5 for z = 1 , . . . , k and a = a\ A . . . A ak. Thus n ^ i = . . . = nkxk = 0. (2) => (3). The result is immediate since Sa C 5* V a G {a G L\{0} |

a < MO)}.

48

(3) => (1). Suppose that n i ( x i ) 0 l + . . . + nk(xk)ak = 0 a , where (xi)ai G S for i = 1 , . . . , &. Then a = ai A ... Aak, ri\X\ -f ... + n/c^fc = 0 and Xi G /i a for i = 1 , . . . , A;. Thus nixi = . . . = n^fe = 0. ■ Definition 2.2.3 Let \x G L(G). An independent system Ai of L-singletons in n is said to be maximal if fl an independent system S of L-singletons in H such that M. C S. Proposition 2.2.4 Let fi G L(G). Every independent system S in fi can be extended to a maximal independent system. Proof. By Proposition 2.2.2, <S* is independent in /i*. By Zorn's Lemma, S* can be extended to a maximal independent system M in /x*. Let M — {xa\ x G M, fj,(x) = a}. Since x G //*, a > 0. Thus A4* = M and so A4 is independent. Clearly, M is maximal since A4* is maximal. ■ T h e o r e m 2.2.5 Lei yu G L{G). S is independent in fi if and only if the L-subgroup of G generated by S in fi is a direct sum of L-subgroups of G whose supports are cyclic, i.e., for S = {(xi)a.\0 < ai < fj,(xi),i G / } , (S) = ©i€l<(*i)a<>.

Proof. (S) = J2iei ((xi)ai) • N ° w S is independent in /i <^> 5* is independent in tf <* (<$*) = 0z6/ (x»> <& (S) = @i€l ((xi)a.) since ((x;) a .)* = (^) V i G

/.■

2.3

Primary L-Subgroups

Definition 2.3.1 Let \i G L(G). fi is called a p-primary L-subgroup of G if 3 a prime p such that V L-singletons xa C /j, with a > 0, 3 n G N such that Pn(xa) = 0 a . Let fi e L(G). Then ^x is p-primary if and only if /x* is p-primary. Also /x is p-primary if and only if \ia is p-primary Va £ {a e L\{0}|a < /x(0)}. For H a subgroup of G and p prime, we let Hp denote the p-primary component of H. Proposition 2.3.2 Let /i G L(G). Let p be a prime. Define the L-subset ^ ofGbyMxeG, iP)(x) = fi(x) if x G (fj,*)p and^\x) = 0 otherwise. Then ji^ is ap-primary L-subgroup of G and (fi^)* = (/x*)p. Furthermore (/x^) a = (Pa)p V a G L, a < /x(0). ■ Proposition 2.3.3 Let /i G L(G). Let p be a prime. Then fi^ maximal p-primary L-subgroup of G in /JL. H

is the unique

49 For p a prime, we call ^

the p - p r i m a r y c o m p o n e n t of /x.

L e m m a 2 . 3 . 4 Let xx, G L(G). Let m G L(/x), i G I. Suppose that /x* = ©» € //z* and that /x* is torsion. Suppose that V x G /x*, /x(x) = /x^x) V i e I. If the order of the elements of \i\ are relatively prime to those of yf- (i ^ j , i, j G I ) , iTien /^ = QiziHi. Proof. Let x G xx*. T h e n £ — 2 J i 6 / ^ where x^ G ^ * and all b u t a finite number of X{ — 0. By the assumption concerning the orders of the elements of fii*, V i G / , 3 ki G N such t h a t kiX = kiXi and A;*, o(xi) are relatively prime. Hence (xi) = (kiXi) and so fifa) = /j,(kiXi) = y{kix) > /JL(X) V i G I. T h u s (Zi€iVi)(*) = V{A{ W (*J) | x - S i e / ^ i < e /x?} | * € J } = A { / i ; ( ^ ) | x = Ylieixi} ( s i n c e ^* = $»€//**) = A{/x(a;i) | x = E z 6 / ^ } ^ x / x ( » . Clearly, ( E ; e / / ^ ) ( ) ^ M*)- T h u s M = J2iei &- H e n c e M = ®ieiVi by Theorem 1.5.9. ■ D e f i n i t i o n 2 . 3 . 5 Let /x G L{G). ji is called a torsion L-subgroup fuzzy singletons xa C /x wzY/i a > 0, E3n G N stxc/i i7ia£ n ( x a ) = 0 a P r o p o s i t i o n 2.3.6 (1) ii is torsion G. (2) /x is torsion (3) There exists r is torsion. ■

of G ifM

Let \i G L{G). The following assertions hold. if and only if \i* is a subgroup of the torsion subgroup of if and only if jia is torsion Ma G {a G L|0 < a < /x(0)}. a unique maximal L-subgroup r of G such that r C /x and

T h e o r e m 2 . 3 . 7 Let fi be a torsion L-subgroup of primary L-subgroups of G.

of G. Then fi is a direct

sum

Proof. Since /i* is torsion, /x* = © p (xx*) p , where (/x*)p is the p-primary component of /x*. By Propositions 2.3.2 a n d 2.3.3, / z ^ is the p-primary com­ ponent of /x a n d (/x^))* = (/x*) p . T h u s xx* = ® p ( x x ^ ) * . By L e m m a 2.3.4, /x =

e P /x^.B T h e o r e m 2 . 3 . 8 Let xx, */ G L(G) such that /x C v. Then there is a unique maximal L-subgroup j of G such that /x C 7 C v and \/xa C 7 with a > 0, iTiere exists n G N and 6 G L \ { 0 } stxc/i that nxb C /i. Proof. There exists a unique maximal subgroup T of 1/* such t h a t T D /x* and T//x* is torsion. Define t h e L-subset 7 of G by Vx G G, 7(2;) = i/(x) if x G T a n d 7(0:) = 0 if x £ T. T h e n 7 is an L-subgroup of G a n d xi C 7 C 1/. Let x a C 7 with a > 0. T h e n x e T and so there exists n G N such t h a t nrr G /x*:. T h u s n x t C /x for 6 = fj,(nx) > 0. Let £ be any L-subgroup of G such t h a t

50

fi C 6 C v. If xa C 6 with a > 0 and nx^ C //, for some n and 6 > 0, then 6(x) < v(x) = 7(x). Hence 7 is imique maximal. ■ The L-subgroup 7 of G in Theorem 2.3.8 is called the torsion closure of /i in v. Theorem 2.3.9 Let {j,,v G L{G) be such that fi C v. Let 7 6e an L-subgroup of G such that /i C 7 C v. Then 7 zs £/ie torsion closure of [i in v if and only if 0. Hence 3n and 6 G £\{0} such that nxb C /i. Then nx E fi*. Thus 7*/^* is torsion. Let a: G z/* be such that 3n such that nx E fi*. Then xa C v with a > 0 and na^ C /i for some 6 G £\{0}. Thus x a C 7 and so a: G 7*. Hence 7*/^* is the torsion subgroup of i/*//i*. Clearly, 7 = z/ on 7*. Conversely, suppose that 7*/^* is the torsion subgroup of I/*//J,* and 7 = 1/ on 7*. Let x a C 7 with a > 0. Then x G 7*. Thus 3n G N and b E L\{0} such that nxb C /i. Now suppose that x a G z/ with a > 0 and n&x C /J, for some n and b. Then a: G 7*. Hence 7(2;) = v(x) > a > 0. Thus x a C 7. Hence 7 is the torsion closure of \i in ZA ■

2.4

Divisible and Pure L-Subgroups

Definition 2.4.1 Let /z G L(G). \x is called divisible ifVxa C /x wzY/i a > 0, and Vn G N, 3y a swc/i £/ia£ y a C ji such that n(ya) — xa. Proposition 2.4.2 Let /z G L(G). \x is divisible if and only if \ia is divisible Va G {a|0 < a < /i(0)}. ■ Proposition 2.4.3 Let \x E L(G). If /i 25 divisible, then /i* 25 divisible. M Proposition 2.4.4 Let \i E L{G). If ji indivisible /z*\{0}, then \i is divisible.

and /i is constant on

Proof. Let xa C jj, with a > 0 and n G N. Then x E \i* and so 3 y E /x* such that rr = ny. If 2/ = 0, then x = 0 and the result follows. Let y ^0. Since // is constant on /i*\{0}, fi(y) = fi(x) > a. Thus ya C /i and x a = n(ya). Hence /x is divisible. I Let n E N and x a C 1{0}. Then x = 0. Hence 3 ya C 1{0} such that w(2/a) = #a- Thus 1{0} is divisible. Let T denote the torsion subgroup of G. Proposition 2.4.5 Let fi E L(G). Vx,y G G and n G N, ny = x implies that fi(x) = n(y) for all divisible L-subgroups uofG if and only if G is torsion-free.

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Proof. Suppose that the condition concerning /J, holds. Let x G T. Then 3n G N such that nx = 0. Since 1{0} is divisible, l{o}(^) = l{o}(0) = 1- Hence x = 0. Thus T = {0}. Conversely, suppose that G is torsion-free. Let \i be any divisible L-subgroup of G. Suppose that nx — y. Let a = ji(x). Then n(ya) = xa. Now 3yf G G such that ny'a = xa and y' C //. Thus ny' — x. Since G is torsion-free y = y' and so ya C /z. Thus a = /z(x) = fi(ny) > /j,(y) > a. Hence

Kx) = Kv)-U

Theorem 2.4.6 Let \i G L(G). Let G = Q,, w/iere Q denotes the additive group of rational numbers. Then \x is divisible if and only if /j, is constant on G. Proof. Suppose that ji is divisible. Let x G G. Then 3n G N such that nx = m G Z. By Proposition 2.4.5, fi(x) = /z(ra). Now m • 1 = m and so /x(m) — /x(l) by Proposition 2.4.5. Thus /z(a;) = /x(l). Conversely, suppose that \i is constant on G. Let xa C fi and n G N. Now 3y e G such that ny = x and so n(y a ) = x a . Since yu is constant on G, /x(y) = /x(x). Thus ya C |x. Hence /x is divisible. ■ Proposition 2.4.7 Let G = Z(p°°), a quasicyclic group [49]. ThenM divisible L-subgroups fj, of G, Vx,y G G\{0}, Vn G N, ny = x implies fi(x) = n(y). Proof. Let x, y G G\{0}, n G N and ny = x. If ^(x) = 0, then fj,(x) = fi{y). Suppose that fi(x) = a > 0. Then xaC p,. Now 3ya C ^z such that n(?/a) = x a . Thus ny — x. Hence a = fi{x) = n(ny) > p,(y) > a. Thus ji(y) = fi(x). Hence it suffices to show that if nw — x and nz = x, then /J,(W) = fi(z) since we would have yb(y') = fJ>(y) = p.{x) whenever ny' = x. Let w = u/pr and z = v/ps, where u,p are relatively prime and v,p are relatively prime. Assume that r < s. Suppose that n{u/pr) = n{v/p8) for some n G N. If pr\n in Z, then n(u/pr) = n(v/ps) = 0 in Z(p°°) and so # = 0, a contradiction. Thus n = pfn', where 0 < t < r and n' and p are relatively prime. Hence n'(u/pr~t) = n^v/p3-1). Thus 0 = riu = n'{v/ps-r) in Z(p°°). If r < s, then p|n'i; in Z, which is impossible. Hence r = s. Now 3a, b G Z such that 1 = au + 6p r and so 1 / y = a(u/p r ) + o = em/p r in Z(p°°). Thus v/p 5 = v / p r = va(w/p r ). Hence Kv/Ps) = Kva(u/Pr)) > Ku/Pr)' % symmetry, fi{u/pr) > fi(v/ps). ■ We note that if G = Z(p°°), then ^(x) = /x(y) VX-subgroups // of G and VX,T/GG\{0}.

Theorem 2.4.8 Le£ /z G L(G). Let G = Zip00), a quasicyclic group [49]. Then \x is divisible if and only if /i is constant on G\{0}.

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Proof. Suppose that /x is divisible. The subgroups of G are chained, say {0} C H\ C . . . C Hi C . . . , where Hi is a cyclic group of order p\ Let x G G\{0}. Then x has order p1 for some i G N. Let y — p 2 _ 1 x. By Proposition 2.4.7, /x(x) = fi(y). Now 7/ has order p and iJi = (y). By comments preceding the theorem, \i is constant, say a, on i?i\{0}. Hence u(x) = a. Conversely, suppose that a is constant on G\{0}. Let xa C /x with a > 0 and let n G N. There exists y e G such that ny = x. Hence ny a = x a . However, /x(?/) = u(x) > a. Thus ya C ytx. Hence /x is divisible. ■ Let L'(/x)=L0x)\{l { o } }. Definition 2.4.9 Let [i G L(G). fi is called reduced ifjBvE v is divisible.

L'(JA)

such that

Proposition 2.4.10 Let \i G L(G). [i is reduced if and only if 3a G {a|0 < a < /x(0)} such that /xa is reduced. Proof. Suppose that \x is not reduced. Then 3 v G L\\x) such that v is divisible. If z/(0) ^ M(0), then define the L-subset v' of G by v'($$) — /x(0) and z/(x) = v{x) Vx G G\{0}. Then v1 G Z/(/x) and v' is divisible. Hence we may assume that v(0) = /x(0). By Proposition 2.4.2, va is divisible Va G {a|0 < a < yLx(O)}. Since z/a C xxa, /xa is not reduced Va G {a|0 < a < /x(0)}. Conversely, suppose that Va G {a|0 < a < /x(0)}, /xa is not reduced. Then 3 a divisible subgroup D^ of xxa Va G {a|0 < a < / J ( 0 ) } . Let D = D^ for a = /x(0). Then £> C xxa Va G {a|0 < a < xx(0)}. Since £> C /i*, /x(x) = /J(0) Vx G D. Define the L-subset i/ of G by Vx G D, u{x) = zx(0) and v{x) = 0 otherwise. Then v is an L-subgroup of G such that ^ C fi. Now Vxa C i/ with a > 0 and Vn G N, 3y e D such that ny = x. Since u(y) = /x(0) > a, ya C v. Thus z/ is divisible. Hence /x is not reduced. I Corollary 2.4.11 Let /x G L(G). /i zs reduced if and only if /i* zs reduced. Proof. Suppose /x is reduced. Then [ia is reduced for some a G {a|0 < a < ^(0)}- Now /x* C /i a and so /x* is reduced. Conversely, suppose that /x* is reduced. Let a = /x(0). I Definition 2.4.12 Let fi G L(G). Let v G L(/x). Then v is said to be pure in fi ifVxa C v with a > 0, Vn G N, Vya C /x, n(y a ) = x a implies 3 za C v such that n{za) = xa. Proposition 2.4.13 Let fi G L{G). Let v G L(/i). Then v is pure in \i if and only if va is pure in /xa Va G {a|0 < a < i/(0)}. I

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Definition 2.4.14 Let x be an L-subset of G and let n G N. Define the Lsubset n\ of G by Vx G G, (n%)(x) = 0 if x £ nG and (n%)(x) = \/{x(y)\y £ G,x = ny}. Proposition 2.4.15 Let \i G L{G). Let n G N. Then (1) n/i(0) = /i(0); (2) nfi C n] (3) n/i is an L-subgroup of G; (4) If /i has the sup property, then nfi(G) C /i(G). Proof. (1) (n/i)(0) = V{/i(y)|0 - ny} = /i(0). (2) If x ^ nG, then (n/i)(x) = 0 < /i(x). Suppose that x G nG and x = ny for some y e G. Then /i(x) = /x(m/) > A*(z/)- Hence (n/i)(x) = V{/i(?/)|2/ G G, x = ny} < /i(x). (3) Let x,y e G. If either x ^ nG or y £ nG, then (n//)(x) A (n/i)(y) = 0 < (n/i)(x — y). Suppose that x G nG and y G nG. Then x — y G nG and (n/i)(x — y) = \/{fi(w)\w e G,x — y = nw} > W{/i(u — v)\u,veG,x = nu,y = nv} > V{fi(u) Afi(v)\x = nu,y = nv} = (\Z{/i(u)\x = nu}) A(V{fi(v)\y = nv}) = (n/i)(x) A (nji)(y). Thus n/i is an L-subgroup of G. (4) Let a G (nfi)(G). Then (n/x)(x) = a for some x € G. Now (n/x)(x) = V{/i(y)|x = ra/,y G G} = a. Since /i has the sup property, 3y G G such that u(y) = a. Hence a G fi(G). Thus {nii){G) C /x(G). ■ Suppose that /i* is torsion-free. If x G nfi*,n G N, then (n/x)(x) = /i(w) for some unique w £ /i* such that x = nu;. Proposition 2.4.16 Let \i G L(G). Lei n G N. Then (1) (nfj)*

=nfi*;

(2) n/ia C (n/x)a Va G a G {a|0 < a < A*(0)}- # either /i has the sup property or /i* is torsion free, then nfia = (n/i)a Va G {a|0 < a < /i(0)}. ( ^ Let v G £(//) and u(0) = fi(0). If fi has the sup property and v is pure in fi, then nva = {nv)a Va G {a|0 < a < /x(0)}. Proof. (1) x G (n/i)* <£4> (n/z)(x) > 0 <£> x = nw for some w e ji* <£> x G n/i*. (2) Let x G n/i a . Then x = nw for some w G /i a - Hence (nfi)(x) > a. Thus x G (nfi)a. Conversely, let x G {nfi)a. Now {np)(x) = V{/i(?/)|x = ra/,i/ G G}. Thus 3?/ G G such that x = ny and /i(y) > a. Hence x G n/i a . (3) Let a G {a|0 < a < /i(0)}. Since i/ is pure in /i, i/a is pure in /ia. Thus nva = vaC\ n\xa. Let x G (nz/)a. Then (^ fl n\i){x) > (nv)(x) > a. Thus x G (^ n n/i)a = var\ (nfi)a = va fl n/i a = ni/ a . Hence {nv)a C nz/a. ■

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Proposition 2.4.17 Let /x G L(G). Let v G L(/x) be such that v(0) = /x(0). (1) Suppose that /J, has the sup property. If v is pure in /x, then Vn G N, nv = v D n/x; (2) Suppose that fi has the sup property. IfVnE N, nv = v D n/x, then v is pure in /x; (3) Suppose that /i* is torsion-free. Then v is pure in /x if and only if Vn G N, nv = v D n/x. Proof. i/ is pure in /x O Va G {a|0 < a < /x(0)}, z/a is pure in /xa ^ Va G {a|0 < a < /x(0)}, Vn G N,ra/ a = ^ a H n/xa ^ V a E {a|0 < a < /x(0)}, Vn G N, (ni/) a = ^ a n(n/x) a oVae {a|0 < a < /x(0)}, Vn G N,nz/ a = (i/nn/z) a <& Vn G N, ni/ = v fl n/x. ■ Proposition 2.4.18 Le£ /i G L(G). Let I / J E L(/x) 6e snc/i that 7 C 1 / and 7(0) = i/(0) - MO). (%) If 7 is pure in v and v is pure in /x, £/ien 7 is p^/re m /x. (2) If v is divisible, then v is pure in /x. (3) Suppose that 11 is divisible. Then v is pure in \i if and only if v is divisible. Proof. (1) Suppose that nya = xa, where ya C /x, xa C 7, and n G N. Since %a Q v, 3za C v such that nza = xa. Hence 3 wa C 7 such that nw 0 = x a . (2) i/a is divisible Va G {a|0 < a < ^(0)}. Thus va is pure in /xa Va G {a|0 < a < /x(0)}. Hence 1/ is pure in /x. (3) /xa is divisible Va G {a|0 < a < /x(0)}. Thus va is pure in /xa Va G {a|0 < a < M(0)} ^ ^a is divisible Va G {a|0 < a < /x(0)}. That is, z/ is pure in /x <=> v is divisible. ■ Proposition 2.4.19 Let /x G L(G). Let v G L(/x) be such that v(0) = /x(0). 7/ Va:a C /x 5^c/i £/ia£ x a 2^j /3 n G N suc/i £/ia£ n(xa) C 1/, £/ien */ is pnre m /x. Proof. If 3a G {a|0 < a < /x(0)}, 3 x G /xa such that x £ va and nx = z G va for some z G va and for some n G N, then we have a contradiction of the hypothesis. Thus /x a /i/ a is torsion-free and so va is pure in /xa V a G {a|0 < a < /^(O)}? [49, P- 114]. Thus v is pure in /x. ■ Proposition 2.4.20 Lei /x G £(G). Suppose that /x* zs torsion-free. Let {va \ a G fi} 6e a collection of L-subgroups of G such that va C /x, i/a zs pure m /x, and va(G) = /x(0) Va G fi. 77ien n a € fti/ a w pure in /x.

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Proof. Let x G /x*. Then (naennVa)(x) = A{nva(x)\a G ft} = 0 if x £ nji* and (n a ennz/ a )(:r) = A{nua(x)\a G ft} = A{i/ a (t/)|a G ft} if a: = ny G n/x*. Now (n fl a € n ^a)(^) = 0 if x <£ n/x* and (n H a € ^ 1^)0*0 = (n a € ni/ a )(s/) = A{ua(y)\a G ft} if z = ny G n/x*. Thus n n a € n i / a = n a € f i nz/ a . Now ( n a 6 f i ( V a n n/x)) (re) = A{(z/a D nji){x) \ a G ft} = A{z/a(a;) A (n/x)(ar) | a G ft} = A{^ a (x) | a: G 0 } A (n/x)(x) = (n a G n^ a )(x) A (n/x)(:r) = ((n a € ni/ a ) fl (n/x))(x). Hence (1 a€ fi(i/ a fl n/x) = (n a € Q^ a ) fl (n/x). Va e (l,Vn G N,nz/ a = i/Q D n/x. Thus Vn G N, naenni>a = n a € n ( i / a H n/x) and so n n a € Q i/a = (fl a€ Q^ Q ) fl n/x. ■ Proposition 2.4.21 Lei /x G L(G). Suppose that /x* is torsion-free. Let v G L(/x) be such that i/(0) = /i(0). T/ien z/ 25 contained in a unique smallest pure L-subgroup in /x. Proof. Let B = {y\ v C 7 C /i,7 is a pure L-subgroup of G in /x}. Then /x G S and so B ^ 0. The desired result now follows since n 7 € s 7 is an L-pure subgroup of G in /x by Proposition 2.4.20. ■ Lemma 2.4.22 Lei {^a|a G ft} and {7<*|a: G ft} 6e chains of L-subgroups of G. Let n G N. Tften ^

n(U a € Qi/ a ) = U a € n n i / a ;

(%> U Q 6 n K n 7 a ) = (U aG Qi/ a ) fl

(Uaenja).

Proof. (1) Let x G G. Then n(U a6 r2^a)(#) = 0 if x £ nG and n(Uaen^a)(x) = y{Uaen^a(y) I x — ny) = W{\/{ua(y) \ a G ft} \ x = ny} if x G nG. Now (Uaenni>a)(x) = V{(nva)(x) \ a G ft} = 0 if x <£ nG and (U aG ^nz/ a )(a:) = V{(ni/ a )(x) I a G ft} = V{V{i/a(y) | x = ny} | a G ft} if x G nG. Thus we have the desired result. (2) Let x G G. Then (U a € n(i/ a n 7*))(&) = V{(i/a n 7 a )(x) | a G ft} = V{z/a(x) A 7 a (x) I a G ft} = (V{i/a(a;) | a G ft}) A (V{7 a (x) | a G fi}) = (U aG ni/ a )(a;) A (U aG fi7 a )(x) = (( U a G ni/ a )n (U a € n7 a ))(x). ■ Proposition 2.4.23 Lei /x G L{G). Let {/xa|aj G ft} 6e a c/iom of L-subgroups of G such that fj,a C zx,/xa is p^xre in /x, and /xa(0) = /x(0) Va G ft. Suppose that either /x and U aG ^/x a /lave i/ie s?xp property or /x* is torsion-free. Then UaG^Ma is fl p^re L-subgroup of G in /x. Proof. Let n G N. Then n U aG n zxa = U a 6 n n / i a = U a € n(/x a fl n/x) = (UaeftMa) fl n/x by Lemma 2.4.22 and Proposition 2.4.17. ■ Proposition 2.4.24 Le£ /x G L(G). Le£ z/, 7,6 G L(/x) 6e s?xc/i #ia£ K i / and 5(0) = /x(0). Suppose that /x = v 0 7. T/ien ^ Vn G N, n/x = nv ® 727; (^) 5 + 7 = fi 0 7 and 1/ fl (5 0 7) = 5.

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Proof. (1) Let x G /z*. Then (n/x)(x) = V{^(2/)|y GG,a; = ny} = V{(i/ 0 7)(x) | x = ny} = V{i/(s/i) A 7(2/2) I z = ny,s/ = 2/1 + 2/2 for 2/1 G ^*,2/2 G 7*} since ^x* = 1/* 0 7 * by Theorem 1.5.9. Now (ni/^nj)(x) = {nv(x\) A (w7)(#2) such that xi G i/", #2 G 7*. Thus (nv 0 727)(x) = (y{v(yi)\x\ = nyi}) A (V{7(2/2)|^2 = ^2/2})- Hence n/x C 77,1/ 0 717. Clearly, nz/ © 717 C n/x. (2) For a U x e G , 0 < (£07) (a;) < (^ri7)(x) = 0. Thus £ + 7 = £ © 7 . Now \fx G G, (i/n(507))(ar) = i/(x)A(«97)(a:) = 1/(2;) A(5(xi) A 7(2:2) if a; = X i + x 2 for xi G (5*, X2 G 7* and (1/ fl (5 © 7))(#) = 0 otherwise. Consider the former case. Suppose that x £ v*. Then (y f) (S © j))(x) = 0 = 6(x). Suppose that x e ^ * . Then x2 = 0 and so (1/ D (5 0 7))(x) = v\x) A 5(a) A 7(0) = v\x) A 5(ar)

= 6{x). m Proposition 2.4.25 Let JJL G L(G). Let v and 7 G L(/x). If JJL = v © 7, £/ien 1/ ana7 7 are pure in \i. Proof. Suppose that nya = :r a , where n G N, y a C fi, and x a C IA Since /x = z/ © 7, ^ a = i/a © 7 a . Thus z/a D n/za = ni/ a . Now ny a = (ny)a. Hence ny = x,y e fj,a, and x G ^G. Thus 3z G ^ a such that nz = x. Hence nza = xaThus v is pure in fi. ■ Example 2.4.26 Le£ L = [0,1]. Let G = Z(p°°) © Z(p°°). £>e/me «fte fuzzy subsets \x,v, and 7 of G by Vx G G, /i((0,0)) = 1, /i(x) = ^ + ^ z/ x G ( ( ^ ) ® { 0 } ) \ ( ( ^ ) ® { 0 } ) / o r n = 2 , 3 , . . . , ^ ) = 3i/a;^Z(POO)©{0}; i/((0,0)) = l , K x ) = 5 * / * G {0} e Z ( p ° ° ) \ { ( 0 , 0 ) } , i/(x) = 0 otherwise; E ( ( ! , ! ) , . . . , ( £ , £ ) , . . . } \ { ( 0 , 0 ) } , 7(*) = 0 7 ((0,0)) = l ) 7 ( x ) = i ifx otherwise. Then \i, v, and 7 are /ILZZJ/ subgroups of G such that v, 7 C /i and p* = ( ? , / = {0} © Z(p~), 7* = ( ( £ , £ ) , - • • , ( £ , £ ) , • • •) • Since [y + 7)* = ^* + 7* = ^* © 7*, v + 7 = v © 7 fry Theorem 1.5.9. 5mce i/* ana7 7* are divisible and 1/, 7 are constants on r/*\{(0,0)}, 7*\{(0,0)}, respectively, v and 7 are divisible by Proposition 2.4.4. iVow (&> 0 7)* = G which is divisible. Let (u,v)eG\{(0,0)}. Then(v®j)(u,v) = i/((0,v-u)) Ay(u,u)) = \. That is, (^©7) w constant on (^©7)*\{(0,0)}. Hence z/©7 25 divisible. jB a nontrivial fuzzy subgroup 6 of G such that ji = {y © 7) © 6 else {y © 7)* © (5* = ^* = (^©7)* ana7 50 <5* = {(0,0)}. Clearly, JJ, z£ v + j . That is, we have shown that a divisible fuzzy subgroup need not be a direct summand (of /i) even though the corresponding result holds for crisp Abelian groups.

2.5

Invariants of L-Subgroups

One of the major goals of group theory is the description of a class of groups by means of invariants. Let G be an Abelian group. If G is finitely generated,

57

then G is a direct sum of cyclic groups of order infinity and powers of primes and thus has a complete system of invariants, [49, p. 81]. The purpose of this chapter is to determine a complete system of invariants for certain L-subgroups fj, of G. In particular, we give a complete set of invariants for those ji which are direct sums of L-subgroups whose supports are cyclic. Our invariants satisfy the criteria that they are easily described quantities and are uniquely determined by the L-subgroup. We recall some notation. For x G G, o(x) denotes the order of x. Let /i denote an L-subgroup of G. Let S([JL) — {xa | x G /i*,n{x) = a).

Let F(li) = {x I X is an L-subset of G and x Q A*}The following definition corresponds to Definition 2.1.2. Definition 2.5.1 Let \i G L(G). Let x € J^AO- Then x (or S(x)) is said to be a minimal generating set for fj, if fi = (x) U 0^(0) and V x a G «S(x)> (x - ^a)UOM(o) C (x), where (x-xa)(y) = (x)(y) ify^x and (x-xa)(y) = 0 Proposition 2.5.2 Let /J, G L(G). Suppose that /z* = (x) = (y). For all u G fji(G)y let nu, mu be the smallest positive integers such that fiu = (nux) = (r^uy) • Then nu = mu V u G fJ>(G). Proof. There exist qi, q2 € Z such that x = qiy and 7/ = #2^- Suppose that /z* has infinite order. Now x = qiq2x and so ^ = q2 — ± 1 . Thus n n = mu V ix G fi(G). Suppose that //* has finite order q. Then g, q\ are relatively prime else o{x) < o(y). Thus (n u y) = (nuqiy) = (nux). Similarly, (m n x) = (m u y) . Hence nu = mu V u G /i(G). I Let /i G L(G) and x E fi*. Let /xx be the L-subset of G such that A** 0*0 = fji(z) V z G (x) and ^ ( 2 ) = 0 V 2 G G \ (a:). Then fix is an L-subgroup of G. Let Lc = {nu\ u G ^ X (G)\{0}, n u the smallest positive integer such that (nx)u = (n u x)}. Suppose that /x* = ©ae.4 (^a) • Let J^ denote the ordered pair, (Uaei{(^x Q , o ( i a ) ) } , /i(G)\{0}). In the remainder of this section, G' denotes an Abelian group. Lemma 2.5.3 Let \i G L(G) and v G L{G'). Suppose that /z* = (x) and v* = (y) . Then 1^ = Iu if and only if 3 isomorphism f of fj,* onto v* such that /(//) = v on is*. If' Ifj, = ii/, then every isomorphism f of fi* onto v* is such that f(ji) = v on v*.

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Proof. Suppose that JM = Iv. Then clearly /J,* ~ v* for some isomorphism / and we may take / so that f(x) = y. Then V u G /i(G), f{nux) = nuy. Since M ( G ) \ { 0 } = i/(G')\{0}, / ( ^ ) = i/u V u G MG)\{0}. Thus since / is one-to-one, V z G /x*, f(fj)(f(z)) = V{zx(w) | /(w) = /(*)} = /x(*) = v(f(z)\ where the last equahty holds since /(/x n ) = vu V u G /x(G)\{0}. Hence /(/x) = v on i / \ Conversely, suppose 3 isomorphism / of /z* onto i/* such that /(/i) = i/. Then V z 6 / i * , K/0*)) = f(fi)(f(z)) = /z(z) as above. Hence /x(G)\{0} = Im(i/)\{0}. Clearly, o(x) = o(y). Now 4 = J y since vu = f(fi)u = f(fjbu) V u G /i(G)\{0}. Thus 1^ = IU. For any isomorphism / of /x* onto */*, where J^ = Iu, the proof that /(/x) = v follows in a similar manner as above since Iv is independent of the generator of v* by Proposition 2.5.2. ■ Let fi G L(G) and v G L(G'). Suppose that /i* = (BaeA (xa) and v* — QcxeA (Va) • By Theorem 2.1.13, ^

= ({(nUaxa)Ua\ua

G fJ.x*(G)\{0}})

and "ya = < { K ^ * ) . > a € ^(G)\{0}}> V a G A We let /xa = xxx« V a G A. The following two results show that 1^ is a complete system of invariants for n when /x is a direct sum of L-subgroups whose supports are cyclic and where the xa below have orders oo or powers of primes. There is no loss in generality in using the same index set A for /x and v. Theorem 2.5.4 Let /x G L(G) and v G L{G'). Suppose that fj, = (BazAf1** an^ v = 0 a e ^ ^ a , where /x* = © a G ^ (xa) , i/* = © a e.4 (2/a) • If ly. — Iv, then 3 isomorphism f of /x* onto v* such that /(/x) = v. Proof. By Lemma 2.5.3, 3 isomorphism fa of (xa) onto (ya) (assuming the xa and ya are suitably arranged) such that fa(xa) = ya and fa(l^a) = a v V a G A. Let / = 0 a € > 4 / a . Then / is an isomorphism of /x* onto v*. Now since / is one-to-one. Also / z ( E a m a x a ) = ( © a e . A ^ X E a m « ^ a ) = A{/x a (ra a x a ) | a G .A} (since /x* is a direct sum) = A{wa \ wa G /i a (G)\{0}, a G A}, where u/ a = /x(0) if m a = 0 and wa is such that maxa G ( n ^ X a ) \ (n U a x a ) V ua G fJ>a(G) such that u Q > w a , n ^ a , n U a G J X a , otherwise. Now nWa, nUa G J ^ , m a y a G (nWaya)\(nUaya) V ixa G i/*(G) such that ixa > w a and so KEm<*2/c*) = ^{wa | w a G Im(i/*)\{0}, a G .4} in a similar manner. Thus M E a m < * z a ) = v(52am*ya)' Hence / ( M ) = i/. ■ Theorem 2.5.5 Let /x G L(G) and v G L(G'). Suppose that f is an iso­ morphism of ii* onto v* such that /(/x) = v. If fi* = (BaeA (xa), then v* = ®*eA (f(xa)), h = I», <™d f(EaeA»a) = E«eA»a-

59

Proof. Clearly, v* = ®aeA (f(xa)), f(jia) = i / \ and o(x 0 ) = o(/(x Q )) V a e A. By Lemma 2.5.3, IXa = If{Xa) V a € A Now »/(E« "*<*/(*«)) = / ( / * ) ( / E a "»«*«*) = M E a " > « x a ) . Thus /i(G)\{0} = v(G)\{0}. Hence JM = J„. NOW ( / ( E a M Q ) ) ( E a ^ a / ( ^ ) ) = ( / ( E Q ^ ) ) ( / ( E a m Q X Q ) ) = ( £ Q M a ) ( E a m a a ; a ) = A { ^ a ( m a x a ) | a e .A} (since fi* is a direct sum) = A{f(fj,a)(f(ma x tt )) | a G .4} = A{*/*(m a /(x Q )) | a e i } = ( E ^ K E ^ / f e ) ) . Thus /(EaM0,)=Ea"a-" Definition 2.5.6 Let // G L(G) and 7 G L(/x) and x a C ji. Then the L-subset xa + y is called the left L-coset 0 / 7 in fi with representative xa. Using the definition of the sum of two L-subsets, it follows easily that V z G G, (xa + 7)(2) = a A 7(2: - x). It also follows that V a G {a|0 < a < /x(0)}, ( / i / 7 ) ^ — {xa + 7 I x a C /i, x G G} is a commutative group under + , where (xa + 7) + (2/a + 7) = (x + 2/)a + 7Theorem 2.5.7 Le£ /x G L(G) and 7 G £(//). Define /i/j = { xa + 7 | x a C /x, x G G, a G L}. TTien (/i/7, +) is a commutative semi-group with identity. Ifj(0) — A*(0), Wien /i/7 is completely regular [210] and /x/7 = UaeM ( / V T ) ^ > where M = {a | 0 < a < /z(0)}. ■ Theorem 2.5.8 Le£ /i G L(G) and 7 G L(/x). T/ien (/i/7) ( a ) ~ /x a /7a V o G {a I 0 < a < /x(0)}. ■ Let / be a homomorphism of G into G'. Then V x a C 1 G , f(xa)(f(x)) = y{xa{y) I /(*/) = / ( * ) } = V{x a (x)} = a = /(x) f l (/(x)). If f(z) ^ / ( * ) , then /(*«)(/(*)) - V{xa(2/) I /(

anrf

* =

QaeAV"-

Suppose further that J 7 = Is- Then 3 isomorphism f of 7* onto 8* such that f(xa) = ya V a G A and f(*y) = 8. Also the following conditions are equivalent. (1) /(M) = v\ (2) V a G {a I 0 < a < ^(0)}, xa C /j, if and only if f(x)a Q v\ (3) V a G {a I 0 < a < /i(0)}, fa defined by fa{xa + 7) = / ( x ) a + 8 is an isomorphism of (/i/j)^ onto (u/8)^a\

60

Proof. That / exists follows from Theorem 2.5.4. (1) <^> (2): If (1) holds, then (2) holds immediately. Suppose that (2) holds. Since x^x) C /x, /(x) M(a .) C v by hypothesis and since / ( ^ ( / ( z ) ) Q v, a^(/(x)) C \x by hypothesis. Thus fi(x) = a implies v{f{x)) > a and v(f(x)) = 6 implies n(x) > b. Hence ji(x) = v(f(x)). Thus z/ = /(/i) since / is one-to-one. (2) <& (3): Suppose that (2) holds. Now xa + 7 = y a + 7 <=> ( - y + x) a C 7 [20] ^ / ( - y + x) fl C / ( 7 ) ^ ( - / ( y ) 4- f(x))a C / ( 7 ) ^ / ( x ) a + / ( 7 ) - /(2/)a + / ( 7 ) & /(*)« + 6 = f(y)a + « ^ / a K + 7) = /«(l/a + 7)Thus fa is single-valued and one-to-one. Now xa + 7 G (fi/j)^ ^ xa C fi ^

f(x)a

(a)

C

^

/ ( X ) a + ^ G (l//«)<°> ^

/ a K + 7 ) <E ( l / / « ) ( o ) . T h u s fa

maPS

(/x/7) onto (i//fi)W since / maps fx* onto u* and so /(x) Q + 6 is arbitrary in ( i / / $ ) K Now fa((xa + 7) +(y 0 + 7)) = /«((* + y)a + 7)) = / ( * + »)o + ^ -

{fix)

+ f(y))a

+ S = (f(x)a

+ 6) + (f(y)a

+ 6)) = / „ ( ! . + 7 ) + / o ( » a +

7). Hence fa is an isomorphism. Conversely, suppose (3) holds. Then fa has domain ( / X / T ) ^ and image (u/6)^. Hence (2) holds since xa C /x <£> x a + 7 e

(/i/ 7 ) (a) ^ /(x) a + S e (v/8)^

<* f(x)a Cu.M

We now illustrate Theorem 2.5.9 and other results.

Example 2.5.10 Let L = [0,1]. Let G — (x) © (y), where o(x) = p and o(y) = p 3 , p a prime. Let H = (x + py) . Define the fuzzy subset fi of G by fj,(z) = 1 if z G H and fi(z) = \ otherwise. Then p, is a fuzzy subgroup of G. The fuzzy subgroups fix and \iv are such that /i x (0) = 1, fJ.x{z) = \ if z G (x) \ (0), fix(z) = 0 otherwise, /j,y(z) = 1 if z G (p2y) , fiy(z) = \ if z G (y) \ (p2y) , and (J,y(z) = 0 otherwise. Now fix + jiy = JJLX © fiy by Theorem 1.5.9. Let v — y = S = \ix © jiy. Then /x* = v* = 7* = 6* = G. Clearly, J 7 = I§ and so 3 an isomorphism f of 7* onto 6* such that f(j) = 6 by Theorem 2.5.9. However, (x + py)\ C fi, but f{x + py)i <£. u else f(H) C (p2y) and so f(H) has order < p. Hence by Theorem 2.5.9, f(fji) ^ v. In fact, jB any isomorphism f of fi* onto is* such that f({i) = v for the same reason. We note that 1^ = Iu. However, by Example 2.1.18, ji is not a direct sum of fuzzy subgroups of G whose supports are cyclic. Thus Theorem 2.5.4 is not contradicted.

2.6

Basic and p-Basic //-Subgroups

The decomposition of G as a direct sum of cyclic groups yields structure results for G. Although not every G is a direct sum of cyclic groups, 3 largest subgroups of G which are direct sums of cyclic groups of order infinity and powers of primes. When these largest subgroups are pure, they prove to be of importance in determining properties of G, [49]. These largest pure subgroups, called basic subgroups, are invariants of p-groups [49, Theorem 35.2, p. 148]. This yields a system of cardinal numbers as a system of invariants of G, [49, p. 148]. The theory of p-groups is based to a large extent on basic subgroups. Also the

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concept of basic subgroups (and other concepts from Abelian group theory) is of importance when carried over to the study of inseparable field extensions, Chapter 9. Let x G /x*. Let \ix be the fuzzy subset of G such that fix = /i on (x) and \ix — OG on G\ (x), where we recall that OG(X) = 0 for all x G G. Then fix is a fuzzy subgroup of G. Let Ix = {nu\u G lm(/i x )\{0}, nu the smallest positive integer such that (/JLX)U = (nux)}. Suppose that /J,* = (BaeA (xa) • Let 1^ denote the ordered pair, (U a e ^{/a; a , o(x a )}, lm(^)\{0}). In the remainder of this section G' denotes an Abelian group and v an L-subgroup of G\ If / is an isomorphism of G onto G', then /(/i) is the L-subset of G' defined by V y G G', f(fi)(y) = fi(x), where y = f{x). This definition coincides with the usual definition of a homomorphism applied to fi since / is an isomorphism here. It is known that /(//) is an L-subgroup of G'. Suppose that fj,* = (Ba(EA (xa) and v* = 0 a G ^ (ya). By Theorem 2.1.13, \?~ = ({(nUaxa)Ua\ua

G

^

e

fiXa(G)\{0}))

and = ({(mvaVa)vJVa

^(G0\{0}))

V a G A. We call an //-subgroup 7 of G cyclic if and only if 3 x G G such that 7 = (xa) , where a = 7(x). Definition 2.6.1 Lei /J, G £(G) and z; G Z/(AO- ITien 1/ zs said to be basic in V if (1) v is a direct sum of cyclic L-subgroups of G whose supports are cyclic of prime power orders or of infinite order; (2) v is pure in JJ, (S)M a G {a|0 < a < ^(0)}, V x a + i/G \±jv, V n G N, 3 yb + v G pjv such that n(yi + v) = xa + v. Proposition 2.6.2 Let {i G L(G) and v G L(fi). Then (3) of Definition 2.6.1 implies ^/v* is divisible. Proof. Let x + v* G /i*/^* and let n G N. Then x a + z/ G /x/z/ for a G {a|0 < a < M(0)}- Thus 3 y^ + z/ G ^/z/ such that # a + z/ = n ^ -I- v. Hence a = 6 < z/(—x -h ra/) by [146, Proposition 1.7 (hi), p. 3]. Thus v[—x + ra/) > 0 and so — x + ny G z/*. Hence x + z/* = -m/ + v* and so /x*/^* is divisible. ■ It follows that (3) of Definition 2.6.1 is equivalent to the exact same state­ ment with b replaced by a. Proposition 2.6.3 Let /i G L(G) and v G L(fi). Then (3) of Definition 2.6.1 holds if and only if V a G {a|0 < a < zy(0)}, Ha/va is divisible.

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Proof. Suppose that (3) holds. Let a G {a|0 < a < ^(0)}. Let x + va G Va/va and n G N. Then ^(x) > a and so x a C p. Thus 3 ya + v such that n(?/a + i/) = x a + v and so (ra/) 0 + v = xa + v. Hence (—x+ ny)a C v. Thus z/(—x + ny) > a. Hence — x + ny G i/0. Thus ny + ^ a = x + z/a. Hence pa/va is divisible. Conversely, suppose that pa/va is divisible V a G {a|0 < a < ^(0)}. Let a G {a|0 < a < ^(0)}. Let xa + v G ^/z/ and n G N. Then xa C p and so /x(x) > a. Thus x e pa- Hence x + i/a G ^a/^a- Thus 3 ?/ + z/a G /x a /^a such that n(y + va) = x + va. Hence ny -\- va = x + va. Thus — x + ny G i/a. Hence */(—a: + n?/) > a. Thus (—x + ny)a C ZA Hence ny a + v = xa + v. Thus (3) holds. ■ Theorem 2.6.4 Let p G L{G) and v G £(A0- If v is basic in p, then \/ a € {a|0 < a < ^(0)}, va is basic in pa. Proof. The desired result holds from the Propositions 2.4.13 and 2.6.3 and Theorem 1.5.10. ■ Theorem 2.6.5 Let p G L(G). Suppose that p has the sup property. Let v G L(p). If v is basic in p, then v* is basic in p*. Proof. By Proposition 2.6.2, p*/v* is divisible. Now v — ©ae.A7a> where 7 a G L{p) and 7 a is cyclic. Thus v* = 0 a e.A7a* by Theorem 1.5.9 and j a * is cyclic V a 6 l Since v is pure in p, we have by Proposition 2.4.15 and Proposition 2.4.16 that V n G N, nv = v fl np implies ni/* = (ny)* = (yHnp)* = v* fln/z*, i.e., v* is pure in p*. M Let p denote a fixed prime. Definition 2.6.6 Let p G L(G) and v G L(p). Then v is said to be p-pure in pifV L-singletons xa C v with a > 0, V k G N, V ya C /x, pk(ya) = xa implies that 3 za C v such thatpk(za) — xa. Definition 2.6.7 Let p G L{G) and v G L(p). Then v is said to be p-basic in p if (1) v is a direct sum of cyclic L-subgroups whose supports are cyclic of order a power of p or of infinite order; (2) v is p-pure in p; (3)V a G {a|0 < a < p(0)}, V xa + v, V k G N, 3 yb + v such that pk(xa + v) = yb + vTheorem 2.6.8 Let p G L{G) and v G L(p). (1) Then (3) of Definition 2.6.7 implies p*/v* is p-divisible; (2) v is p-pure in p if and only ifva is p-pure in paV a G {a | 0 < a < /x(0)};

63

(3) (3) of Definition 2.6.7 holds if and only if' \xajva is p-divisible V a G {a I 0 < a < ,x(0)}; (4) If v is p-basic in p, then va is p-basic in pa V a G {a | 0 < a < p(0)}. (5) Suppose that p has the sup property. If v is p-basic in /z, then v* is p-basic in //*. Proof. The proofs are similar to those of basic 1,-subgroups. ■ We let Z denote the additive group of integers and Z(p n ) a cyclic group of order pn, n G N. We recall that for H a p-basic subgroup of G, H = ®™=0Kn, where K0 = 0 m o Z and Kn = 0 m n Z ( p n ) , n = 1,2,..., and {m 0 , m i , . . . , m n , . . . } is a system of invariants for G, [49, p. 148]. Theorem 2.6.9 Let p G L(G) and v G L(p) be such that v is p-basic in p. (1) UaeA{m0a , m i a , . . . , m n a , . . . } is a system of invariants for p, where {mQa,mia,... , m n a , . . . } is the system of invariants determined by va in pa andA = {a | 0 < a < i/(0)}. (2) Suppose that p has the sup property. Then { m o , m i , . . . , m n , . . . } is a system of invariants for p, where {mo,mi,... , m n , . . . } is the system deter­ mined by v* in p*. Proof. The proof is immediate from Theorem 2.6.8 (4) and (5) and the preceding comments. ■ Definition 2.6.10 Let p G L{G) and x € Fd1)- Then x is so>id t° be pindependent in p i/V xi, ...,xk G x* and V ai G {a|0 < a < x(xi)} for r i = 1,..., k and V r G N, ni(xi)ai + . . . + nk{xk)ak Q P P (niXi ^ 0, i — 1 , . . . , k) implies pr | r^ for i — 1 , . . . , k. x is called a p-basis of p if x is p-independent in p and p = pp + (x) . Proposition 2.6.11 Let p G L(G) and x £ F(l*)- Then x ^ in p if and only if x* is p-independent in p*.

p-independent

Proof. Suppose that x 1S p-independent in p. Let #i, . . . , Xk G x*- Suppose that n\X\ + . . . + nkXk G prp* (niXi ^ 0,2 = 1 , . . . , k). Then n\X\ + . . . + nkXk = prx for some x G /i*. Let a = ai A . . . A a k A p ( x ) , where a; = x(xi) f° r z = 1 , . . . , k. Then ni(xi)a + . . . + nk(xk)a = prxa C pr/x. Thus p r | n; for i = 1 , . . . , k. Conversely, suppose that x* is p-independent in /i*. Let x\, . . . , xk G x*- Suppose that n i ( x i ) a i + . . . + nk(xk)ak Q PrP (niXi ^ 0,i = 1 , . . . , k). Then {nxxi + . . . + nkxk)a C pr/x, where a = a x A.. .Aak > 0. Thus (prp)(n1x1 + . . . + nkxk) > a > 0. Hence 3 X E G such that 7ii#i -f- . . . + n^x*; = p r x. Thus (prp)(prx) > 0 and so ^(x) > 0. Hence x € p* and so p r x G p r ^*- Thus pr | n; for i = 1 , . . . , k. ■

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Definition 2.6.12 Let p G L(G) and x G ^"(AO- Then x ^ send to be maxi­ mally p-independent in p if x is p-independent in p and fl rj G ^{p) such that 77 is p-independent in p and x C n. Proposition 2.6.13 Let p G L{G) and \ £ T{p). Then x is maximally pindependent in p if and only if x* is a p-basis for p* and V x Gx*, x(x) — p(x). Proof. Suppose that x is maximally p-independent in p. Then x* is pindependent in p* by Proposition 2.6.11. Suppose that x* is not a p-basis of p*. Then 3 y G M*\x* s u c n that X* U{y} is p-independent in p*. Thus by Proposition 2.6.11, rj G ^F(p), where 77 = p on x* U{y} and 77(2) = OV 2 G G\(x* U {y}) is p-independent in p. However, this contradicts the maximality of x- Conversely, suppose that x* is a p-basis of p* and V x G x * , x(x) — M x )Suppose that x is not maximal. Then 3 77 G ^(AO such that x C 77 and 77 is p-independent in ^. Since x = A* o n X*> X* C 77*. Since 77* is necessarily pindependent in p, this contradicts the maximality of x*- Thus x is maximal. ■ It follows easily that if x is a p-basis of p, then x is maximally p-independent in p. However, the converse is not true as the following example demonstrates. Example 2.6.14 Let L = [0,1]. Let G = (x) , where x has order p 2 , p a prime. Define the fuzzy subset p of G by p(z) = 1 if z G (px) and p(z) = | if z G G\ {px}. Then p is a fuzzy subgroup of G and p* = G. Now (pp)(px) = p{x) = \. Thus (p/x)(0) = 1, («*)(*) = £ */* G (px) \ (0), and (pp)(z) = 0 if z G G\ (px). If x is a p-basis of p, ^/ien x* = I 2 } / o r 5ome z G G\ (px). iVow /z(,z) = ^. TTITJS p D pp + (x) since M(P X ) = 1> ^ (PM + (x))(P x ) = V{(p/i)(w) A (x) (TJ) I px = 7j + ?j} = V{(pp)(pix) A (x) (p(l - *)x) | 0 < z < p} < ^- XTms ^x does no^ have a p-basis even though p2p* = (0). Now x is maximal p-independent in p since x* is a p-basis for p*. Lemma 2.6.15 Let p. G L{G). V a G (0,/z(0)], p r (^a) C (p r ^) a . / / eitfier /x /ias the sup property or p* is torsion-free, thenpr(pa) = (prp)a. ■ Proposition 2.6.16 Let p G L(G) and x G ^"(/i). ^ / / x w p-independent in p, then Xa is p-independent in pa V a G {a | 0 < a < /x(0)}. (%) Suppose that p has the sup property or p* is torsion-free. If Xa is p-independent in pa V a G {a | 0 < a < ^(0)}, £/ien x ^5 p-independent in p. Proof. (1) Let xi, . . . ,x^ G Xa- Suppose that nixi + . . . + n^Xfc G pr(pa) {riiXi ^ 0,i = 1 , . . . ,fc). Then xi, . . . , x*; G x* and nixi + • • • + nkxk £ prA**By Proposition 2.6.11, p r | n; for i = 1,..., /c.

65

(2) Let xi, . . . , xk G x*- Suppose that n i ( x i ) a i + . . . + nk(xk)ak C p> (riiXi ^ 0, i — 1 , . . . , k). Then (nixi + . . . + nkxk)a Q Pr^, where a = a\ A ■ ■•Aaife. By Lemma 2.6.15, (;/>)« = p r (/x a )- Thus n ^ i + . . . + nkxk G / ( / i f l ) . Since also # i , . . . ,#£ € Xa,P r |^i for z = 1 , . . . , &. ■ P r o p o s i t i o n 2.6.17 Let \i G £(G). Suppose that ji has the sup property or /i* 25 torsion-free. Let x € ^(/-O &e 5 ^ c ^ ^ ^ x0*0 = /^(^) V x G x* • If Xa is a p-basis of /xa V a G {a | 0 < a < /x(0)}, £/ien x 25 maximally p-independent in /x. Proof. By Proposition 2.6.16(2), x is p-independent in (i. Suppose that x is not maximally p-independent in \i. Then 3 r\ G ^(AO such that x C 77 and 77 is p-independent in /x. By Proposition 2.6.16(1), T)a is p-independent in /xa V a G {a G L\{0}|a < ^(0)}. Since x* C 77*, 3 a such that Xa C 7?a- However, this contradicts the maximality of Xa- B The following example shows that the converse of Proposition 2.6.17 does not hold. E x a m p l e 2.6.18 Let L = [0,1]. Let G = Z(p°°). Let x e G have order p. Define the fuzzy subset n of G by /x(z) = 1 if z G (x) and fi(z) — \ other­ wise. Then \i is a fuzzy subgroup of G and /x* = G. Now 0G is maximally p-independent in \x since 0 is a p-basis o//x*. However, fii = (x) and (0G)I = 0 is not a p-basis of (x). Note also that 0G is not a p-basis of /x. Let fi G L(G). If x is maximally p-independent in /^, we determine in the following to what extent (x) is a p-basic L-subgroup of /x. P r o p o s i t i o n 2.6.19 Let /x G L(G) and x € ^(AO- If X is p-independent in /x, then x is independent in \i. Proof. x is p-independent in /x <=> x* is p-independent in /x* by Proposition 2.6.11. If x* is p-independent in /x*, then x* is independent in /x*. Now x* is independent in /x* <3> x is independent in /x. ■ T h e o r e m 2.6.20 Let \i G L{G) and x £ ^(AO- Then x is independent in /x if and only if (x) = ©x a es x ( z a ) , where Sx = {xa \ x G x*, x0*0 = «}• Proof.

See Theorem 2.2.5. ■

T h e o r e m 2.6.21 Let /x G L(G) and x € Fin). If X is p-independent in fi, then (x) = ®xaesx (xa) , ™^re <SX = {xa\ x G x*, x(s) = «}•

66

Proof. 2.6.20. ■

The desired result follows by Proposition 2.6.19 and Theorem

Lemma 2.6.22 Let /x G L(G) and x € (Xa) = (X)a V a 6 {a|0 < a < ^(0)}.

«^"(A0-

If X is independent in xx, then

Proof. If a > V{x(#) | # G x*}> then the result is immediate. Since Xa Q (X)a» (Xa) C ( x ) a . Let x G (x) a • Then (x) (x) > a. Now (x) - ®Xsesx {xS) by Theorem 2.6.20. Thus x has an unique representation, x = niXi + . . . + nkXk, where X{ G x*- Thus a < a\ A . . . A a^, where xO*^) = fli for z = 1 , . . . , fc. Hence x G (xa) • Thus ( X ) a C (Xa). ■ Theorem 2.6.23 Le£ /x G £(G) and x £ then (x) ^s p-pure in xx.

-^(AO-

7f X ^5 p-independent in

/JL,

Proof. Since x is p-independent in /x, Xa is p-independent in xxa V a G {a|0 < a < xx(0)} by Proposition 2.6.16. Thus (x) a = (Xa) is p-pure in xxa by Lemma 2.6.22 and [49]. Hence (x) is p-pure in xx. ■ Let /x G L(G). We also note that if x £ ^"(AO i s maximal p-independent, then (x*) = (x)* is a p-basic subgroup of /x* since x* is a p-basis of xx* by Proposition 2.6.13. Theorem 2.6.24 Let a G L(G) and x e F{p)(1) Suppose that x is p-independent in xx. If (x) is a p-basic in /x, then x is a p-basis of /i. (2) Suppose that xx is finite-valued. Ifx is a p-basis of xx, then (x) is p-basic in u. Proof. (1) Let xa C xx. Then by (3) of Definition 2.6.1, 3 y a C xx such that #a + (x) =pya + (x) • Thus xx = pxx + (x) • (2) By Theorem 2.6.21 and Lemma 2.6.22, it suffices to show that /x a / (x) a is divisible V a £ {a\0 < a < fi(0)}. Since xx is finite-valued, p(xxa) = (p/x)a by Lemma 2.6.15 and (pxx -h (x))a = (PA*)a + (x) a s i n c e P/^ a n d (x) a r e necessarily finite-valued. Thus from /x = pp, + (x), we deduce that /x0 = p(/xa) + (x a ) by Lemma 2.6.22. Hence by Proposition 2.6.16, Xa is a p-basis of /xa and so Hal (x)a i s divisible V a G {a|0 < a < xx(0)}. ■ Theorem 2.6.25 Let xx G £(G) and i/ G L(p). If v is p-basic in /x, then x is a p-basis of xx, where v = ® a €A ((^Q)oa) andx is the fuzzy subset of G defined by x(z) = f l a if z = xa and x(z) = 0 otherwise.

67

Proof. Now -v* = (BaeA (#<*) • Thus x* is a p-basis of i / \ Since v* is p-pure in /J,*, x* is p-independent in /x*. Since /i*/z/* is divisible by Proposition 2.6.2, X* is a p-basis of /x*. Thus x is maximally p-independent in /x by Proposition 2.6.13. Now v = (x) and so x is a p-basis of /x by Theorem 2.6.24(1). ■ The proof of Theorem 2.6.25 also shows that if v is p-basic in /x, then ji = v on v*. Theorem 2.6.26 Let ji G L(G). If /i* has a unique p-basic subgroup, then /x has a unique p-basic subgroup. Proof. Let v and 7 be p-basic subgroups of /x. Then 1/* and 7* are p-basic subgroups of fi* and so is* = 7*. Now v = (x) and 7 = (77) for p-bases x and 77 of /x by Theorem 2.6.25. By Proposition 2.6.13, x = M o n X* a n d 77 = /x on 77*. Thus v — [IOVLV* and 7 = /x on 7*. But 1/* = 7* and so v = 7. ■ Let /x € L(G). Criteria for /x* to contain exactly one p-basic subgroup can be found in [49, Theorem 35.3, p. 149]. Theorem 2.6.27 Let /x € L{G). (1) Suppose that /x* has a unique p-basic subgroup and that /x is finitevalued. If x and 77 are p-bases of /x, then 1^ = 1^. (2) Suppose that pe\± = 9 for some positive integer e. If'x and 77 are p-bases of fi, then I\x) = I fry Proof. For assertion (1), (x) and (77) are p-basic subgroups of /x by Theorem 2.6.24(2). Thus (x) = (v) • P ° r assertion (2), since pe/x = OG, we have that zx — (x) = (v) • I n either case, let / be the identity map on (x)* • Then /((x)) = (77). Now (x)* = (X*> = © (xa) and so (77)* = (77*) = © (f(xa)) and / <x> = I{TJ) by Theorem 2.5.5. ■ Suppose that G is torsion. Suppose that G has a finite p-basis. Let G' and K be subgroups of G, K' a subgroup of K D G', and H a subgroup of G'. Suppose that K' C K and C ' c G . We note that it is impossible for K' and H to be p-basic subgroups of G' and for K and H to be p-basic subgroups of G : Suppose otherwise. Then we that have G' = pG' + K1', G' = pG' + H, G = pG + K D pG -f K' (since G has a finite p-basis), and G = pG + H. Thus G = pG + G' = pG + pG' + K' = pG + if', a contradiction. Theorem 2.6.28 Suppose that G is torsion. Let fi G L(G). Suppose that /x* /ms a finite p-basis and that p, is finite-valued. Let i / , ) G L(/i). If v and 7 are p-basic in /x, then ^(G)\{0} = 7(G)\{0}.

68

Proof. Let B = v(G)\{0} and C = j(G)\{0}. Since [x = v on i/* and /x = 7 on 7*, we have that £ C /i(G) and C C ^x(G). Since \i is finite-valued, i/ and 7 are finite-valued. Suppose that B ^ C. Then either 3 a € B such that a ^ C or 3 6 G C such that b <£ B, say a exists. Then a < ^(0). Since B and C are finite sets, either (1) 3 largest b e C such that b < a or (2) a < A{b | b e C}. Suppose that (1) holds. Then 3 smallest v G B, v £ C, such that b < v < a. Now 3 smallest u G C such that b < v < u since 7(0) > v. Thus 7^ = 7 n and uv D isu. By Theorem 2.6.8(4), 7V and vv are p-basic in \xv and 7^ and z/n are p-basic in \iu. However, as we have noted above, this is impossible since [J>v D fi>u, where the latter inclusion holds since v, u G //(G). Suppose that (2) holds. Let u = A{b \ b e C}. Then 7 a = j u and z/a D vu and we arrive at a contradiction as just argued. ■ Theorem 2.6.29 Suppose that G is torsion. Let // G 7(G). Suppose that /x* has a finite p-basis and that fi is finite-valued. Let is, 7 G L(fi). If is and 7 are p-basic in /i, then 3 isomorphism f of is* onto 7* such that f(is) = 7 and Iv

Proof. By Theorem 2.6.25, v and 7 are finitely generated. Now is = ®*eA ((xa)aa), where is* = © a € ^ (a;a) and 7 = © / ^ ((yp)bp) , where 7* = ©^ES (yp) by Theorem 2.6.25. By Theorem 2.6.25, \A\ = \B\ < 00. By Theorem 2.6.28, is(G)\{0} = 7(G)\{0} and these sets are finite. Thus is{G)\{0} = { a i , . . . , an} = 7(G)\{0}, where a\ < . . . < an. Since is is finite-valued, z/a = Q>aeA ((x<x)aa)a V a G v(G). Now A = AiU • •. LlAn, where a G A if and only if aa = a{. Thus i/0, = © a G ^ , (x a ) © 0a€A+iU...u-4n (^a) and 0 a G ^ . (x a ) ~ Vajvai+X and ©ae.4i+1u...u.Ari (#<*) ^ ^a i+1 for z = 1 , . . . , n - 1. Thus i/a. ~ ( ^ / ^ o i + i ) © ^ ^ ! a n d similarly 7 a i ~ (jaihai+1) ©7a i+1 . Now there exists an isomorphism gi of i/fl. onto 7 a i for i = 1 , . . . , n by Theorem 2.6.4. Since i/ ai+1 is a direct summand of i/0i and yai+l is a direct summand of yai and these groups are finitely generated, there exists an isomorphism ft of isai/isai+1 onto 7a,/7a i + 1 such that g{ = ft © 0 i + 1 , i = 1 , . . . ,n - 1. Let fn = gn. Then / = / 1 © . . . © / n is an isomorphism of is* = isai = (isaJisa2)<§> . . . ®{yan_1/van)@van onto 7* = 7 f l l = (naiha2) © . . . © (7a n _ 1 /7a n ) © 7an such that /(1/) = / ( 7 ) . Hence Iv = 7 7 by Theorem 2.5.5. ■

2.7

EXERCISES

1. [190] Let G be a group and /1 an 7-subgroup of G. Let 7 be a subset of G and x an 7-subset of G such that x* S I. Then (G, /x) is said to be generated by (/, x) if G = (I) and (fi) = (x). (G, /x) is said to be finitely generated by (J, /x) if (7, /x) generates (G, /x) and 7 is finite. Let X be a free group on the set B with respect to the function / : B —► X. Let (3 be an L-subset of R Let

69 £ be an L-subgroup of X. Then £ is said to be free with respect to ft if /(/3) = £ on f(B) and V groups H and L-subgroups rj oi H with g : B —> H and g(/J) = 77 on p(-B), there exists a unique homomorphism h oi X into i7 such that g = ho f and /i(£) C 77. Prove the following statements: (a) Every L-subgroup is the homomorphic image of a free L-subgroup. (b) Every L-subgroup is isomorphic to a quotient of a free L-subgroup. (c) Suppose that G is commutative. Every finitely generated L-subgroup is isomorphic to a quotient of a finitely generated free L-subgroup. (d) Suppose that G is commutative. An L-subgroup is free if and only if it is a direct product of a family of free cyclic L-subgroups. (e) Suppose that G is commutative. An L-subgroup free over (/, x) 1S a direct product of free cyclic L-subgroups. (f) Suppose that G is commutative. Every finitely generated L-subgroup is a direct product of cyclic L-subgroups.

70

Chapter 3

L-SUBRINGS AND L-IDEALS Abstract algebraic geometry has been the main area of applications of com­ mutative algebra. It has also been the principal incentive of research in com­ mutative algebra. Our approach in the remainder of book is to carry on with this theme for the fuzzy situation.

3.1

Basic Concepts

In this section, we first introduce some operations on L-subsets of a commu­ tative ring R. Definition 3.1.1 Let n, v G LR. Define fj, + v, —fi, fi — v, /J, o v G LR as follows: {n + v)(x)

=

V{ti(y)Av(z)\y,zeR,y

(-fj)(x)

=

/i(-x),

(fj, - u)(x) (fi o v){x)

= V{/x(y) A u(z) \y,zeR,y-z = \/{fi(y) A v{z) \y,zeR,yz

+ z = x}1 = x), = x},

\/ x € R. fi + i/, fJ> — v and \iov are called the sum, difference, of \x and v, respectively, and —u is called the negative of \i.

and product

By definition, it follows that /x + ^ = i/ + //, fi — v = fi-\- (—i/), and since R is commutative, ^ o i / = z/o^V/z, v E LR. Theorem 3.1.2 Let /x, i/, £ e LR. Then /xo(j/ + £ ) C / j o z / + /io£.

71 Proof.

Let w G R and let u, v G R be such that uv = w. Then

U(U)A(IS

+ £)(V)

= = < < <

n(u)h(V{v(y)A£(z)\y,zeR,y + z = v}) V{(u(u) A 1/(2/)) A fr(u) A f (*)) | y, z G iJ, y + * = i;} V{(M(U) A 1/(2/)) A (fi(u) A £(*)) \y,zeR, uy + uz = uv} V{(iiov)(uy) A (uo£)(uz) \ y,z G R,uy + uz - uv} (fiov + fj,o£)(w).

Thus (a o {y + i)){vS)

=

V{fx(u) A(v + £)(v) \u,v G R,uv = w}

<

(/JL0 1/

+iio€)(w)

Vw G R. Hence ^ o ( i / + £ ) C / i o j / +/i o £. ■ Definition 3.1.3 Le£ /x, 1/ € l A £>e/me uv e LR byV x £ R, n

(uv){x) = V{Ag=1(M(l/i) A 1/(2*)) I 2/i, Zi G # , 1 < i < n, n G N, $^2/i*i = a;}. Since R is commutative, /xi/ = V/J, V/i, z/ G Z/**. Theorem 3.1.4 Let //, z/, £ G L^. TTien tVie following assertions hold. (1) /i o v C /iz/. (2) v C £ => fju/C fj£(3) (/xi/)£ = / x K ) . M (/-"'X* + 2/) > 0^)0*0 A (xxi/)(2/) Var, 2/ e -R. ^ 7/i? /ms an identity 1 and 1^^ C z/, ^/ien /x C fiv. (6)

l

R

o fj,C

fji.M

Note that for any w e R, ((fiiy)^)(w) = V{ A"=1(/x(xi) A z/(2/i) A £(£»)) | £;, 2/i, 2f» G R, 1 < z < n, n G N, £ ? = 1 a ^ ^ i = w}Definition 3.1.5 Le£ /x G ZA Define an and xx(n) G Z ^ as follows, where n G N, n > 1 : xx1 = ix xxn = axoan-x\ and

»w

M
= p =

M(V"_1)-

Definition 3.1.6 Let /t 6 L f l . TTien fi is called an L-subring (RI) n(x -y)> n{x) A n(y) Vz, y e R, and (R2) n\xy) > ft(x) A n(y) Vx, y € R.

of R if

72

The set of all L-subrings of R is denoted by L(R). Definition 3.1.7 Let /i G LR and /x satisfy condition (Rl). Then /i is called an L-ideal of R if it also satisfies the condition: (R3) fj,(xy) > /x(x) V fi(y) V x, y G -R. We denote the set of all L-ideals of R by LI(R). If ji G LI(R), then we let /x* = {x G R | /x(x) = /i(0)}. Let /i G 1/^. Since i? is commutative, /x satisfies (i?3) if and only if /x(xy) > fi(x) Vx,y G i?. Theorem 3.1.8 Let A C R. Then A is an ideal of R if and only if 1A ™ an L-ideal of R. ■ Theorem 3.1.9 Let /J, G LR. Then fi G LI(R) if and only if /xa is an ideal of U V a G fi,(R)U {beL\b< /x(0)}. Proof. Let /x G LI(R) and a G L be such that a < ^x(O) or a G fJ>(R). Let X ,V € t^a and r € R. Then /x(x — y) > /x(x) A /x(y) > a and xx(rx) > /x(x) > a. Hence x — ?/, rx G xxa. Thus /xa is an ideal of R. Conversely, let jia be an ideal of R Va G /x(i?)U{& G L \ b < /i(0)}. Let x,?/,r G i? and a = /i(x) A/x(y). Then & < A^(0) a n ( i x> 2/ € A^a- Thus x — y G /xa. Hence /i(x — 2/) > a = /x(x) A /i(y). Let b = /i(x) G M(-R). Then x G /i6- Thus rx G /i6 since /i& is an ideal of R. Hence /i(rx) >b = ji{x). Thus fi is an L-ideal of R. ■ Theorem 3.1.10 Let /x, */ G L/(i?). fij /x(0) > ^(x) V x G fl. (%) 7/i? is with identity, then /x(l) < /i(x) V x G R. (3) Let x,y e R. If fi(x — y) = /x(0), then /x(x) = n(y). (4) /i* is an ideal of R. (5) If L is regular, then /x* is an ideal of R.

(6) fj,*nv* c (/xni/)*. Proof. (1) /j,(0) = fi(x — x) > fj,(x) A n(x) = ji(x). (2)Ax(x) = /i(xl)> A x(l). (3) fi(x) = /i(x - y + y) > /x(x -y)A n(y) = /x(0) A /x(y) = zx(y). Similarly, Kv) > Kx)- Hence fi(y) = /x(x). (4) and (5) are straightforward. (6) Let x G /x* n v*. Then /i(x) = /i(0) and z/(x) = i/(0). Now (/x D iv)(x) = /x(x) A i/(ar) = /LX(0) A i/(0) = (/x H i/)(0). Thus x G (/x n i/)*. Hence /i* D v* C (/ifl ^)*. ■

73 In general, equality in Theorem 3.1.10(6) need not hold, as shown by the following example. E x a m p l e 3 . 1 . 1 1 Let L = [0,1]. Let /x, v E LR be such that /JL(X) = ^ V x G R and u(x) — \ if x ^ 0 and v(Q) = 1. Then \i,v G LI(R). Now ^ f l i / * = R n {0} = {0} and (/x D i/)* = R. L e m m a 3 . 1 . 1 2 Let [i,v G LI(R)

and let /x(0) = v(0). Then fjL+nv* = (/xflz/)*.

Proof. Let x G (/x D z/)*. T h e n (fj, (1 u)(x) = (/x n ^)(0). T h u s /x(x) A i/(x) = fi(0) A 2/(0). Hence zx(:r) = /x(0) = i/(0) = v(x). Therefore, x G /x* fi z/*. Hence (/ifli/)* C /x* D v*. Now /i* D z/* C (/x n z/)* by Theorem 3.1.10. T h u s jtx* fl v* = (/xflz/)*. ■ T h e o r e m 3 . 1 . 1 3 Lei /i G LI(R). (1) fii(0)=fi(0) VzGN; W (jg) ^ ( 0 ) = M ( 0 ) V i G N .

Then

Proof. (1) Clearly, t h e result is true for i — 1. Assume t h e result is true for i > 1. Now /x* +1 (0) = V{/i*(x) A xx(?/) | x , y G # , 0 = xy} = xx(0) since t h e supremum is attained when x = y — 0. T h e result now follows by induction. (2) T h e proof follows from (1) and t h e fact t h a t /x2 C ^ C /i. ■ L e m m a 3 . 1 . 1 4 Let ^i e LI(R). (1) (fjL% C ^ V i G N ;

Then

Proof. (1) Let x G (/x*)*- T h e n /x*(x) = /x^O) = /x(0). Since /x*(x) < /i(x), fi(x) = /x(0). T h u s x G /x*. Hence (/x1)* C /i*. (2) T h e proof is similar to (1). ■ L e m m a 3 . 1 . 1 5 Let fi G LI(R), and let k G N. i / a r i , #2, . . . , #fc E i?, tfien ^ fik(xix2 . . . x fc ) > A^L 1 /i(x i ); ^/x(fc)(a:iX2...xfc)>AtiM(^). Proof. (1) Clearly, t h e result is t r u e for k = 1. Assume t h a t it is t r u e for /c > 1. Now /i f c + 1 (xia;2 . •. xk+1)

= > >

(/xfc o fj)(x1x2^xk+1) iik{x1x2 . . . xk) A /i(x f c + i) ( A ^ / x ^ ) ) A/x(x f c + i)

(2) T h e proof follows from (1) a n d t h e fact t h a t /xfc C /x(fc) C /i. ■

74

Theorem 3.1.16 Suppose that L is a chain and that /J, is a finite-valued Lideal of R. Then (1) (fi% = {^y Mien(2) (,*«)„ = ( ^ ) * v i G N. Proof. (1) Clearly, the result is true for i; = 1. Assume that it is true for i > 1. Let x G (^ i + 1 )*. Then ^ i + 1 ( x ) = /x*+1(0) = /z(0). But ^ + ! ( x ) = V{n\y)An(z)

\y,zeR,x

= yz} = jz(0).

Thus ^(y) = fi(0) = n(z) for some y,z e R since ^(y) < 1^(0) = /z(0), L is a chain, and /z2 is finite-valued Vi > 1. Thus x = yz for some y € (/i2)* = (/i*)1 and z G //*. Hence a: 6 (/x*)>* = (A**)i+1- Thus (^ i + 1 )* C (/i*) i+1 . Now let x = Y%=i x3ixJ2 • • • xji+i ^ (A 4 *)^ 1 * where Xjk G /** Vfe= 1, 2 , . . . , i + 1 and jf = 1 , . . . ,r. Then

//x^i+1 ( x(x) )

> > A j5= =1 /1 i^ + 1 (x [Xjh±Xjx2j2 .... . .Xj xji+1 i+1) r (Aj£/i(ay )) > A$ fc A=1j=1(Ait\»(xJk = = M(0). MO).

Thus ^ + 1 ( * ) = M(0) = M i+1 (0)- Hence x G (^ i + 1 )*. Thus (/i«,)i+1 £ (Mi+1)*Hence (/ii+1)«. = ( ^ ) i + 1 (2) Clearly, the result is true for z = 1. Assume that it is true for i > 1. Let x G (IJ?+1)+. Then ^ ( i + 1 ) ( * ) = M (i+1) (0) = /x(0). But ^^(x) = V{A^ =1 (^)(2/ fc ) A fi(zk)) | yfc,zfc G J l , l < k < n,n G. N,££ = 1 y f c z f c = x} = /x(0). Thus 3yk,zk G # , 1 < k < n,n € N, x — Ylk=i Vkzk such that since /zW i s finite-valued Vz > 1. This implies that ^\yk) = /x(0) = /x(2?fc), Vfe, 1 < k < n. Thus zk G /z* Vfc, 1 < k < n. Also /i(O) > M(i)(0) > /i(i)(2/fc) = M(0) and so MW(0) = fi^(yk), i.e., ^ G (fi% \/k,l. Thus (// 2 + 1 ))* C (M*)(* + 1 ). Now let x = ^ = i xjlxh . . . xji+1 G (/x*) (i+1) , where xjk G //* Vfc= 1, 2 , . . . , 2; + 1 and j = 1 , . . . , r. Then (i+1) ^^ ( x(*)) M

> > >

AA^^^/ + i *D*C1 ^) ^x -^,. .. .• .x i*i +^i )) Aj (AJi11M(xiJ) Aj=1 =1(AJ±i/iteJ)

- M(0). /«(0). =

Thus // i + 1 >(x) = n(0) = ^ i + 1 ) ( 0 ) . Hence a: € (^ i + 1 >)„. Thus (^») ( i + 1 ) C ( / / i + 1 ) ) , . Hence ( ^ i + 1 > ) , = ( M ») (i+1 >. ■ Theorem 3.1.17 Suppose that \x is a finite-valued L-ideal of R. Let L be a chain. Then

ft) (ngx/i'^ng!(/*.)•;

75

Proof. (1) Let x G R. Then x G ( f l g ^ ) * if and only if ( 0 ^ ) 0 * 0 = (n»/x*)(0) if and only if A ^ x ) = A ^ ( 0 ) = /i(0). Since ^(x) < /i(0), the latter condition is equivalent to ^(x) = /J,(0) Vz > 1 which in turn is equivalent to fi*(x) = /z*(0) Vz > 1. But this is equivalent to x G ng^/i*,)* since (/x*)„ = (/z+)* Vz > 1 by Theorem 3.1.16(1). Hence (ngi/z*)* = n g ^ / i * ) * . (2) The proof is similar to (1). ■ Theorem 3.1.18 Let JJ,, v, n G LI(R).

Then fi o v C 77 z/ and on/z/ z//izy C 77.

Proof. Clearly, \xovC. [iv. Thus if /zz/ C 77, then /i o z/ C 77. Conversely, suppose 110 v C. r). Let x £ R and let £ = X ^ = 1 Vizii 2/i» ^i € i?, z = 1 , . . . , n. Then 77(2:) = ^(£"=12/***) > A j L ^ ^ Z i ) . Now, 77(7;^) > ( / i o i / ) ( ^ i ) > Mlfc) A 1/(2*) V i = 1 , . . . ,n. Thus A?=1{/i(2/<) A v(Zi)} < A? =1 {77(^2*)} < */(*)• Hence n

V { A ? = 1 ( ^ ) A 1/(2*)) I x - $^2/<*i, n e N } < r](x). 2=1

That is /zi/(x) < r](x). Hence \iv C 77. ■ Corollary 3.1.19 Le£ /i, z/, 77 G LI(R). Then and only if /J,V C 77 => /z C 77 or v C 77. ■

fioiyCn^fiCrjorvCrjif

Lemma 3.1.20 Le£ /z arzrf z/ 6e L-subrings of R. Then (1) for all a G L, /i a + zva C (/z + z^)a; ( ^ if L is a chain and /J, and v are finite-valued, then V a G L, /za + z/a = (/i + ^)a. Proof. x G /i a + va ^ 3 7/ G /i a , 2: G z/a such that x = y + z => V{fJ>(y) A v(z) \ x = y + z}>a<&(n + v)(x) > a <=> x G (/i + z/)a. If L is a chain and zz and v are finite-valued, then the imphcation in the previous sentence becomes an equivalence. ■ Example 3.1.21 Let L = [0,1]. Let Z denote the ring of integers. For n G Z, Ze£ (n) denote the ideal generated by n. Define the fuzzy subsets fi and voflL by\/x£ Z\{0}, fi(x) = 1 — 4-, ix is the smallest i such that x £ (2 2 ) , /i(0) = 1 and u(x) = 1 — 4-, j x is the smallest j such that x £ (3 J ) , z/(0) = 1; i,j = 1,2, Then /z and v are fuzzy ideals of Z. Clearly, /z* = {0} = z/*. There exists sn, tn £Z such that 1 = sn2n + £ n 3 n /or n = 1,2, T/izzs (/i + z/)(l)

= >

V{/z(z/)AKz)|l = z/ + z} V{zz(sn2") A z/(t n 3 n )|n = 1,2,...}

—► 1 as n —> 00.

iJence (/i + z/)(l) = 1. T/izzs (/i + i^)* = Z. Therefore, /i* + z/* C (/i + z/)*.

76

Example 3.1.22 Let L = {1, a, 6, c, 0}, where 1 > a > c > 0 and 1 > 6 > c > 0, a A 6 = c, and a and 6 are noncomparable. Then L is regular. Let R — 1L. Define fi, v G LR as follows: fi(x) = 1 if x e (&) , fi(x) = a if x G (2) \ (6), ^(#) = b if x G (3) \ (6) , /J,(X) — c otherwise and v(x) = 1 if x G (6), v(x) = a if x G (3) \ (6), */(#) = b if x G (2) \ (6), z/(#) = c otherwise. Then /J, and v are L-ideals of R. Now (/i + z/)(2) = W{fi(y) A v{z) \ y + z = 2} > (/i(0) A i/(2)) V (^(2) A i/(0)) = 6 V a = 1. 77ms 2 e ( / i + i/)i, Jm* 2 £ (6) = [i\ + v\. Thus Lemma 3.1.20(2) zs nc?£ £nze if L is not a chain. Proposition 3.1.23 Let /J, be an L-ideal of R. Suppose that R has an identity. (1) Let x G R be a unit. Then /JL(X) = i^{x~l) = /i(l). (2) If x and y are associates, then fi{x) = fi(y). Proof. (1) Now fi(l) = ji(xx~l) > fi(x) V n(x~l) and fi{x) = fi(lx) > /x(l) V ii(x) > n(X). Thus fi(x) = /J>(1). Since x_1 is also a unit, n(x~l) = /i(l). (2) Now x = uy, where u is a unit. Thus ji{x) = fi(uy) > fi(u)\/fi(y) > /x(y). Similarly, /j,(y) > /i(x). ■ Theorem 3.1.24 Let m G LI(R), i G L Then ni€lm

G LI(R).

■

Definition 3.1.25 Let /J, G LR. Let

(^ = n{v | fiCis,veLl(R)}. Then (/i) is called the L-ideal of R generated

by /i.

By Theorem 3.1.24, (/i) is an L-ideal of R. Clearly, (/j) is the smallest L-ideal of R containing /J,. Theorem 3.1.26 Let ji, v G LR. Then (1) fi G LI(R) & (fi) = & (2)nCv^ (fi) C (i/); (3) (fJ>\s) Q (t1) \s £ L(S), where S is a subring of R, fj,\s and (/i) \s are the restrictions of fi and (/J,) to 5, respectively. I Theorem 3.1.27 Let A be a nonempty subset of R. Then (aA) = (A) is the ideal of R generated by A and a G L. ■ Theorem 3.1.28 Let /J, G L

R

u(x) = W{AyeAfi(y) V xeR.

Then (/j) = v.

and define v G LR as follows: | A C R, 1 < \A\ < oo,x G (A)}

Q>(A)I

'where

77

Proof. Clearly, by definition, \x C v and v(—x) — v{x) V x G R. Let x, 2/ G R. Now if x G (A), y G (B), A, B C ii, 1 < |A| < oo, and 1 < |B| < oo, then x+y e ( A U B ) . Thus i/(x+i/) > A^^ufiM*) > ( A U G A M W ) ) A ( / W B M ( V ) ) Hence i/(x + 3/)

> =

=

V { ( A t t 6 ^ ) ) A ( A ! ; € B ^ ) ) I A,BCR,1<\A\

< oo,

1<|B|
Thus z/(x -f y) > v(x) A v(y) Vx, y e R. Now iiy e (B) , B C R, and 1 < |B| < oo, then xy G (B) . Thus i/(xy)

= >

V{Ax€Bri*) \BCR,l<\B\
= Ki/)This shows that v{xy) > v(y) Vx, y e R. Hence i/ G LI(R). Now let 77 G L/(i?) with /x C 77 and let x G R. Let A = {7/1, T/2, ■ • ■, 2/n} S B be such that x G (A) . Then there exist x\, X2, . . . , x n G B and integers m i , 7722,..., mn such that x = E?=i ^2/i + E?=i mi2/»- S i n c e ^7 € ZJ(B), T?(X) = 77(^=1 ^2/* + E?=i ^i»2/i) >

v(Y17=i xiVi) Av(E7=i

m

iyi) >

(A-U^X^)) A ( A ^ ^ ™ ^ ) )

>

A^T?^).

Thus T/(X)

> >

W{AyeAr](y) I A C B,1 < |A\ < 00, x G (A)} V{Ay€AM(3/) | ^ C B, 1 < |A| < 00,x G (A)}

=

i/(x).

Hence v C 77 and consequently (/i) = JA ■ Corollary 3.1.29 Le£ ^ G L^ and define 77 G L^ as follows: Vx G B, 7?(X)

V { E t l ( r i ) l ( ^ ) a i + E^=l nj(yj)bj)(x)

=

KVj)

> bj,Ti,Xi,yj

G R,ai,bj

I /i(Xi) > a»,

G L,TT,J G Z, A:,m G N } .

Tften (/x) =77. Proof.

Let x e i ? . Then (/z) (x) = V{AjL1/i(xi)A Af=1fj.(yj) n

^a* + Y!j=i jyj, n

r x

u uVj

x

x

e R,nj a

eZ,k,m

eN}

\ x = £*=1

= V{(^==1(ri)1(xi)a. +

r

X ^ i j ( % K ) ( ) I M ») > *> M(%) > &j> *> ^i 5 % £ R, a>i>bj G L j n j G Z,

/c, m G N} = 77(x). ■ The following theorem is an easy consequence of Theorem 3.1.28.

78

T h e o r e m 3.1.30 Let x G R and a G L. Then (xa) = a^xy ■ T h e o r e m 3.1.31 Let /x, v G LI(R) LI(R) and /i + i / = ( / i U i / ) .

be such that /x(0) = i/(0). 77ien /x + i/ G

Proof. Let x,y e R. Then (ZJ+J/)(:E) = V{/x(?x) A^(t>) | u, v G i?, w+v = x} > /JL(X) A ^(0) = ii(x). Hence fj, + i/D fi. Similarly, /i + i/Di/, Thus fi + v D [lUv. Clearly, {ji + ^)(-a?) = (/x + ^)(^)- Now (xx + z/)(x + y)

=

V{/x(iz) A i/(v) | u, v G R, u + v = x -f y}

>

V{/x(tZi + t>i) A J/(tX2 + ^ 2 ) I U>\,U2,V\,V2 G i?,

> = = =

^1 + ^2 = z, v1+v2 = y} V{(/i(ui)A/x(vi))A(i/(u 2 )Ai/(i;2)) | ui, u 2 ,i>i, v2 G # , u\+U2 = x, vi-\-v2 = y} V{(/X(TAI) AI/(TX 2 )) A(/i(vi) Ai/(v 2 )) I wi,^2,vi,t; 2 G-R, Ul + ^2 = #, Vl + f 2 = 2/} (V{/z(ui) A 1/(^2) | u i , u 2 G i?,ui + i / 2 = a;}) A(V{^(vi) Ai/(v 2 ) I vi,v 2 G-R,t;i + v2 = y}) (fM + i/)(x)A(fjL + i/)(y).

Also, (xx-h v)(xy)

> >

V{/x(x?x) A u(xv) \ ix, v G i?, w H-1; = y} V{/x(?x) A z/(v) I u, v G i?, u + v = y}

=

(/*+ *)(!/).

Hence /x + v G LJ(ii). Now let f G L/(iJ) and / i U i / C f . Then (/i4-z/)(x)

= < =

V{xx(?x) A v{y) I it, v G R, u + i; = x} V{£(u) A f (v) I u, v G # , u + v = x} * ( * ) ■

Thus xxH-z/ C £. Hence ji+v is the smallest L-ideal of i? such that /xUz/ C \i+v. Thus /x + v = (/i U 1/). ■ Theorem 3.1.32 £e£ /x, 1/ G LR. (1) If ^e LI(R), then / x o i / C / i . (2) If fie LI(R) and v G LI(R), then / i o i / C / i f l i / . i Theorem 3.1.33 Let fi, v G L(i2). TTien /xi/ G L(i?).

79 Proof.

Clearly, {/J,v)(-x) = (/xi/)(ar) V x G R. By Theorem 3.1.4(4), (fiiy)(x + y) > (fja/)(x) A ( ^ ) ( l / )

Vx, y Ej=i W = fiv, it

G i?. Let x, y G i? and consider (fiu)(xy). If x = £ ™ x x*x< and y = 2/j3/j> w n e r e xu xj G iJ for 1 < i < m and y i5 ^ G i? for 1 < j < n, then (Z?=i Xix'i)(ZU VjV'j) = E™ i E?=i(*tf;)(*fr;)- By the definition of follows that

> =

A£x A?=1 ((M(*;) A /ifo)) A (*(*<) A i/(y<))) A ^ A?=1 ((^(xi) A «,(*$)) A (Ufa) A i/(yj.)))

=

( A - ^ / z ^ i ) A !/(*$))) A (A7 =1 ( M (y j ) A i/(yj.))).

Consequently,

(H(^)

>

(VJA^Mx^A^O)!^ (V{A- = 1 ( M ( % ) A % ; ) ) l % , 2/; € R, 1 < J < n, £ ? = 1 %2/• = 2/})

= Hence ^

(HWA(F)W-

G L(R).

■

Theorem 3.1.34 Let \x G LI(i?) and i/ G L(U). T/ien /ZL> G LI{R) and in fact \xv = (fi o v). Proof. Let x,y G R. Clearly, (fiv)(-x) (fjLi/)(x + y)> 0^/)(x) A (/xi/)(y). Now (/j,u)(xy)

> >

= (i*v){x). By Theorem 3.1.4(4),

V{A- l =1 (^(x^) A J>(u*)) I ^t, v* € iJ, 1 < i < n, n G N, E£=i <W = ?/} V{Af=1(//(wi) Au(vi)) \ui,VieR,l
Er=i *w = y} Vx, y e i ? . Hence fiis G LI(R). By Theorem 3.1.4(1), /i o *y C /ZIA Let £ G £/(.#) and fio v C £. Let x e R. Then (/^)(x)

=

V{(A?=1(M(yi) A Ufa))) | 2/i, Zi G iJ, 1 < i < n, n G N,

Er=i 2/i^i = x l <

V{(A?=1(/zoi/)(2fcZi)) | ^ , 2 i G i*,l < % < n,n G N,

E?=i ^ < =

= *}

V{A?=1^(2/iZi) | 2/i, * G i?, 1 < i < n, n G N, £ ? = 1 y;Z; = x} * ( * ) ■

Hence \xv C f. Thus /iu = (fi o iy) . M

80

T h e o r e m 3.1.35 Let x, y G R and a, 6 G L. Then (xa)

O (yb) = (xa)

{yb) = ((xy)aAb)

■

Proof. By Theorem 3.1.30, {(xy)aAb) = (a A b)(xyy Let w G R. Suppose w G (xy). Then w = xyz + mxy for some z G R,m G Z. Hence ((z a ) o (yb))(w)

> > =

(xa) (x) A (yb) (yz + my) (x a ) (x) A (j/6> (2/) a A 6 by Theorem 3.1.30.

Now for any u, v G R such that uv = w, (xa) (u) A (yb) (v) < a A 6. Hence ((xa) o (yb))(w)

=aAb.

If w ^ (xy). Then V u,v £ R such that wv = w, either u £ (x) or v £ (y). Hence ((x a ) o (yb))(w) = 0. Thus we have proved that (xa) o (2/6> = ((xy)aAb) . By Theorem 3.1.34, (rr0> (y6) = ((x a ) o (yb)) . Since (x a ) o (yb) = G L/(iJ), (xa) (yb) = (xa) o (yb) . ■

((xy)aAb)

T h e o r e m 3.1.36 Let fi, v G L/(iQ. Then JAV C /z D ZA Proof.

The result follows from Theorem 3.1.18 and Theorem 3.1.32(2). ■

T h e o r e m 3.1.37 Let //, i/, f e LJ(#) and i/(0) = £(0). Then /x(*/ + f) = /iz/ + zz£. Proof. Since z/ C v + £ and ^ C i/ + ^ ^ Hence [iv + xx£ C //(z/ + £). Let x e R. Now (/i(i/ + 0 ) ( * )

C /i(z/ + £) and //£ C /z(z/ + £)•

-

V{A?=1(/x(i/i)A(i/ + 0 ( ^ ) )

=

V{Af=1(/x(yi) A (V{i/(iii) A ^(Vi) | T ^ , ^ eR,Ui + Vi = *i})) \yi,ZieR,l
=

ll/^eii,l
< v{(AU(MyJ) K^lVi^k = =

A v(<))) A ( A U ( M ^ ) A £K)) I e # , 1 < < < U < k < 8,8,n G N,

V{(/zz/)(a) A 0*0(6) \a,b€R,a ((jtv) + (tf))(x).

+ b = x}

Hence fi(u + 0 ^ ^ + A*f • Thus ^(z/ + 0 = /zz/ + /z£. ■

81

Definition 3.1.38 Let /x, v G LR. Define \JL\V e L /x:i/ = U{f

R

as follows:

| £GLJ(.R),£OZ/C/X}.

Theorem 3.1.39 Let \±,v G LI(R).

Then /i C /x : i/ and /J, : v E LI(R).

Proof. Since /i o // C ^, ix : z/ = U{r] | 77 G LI(R),r] o 1/ C //} D /i. Let x,y e R. Clearly, (ix: v)(—x) = (ix : v)(x). Now, (ix : u){x) A (/i : i/)(y)

=

(V{f (x) | £ G £ / ( # ) , £ ° ^ ix}) A(V{7y(2/) | r; e £ / ( # ) , 77 oi/C^x}) V{Z{x) A 77(3/) | £, 77 G L/(i2), (£ o 1/) U (77 o 1/) C xx} V{(£ -f 77)(x + y) I f, 77 G LI(R), (£ + 77) o v C 11} (/x:i/)(x + y).

= < < Also, Ox : I/)(X2/)

=

> > Hence \x\vE

LI(R).

V{£(x2/) I £ G L/(i2), (£ o 1/) C ix}

V{^)KEL/(i?),Ko^/i} (/x : i/)(x).

■

Lemma 3.1.40 Le£ /x, x/,f G LI{R). Then (1) ix : /x = lie; (%) if ft C is, then zx : £ C x/ : £ ana1 £ : 1/ C £ : /x. Proof. (1) /x : /x = U{77|T7 G LI{R),n o zx C /x} = 1# since 1^ o ix C /x by Theorem 3.1.32. (2) Let 77 G LI(R) be such that 77 o £ C ii. Then 77 o £ C ix C ZA Hence i/:£

= D =

U{77 I r}eLI(R),r]ot
Also, V 77 G LI(R) if 77 o 1/ C £, then 77 o zx C 77 o 1/ C £. Hence £:/x

= 3

U{77 I 77 G £ / ( # ) , 77 o zx C £} U{77 I 77 G £ / ( # ) , 77 o z / C £ }

= (:!/.■

Theorem 3.1.41 Le^ /x,z/,£ G L/(iJ). 7%en (2) if xx C £ and £1/ C /x, iTxen £ C ix : XA

82

Proof. (1) Let x G R and let x = YJi^iVi^ Vi^i E R, 1 < i < n. Let f G LI(R) be such that f o i/ C jz. Then /z(x) = M E i L i 2/*^i) > A ? = 1 / x ( ^ ) > A?=1(£ o v)(yiZi) > A?=1(f (yi) A i/(z<)). Hence /x(x) > A? = 1 ((/i: i/)(y<) A i/(*)). Thus ((/A: I/)I/)(X)

=

V{A? =1 ((/i: v)(yi) A i/fe)) | yh Zi G # , 1 < i < n,

<

/x(x).

Hence (/x : v)v C /i. (2) Since f o i / C f i / C ^ ,

JX:I/D£.

■

Theorem 3.1.42 £e£ //*, v G L/(iJ), 2 = 1,2,... ,ra. T/ien (n? = 1 /i i ):i/ = n ? = 1 ( / i i : i / ) . Proof. We prove the result for n = 2. Clearly, (/ii fl/z2) : ^ C (/^ : z/) fl (^i : i/). Let x G i?. Let ( ( ^ n ^ ) : v)(x) = V{£(x) | £ G LI{R)^ou C ^ i f l ^ 2 } = a. Now ((Ml:i/)n(M2:^))(*)

= =

(V{T7(X)|T7GLJ(B),77OI/CMI})

AMCWKGL/^CO.C^}) V{T7(X) A C(X)|T7, C eLI{R),77 Cot/ C /z 2 }.

oi/C/n,

Let 77, C G LI(R) be such that 77 o z/ C // l5 £ o 1/ C // 2 . Let 77(2;) A £(x) = 6 and let J4 = rjb fl C&- Then >1 is an ideal of R. Let f G JL H be such that £(x) = 6 if x G A and £(x) = 0 if x £ A Then f G LI(R). Clearly, £ C 77 and £ C £. Hence ^ o z / C ^ o z / C / i ! and ^ o i / C ( o z / C / i 2 . Thus £ o z/ C /iX n // 2 . Hence a

> = =

V{7(x)|7GL/(i?),7oz/CMln/i2} «*) 6 770*0 A C(x).

((//ifl^ 2 ) : z/)(x)

= > =

a V{77(x)AC(x) | 77, £ G LJ(i?), 77 ((Mi : * 0 n ( / z 2 :^))(a:).

Thus OZ/C/^COZ/C^}

Hence (n?L l W ) : 1/ D n ? = 1 ( W : 1/). Thus (fl? = 1 ^) : 1/ = n ? = 1 ( W : 1/). ■ Definition 3.1.43 Ze£ ^ G L/(iJ). T/ien \i is called proper if p ^ /x(0)u, i.e., if fi is not constant.

83

Definition 3.1.44 Let R have an identity and let fi, v G LI(R). \i and v are said to be comaximal (or coprime) if both /x and v are proper, xx(0) = ^(0), and xx + v = /x(0)#. Theorem 3.1.45 Let R have an identity. Let xx, v, £ G LI(R). \i and v are comaximal with £. Then (1) \iv and £ are comaximal. (2) [iV\v and £ are comaximal.

Suppose that

Proof. (1) Since /x and v are comaximal with £, xx(0) = £(0) = ^(0). Clearly, xx D v is a proper L-ideal. Since \iv C xx D v, fiv is a proper L-ideal. Now n(0)R = (fji + 0 ( ^ + 0 = (l* + 0 " + (M + Of = W + Z" + M0)a£ C /xi/ + (^ fl i/) + (M(0)H n f) = /ii/ + £ C xx(O);?. Hence xxz/ + £ = ^(0)^. Thus xxiv and £ are comaximal. (2) The proof follows from part (1) since xx o i/ C xx fl IA ■ Theorem 3.1.46 Let R have an identity. Let xi, v, £ G LI(R) and £(0) = /i(0). Suppose that /x and v are comaximal and xx 2 ^f • TTien / i D ( . Proof.

Now xx = xx + /x D xx£ + i/f = (/i + i/)f = f (0)j*f = f. ■

Theorem 3.1.47 Lei i2 /lave an identity. If/j, and v are comaximal L-ideals of R. Then xx f] v = XXXA Proof. Now xxDz/ = (A*ni/)/z(0)ji = (p,r\v){p. + v) — ((/xni/)/x) + ((^ni/)i/) C j//x + /JUS — y>v + fiv = XXJA Thus /J,V = fi H v. M Theorem 3.1.48 Lei i2 /iave an identity and let xx, i/ G LI(R). (1) If n* and v* are comaximal ideals of R, then xx and v are comaximal. (2) If /x and v are comaximal and fi{R) U v(R) is a finite subchain of L, then /x* and v* are comaximal. I

3.2

Quotient L-Subrings

Definition 3.2.1 Let \i G L(R) and let x G R. Then /x(0){x} + fi is called a coset of fi. Let x,y E R. Then (xx(0) {x} +/x)0/)

= = =

V{/i(0) { s } (Ti)A/i(i;)|u,t;€/J,u + t; = y} M°){^}(^) Axx(x-y) /*(*-2/).

We write a: + xx for /i(0){x} + xx. Thus (x + //)(?/) = /x(:r — y) V y G R.

84

Lemma 3.2.2 Let fi G LI(R) only if ji(x-y) = /x(0). Proof. Suppose x + fi = /x(0) = fj,(y — x) = fi(x — y). [i{z - x) = /x(z - y + 3/ - x) z e R. Hence x + fiDy + fi.

and let x,y G R. Then x + ji = y -f /x if and

y + fj,. Then (x + [J){X) = (y + M)( X ) an< ^ hence Conversely, let /x(0) = fi(x — y). Then (x + fi)(z) = > [i[z - y) A /x(x - y) = fi(z - y) = (y + /x)(z) V Similarly, y + JA^) x + fi. Thus £ + /x=:2/-f-/x. ■

Let i?//x = {x + [i \ x € R}. Define + and • on R/fi as follows: (x + p) + (y + p) (x + xx) • (y + /x)

= (x + y) + /j, = xy + fi

Vrr, y e R. We now show that -h and • are well defined. Let x,y,u,v G R and let x + /x = u -f /i and y + /x = v + /x. Then /i(x — ix) = /i(y — v) = /x(0). Hence /x(x + y - u — v) > /J,(X — u) A /x(?y — v)= /x(0). Thus /x(z + y — u — v) = /i(0) and so (x + y) + /i = (u + v) + fi. Now fi(uv — xy)

=

fx(uv — uy + uy — xy)

>

/JL(U(V - y))

>

(/x(ix) V n(v - y)) A (n{u -x)y

A fjb((u -

x)y)

fi(y))

= MO)Hence xy + fi — uv + /x. Thus -(- and • are well defined. Clearly, —(X + /JL) = (—x) + /J,V x £ R and 0 + /x = fj, is the zero element of i?//x. It is now a routine verification to show that (i?//x, -f, •) is a ring. Definition 3.2.3 Let /i G LI(R). Then the ring R//J, = {x -f fi \ x G i?} is called the quotient ring of R by u. Theorem 3.2.4 Let /i G LI{R).

Then R/fj, ~ R/fi*.

Proof. Define f : R -> R/fi by V x G ii, / ( x ) = a; + /x. Then / is a homomorphism of R onto i?/xx and Ker / = /x*. Thus i?//x ~ i?//x*. ■ Let R and 5 be rings, /x G LI(R), v G LI(S), and let / : i? —► S be a ring homomorphism. Let 0' denote the zero element of S.

85

Lemma 3.2.5 Let R, 5, /J, and f be defined as above. Then

(i)/Miff) = IM(O)\ (2) /(/**) C (f(v)U (3) Iffi has the sup property, then /(xx*) = (/(/A))*; (4) If fi is constant on Kerf, then (/(/x))(/(x)) = /x(x) V x G R. Proof. (1) (/(/i))(0') - V{/x(x) | x G i?, / ( x ) = 0'} = xx(0) since /x(0) > fi(x) V x G i? and /(0) = 0'. (2) Let x' G /(/x*). Then x' = / ( x ) for some x e R such that xx(x) = xx(0). Hence (/(/i))(x')=M(0) = (/(Ai))(0'). Thus ar' e (/(**)).• Hence / ( ^ ) C (/( M )),. (3) Let a; e i? be such that f(x) € ( / ( / / ) ) * . Then MO) = /(/*)(/(*)) = V{M(i) | f(t) = f{x)} = M (t) for some t 6 R such that /(£) = / ( x ) . Hence t e /J,*. Thus / ( x ) G /(//*). (4) Now (/(/x))(/(x)) = V{/x(y) I 2/ € J2,/(y) = / ( * ) } • Let /(y) = / ( x ) . Then / ( x - y) = /(0) and hence x - y G Ker / . Thus /x(x - y) = /x(0). Then xx(x) = xx(y). Hence (/(xx))(/(x)) = /x(x). ■ Theorem 3.2.6 Le£ i£, 5, /x, i/ and / 6e defined as above. Then (1) f~x{y) is an L-ideal of R which is constant on Ker / ; ( ^ / / / is onto, then ( / o / - 1 ) ( ^ ) = i^; ( ^ ///x is constant on Ker / , then (/~ 1 o /)(/x) = /x. Proof. (1) Let x, y e R. Then / " ^ ^ ( x - y) = i/(/(x - y)) = i/(/(x) /(y)) > K/(s))Ai/(/(y)) = / - V ) ( z ) A . T V ) ( 2 / ) a n d / - » ( * y ) - i/(/(*y)) i/(/(x)/(y)) > i/(/(x)) V i/(/(y)) = f-\y){x) V / " » ( y ) . Hence / " » G Z J ( # ) . Let x G Ker / . Then /" 1 (z/)(x) = i/(/(x)) = i/(/(0)) = i/(00- Hence f~x{v) is constant on Ker / . (2) Let x e R. Now x G / _ 1 ( ^ * ) ^ K/0*0) = K0') = K / ( 0 ) ) <^ / - » ( x ) = / " » ( 0 ) ^ x e ( / " > ) ) * . Hence / - > * ) = ( / " > ) ) * . (3) Let y G S. Then y = / ( x ) for some x e R. Now ( ( / o / _ 1 ) ( ^ ) ) ( y ) =

r1)M = ^(4) Let xeR. Then ( ( / " V X / M * ) = f'Hf (»))(*) = / M ( / ( * ) ) = /*(*) since /i is constant on Ker/. Hence ( / _ 1 o /)(//) = /x. ■ Theorem 3.2.7 Le£ / , i?, 5 and /x be define as above. Suppose f is onto. Then /(/x) G LI(S). If fi is constant on Ker / , then /(/x*) = (/(/x))*.

86 Proof.

Let u,v G S. Then u = f(x), v = f(y) for some x,y G R. Now

f(ri(u-v)

= > > = =

V{n(z)\zeRJ(z)=u-v} V{/x(x -y)\x,yeR, f(x) = u, f(y) = v} V{/x(x) A fj,(y) \x,yeR, f(x) = u, f(y) = v} (V{/x(x) \xeR, f(x) = u}) A (V{/x(?v) | y G iJ, /(y) = v}) f(l*)(u)Af(vL)(v).

Also, /(/z)M

>

V{AX(2)

>

V{AX(X) V /a(y) I x, y G iJ, / ( x ) = u, /(y) = v}

=

(V{/x(x) | x G U, / ( x ) = u}) V (V{/x(2/) | 2/ G iJ, /fa) = v})

=

I z G 12, /(*) = uv} V{n(xy) | x, y G iJ, / ( x ) = u, /(y) = v}

fWMvfOMv).

Hence /(/x) G LI(S). Let u G (/(/i))*. Then /(/x)(u) = / ( M ) ( 0 ' ) = /z(0). Since / is onto, u = / ( x ) for some x £ R. Hence by Lemma 3.2.5(4), f{fi)(u) = /i(x). Thus fi(x) = n(0) or x G /x*. Hence ?x = / ( x ) G /(/x*). The result now follows by Lemma 3.2.5(2). ■ Theorem 3.2.8 (Correspondence Theorem for L-ideals) Let R and S be rings and let f : R —> S be an epimorphism. Then there is a one-to-one order preserving correspondence between the L-ideals of S and the L-ideals of R which are constant on Kerf. Proof. Let ^(R) denote the set of L-ideals of R which are constant on Kerf. Define $ : F{R) -+ LI{S) and * : LI(S) -> F{R) by <1>(AX) = /(/x) and *(iz) = f~l{v) V /x G F(R) and V v G LI(S). Then <S> and * are weU defined and are inverses of each other by Theorems 3.2.6 and 3.2.7, thus giving the re­ quired one-to-one correspondence. Clearly, this correspondence preserves the order. ■ Let /x G LI(R) and define /x<*) G LR/* as follows: /X(*}(X + M) =H(X)

VxeR.

By Theorem 3.1.10(3), it follows that ^(*) is well defined. Let x,y G R. Then ^W(( r c +

Ai)

+ (2, +

/ i ))

^W(( a ; + y ) +

=

= >

/*(* + 2/) /x(x) A xx(y)

M)

87 and

/,<•)((* + A*)'

■ ( »

+ /*))

/x(xy) = Kw) > =

/i(z) V ^(j/)

/xW(a: /iW(x ++ A«)V /i)V/iW(j/ /i). MW(2/ ++ M)-

Hence //(*) G LI(R/fi). We have thus proved the following. T h e o r e m 3.2.9 Let /z G £ / ( # ) . TTien /*W G LI(R/n), defined by i^*\x + ji) = /x(x) V x E R. ■

where /*W G LRI^ is

T h e o r e m 3.2.10 £e£ z/ € L(R) and let A be an ideal of R. Define £ G LR>A as follows: *([*]) = V{v(z)\ze[x]} V x € R, where [x] = x + AV x € R. Then f G L{R/A). Proof.

Let x,y £ R. Then

*(M --[y]) = Z([x-y)) = = > > > = = = =

V{v{x-y V{i/(ar-y + + z) 2?) | z2 G A} - y + a - 6) a-b)\a,beA} I a, 6 G A} V{i/(x -y V{v(x + a) A v{y 1/(2/ + 6) b) |I a, 6 G A} V{v(x (V{i/(x + oa)) |IaaGGAA}) } ) AA (V{i/(2/ (V{>(y + 6) | b 6 G A}) «W)A«[y])«W)A«[y])-

Also,

*(MM) = t(l*y]) = > > = > = = = =

V{i/(xi/ + + z) \I 2: G A} y{v{xy zeA} (xi; + uy + uv)) i^)) \| u, ^, vv G A} ^4} \/{v{xy + (xv V{v((x V{i/((x + i/)(y + v ) ) | u , v G A} \/{u(x W{u(x + u) A u(y v{y + v)) \u,v E e A} (V{i/(x + u) |\ueA})A ix G A}) A (V{i/(i/ G A}) (V{v(y + + v) | vv)\veA}) £([*]) A A «M). «[y]). €(M)

Hence £ G L ( i ? / ^ ) - ■ The L-subring £ defined in Theorem 3.2.10 is called the q u o t i e n t (or dif­ ference) L-subring of v relative to A and denoted by vjA. We now define the notion of an L-ideal of an L-subring. Definition 3.2.11 Let fi G LR and v G L(R) with fi C v. Then /z is called an L-ideal of v (or in v) if x - I/) > M(X) /x(x/x(y) K -y)> Kx)A A Kv)

88

and xx(xy) > (z/(x) A fi(y)) V ((/i(x) A v{y)) Vx, y E R. By definition, it is immediately clear that (1) /x G L H is an L-ideal of the ring R if and only if ji is an L-ideal of the L-subring \R\ (2) if ii G LI(R), v G L(i2), and zx C z/, then /x is an L-ideal of z/; (3) an L-ideal of z/ G L(-R) is an L-subring of R. Since i? is commutative, it is clear that in Definition 3.2.11 fi(xy) > (z/(x) A y.{y)) V (xx(x) A v{y)) Vx, y € Rif and only if /x(xy) > z/(x) A xx(y) Vx, y G R. The proofs of the Theorems 3.2.12 and 3.2.13 are straightforward. Theorem 3.2.12 Let \i G LR and v G L(R). Then /J, is an L-ideal of v if and only z/V a G f^(R) U{6 G L | 6 < /x(0)}, /xa 25 an ideal of va. ■ Theorem 3.2.13 Let v G L(i?) and /x be an L-ideal of v. Then xx* is an ideal of v*. If L is regular, then /x* is an ideal of is*. H Theorem 3.2.14 Let /x G LI(R) v.

and v G L(R). Then /x fl z/ is an L-ideal of

Proof. Obviously, /x C\ v C z/ and /x H z/ is an L-subgroup of the additive group of i?. Moreover, (xx D v){xy) = xx(xy) A v(xy) > /j,(y) A z/(x) A v(y) = v(x) A (/i D v){y) Vx, y e R. Hence xx n v is an L-ideal of v. ■ Theorem 3.2.15 Let £ G L(i?) and fi and v be two L-ideals of'£. T/ien xx Pi z/ zs a/so an L-ideal of £. Proof. Clearly, /xDi/ C f and xxflz/ is an L-subgroup of the additive group of ringi?. Moreover, (xxnz/)(x?/) = fi(xy) Av(xy) > (£(#) A/i(?/)) A(£(x) Az/(y)) > £(x) A (xx(y) A 1/(2/)) = £(x) A(iiC\v)(y) Vx, y e R. Therefore, /xflz/ is an L-ideal

off.B Let /i G L(i£). Assume that 5 is a commutative ring and v G LC^S). Then it follows that if / is a homomorphism from R onto 5, then /(/x) G L[S) and f~x{v) G L(i?). Furthermore, we have the following result. Theorem 3.2.16 Let v G L(R) and /x be an L-ideal of v. Suppose that S is a ring and f is a homomorphism of R into S. Then /(/x) is an L-ideal of f(v).

89 Proof. Note first that both /(/x) and f{v) are L-subgroups of the additive group of R and f(fi) C f(v). Next, we observe that f{fi)(xy)

=

V{[i(w) \weR,

f(w) = xy}

> V{n{uv)\u,veR,f(u) > = =

= x,f(v) = y}

V{(i/(u) An(v)) \u,v€R,/(«) = x, f(v) = y} (V{i/(u) \u€R, / ( « ) = x}) A (V{M(V) h € i J , /(«) - 2/}) /(i/)(x)A/( M )(y)

Vx, y G 5. Therefore, /(/x) is an L-ideal of / ( ^ ) . H Theorem 3.2.17 Assume that S is a ring and f is a homomorphism from R into S. Let v G L(S) and ji he an L-ideal of v. Then / _ 1 (/x) is an L-ideal of

r»- ■ Definition 3.2.18 Let S be a commutative ring and let /x G L(R) and v G L(S). (1) A weak homomorphism from \i into v is an epimorphism f of R onto S such that /(/x) C v. If f is a weak homomorphism of ji into v, then we say that fi is weakly homomorphic to v and we write \± ~ v, or /i ~ v for short. (2) A weak isomorphism from [i into v is a weak homomorphism f from fi into v which is also an isomorphism of R onto S. If f is a weak isomorphism from n into v, then we say that fi is weakly isomorphic to v and we write /i ~ z/, or /i ~ v for short. (3) A homomorphism from \i onto v is a weak homomorphism f from \i onto v such that /(/i) — v. If f is a homomorphism of fi onto v, then we say that \x is homomorphic to v and we write \i~v, or fi w v for short. (4) An isomorphism from /x onto v is a weak isomorphism f from [i into v such that /(/x) = v. If f is an isomorphism from /x onto v, then we say that fi is isomorphic

I

to v and we write fi = v, or /x = v for short.

Let v G L(R) and /x be an L-ideal of v. Assume that L is regular. Then, it is clear that /x* is an ideal of ^* and v\v* is an L-subring of v*. Thus by Theorem 3.2.10, we can talk about the quotient L-subring of v\v* relative to /i*, v\„+ /n*. We call this L-subring, the quotient L-subring of v relative to H and denote it by V//A. Theorem 3.2.19 Let v G L(R) and fi be an L-ideal of v. Suppose that L is regular. Then v\u* w z///x. H

90

Theorem 3.2.20 Let v G L{R) and £ G L(S), where S is a ring such that v & £. Suppose that L is regular. Then there exists an L-ideal fi of v such that

"ft* = Z k * - m Theorem 3.2.21 Let /J, G LI(R) Then vj{\x n v) ~ (/j, + z/)//x. ■

and v G L(R). Suppose that L is regular.

Theorem 3.2.22 Let fi, is, £ £. L(R), V> be an L-ideal of v, and /J, and v be L-ideals of £. Suppose that L is regular. Then (£//i)/(vjfi) = £/v. ■

3.3

Direct Sums

Let Q. denote a nonempty set. Let x, xa G R, where a G Q. By the summation x = J2ae& X Q ' w e m e a n t n a t a n DU^ a n n i t e number of the xa are zero. If {A a | a G f2} is a collection of ideals of i?, then we let Eaeft Aa denote the set (Eaeft xa \ xa € Aa, a E ft} and we let 0 a G n A a denote the weak direct sum of the Aa. For (3 G ft, we let Qp denote Q\{/?}. Definition 3.3.1 Let {/xa | a G ft} C l A Define the L-subset EaeftMa of R byV x e R, (52 Ma)(z) = V{A a€ ^/x a (a: a ) \ x=52

x

«}-

Proposition 3.3.2 Let {fia \ a G ft} C £,/(#) and jz a (0) = /^(O) Va,/3 e ft. Then E a e n V<* e LI(R) and na C E a e n /ua V a G ft. ■ Definition 3.3.3 Le£ {/ua | a e fi} U {fi} C L(i2). TTien // is 5azrf to 6e £/ie weak L-direct sum of the ^ a if fi = E<*eft Ma and V x e iJ, x / 0, (M/3 H X^aen/3 / i a)( x ) = ^- -V M ^5 ^ e ^eaA; L-direct sum of the /z a , then we write n = ©aenMaProposition 3.3.4 Let {fia | a G ft} C L(i?). Suppose that AaenjLia(0) > 0. Suppose that L is regular. Then

$>a = (£>*«)*• a£fl

a€H

Proof. Now x G ( E a € ^ a ) * & ( E a e n ^ X * ) > ° ^ V { A a G ^ a ( a ; a ) | x = E a € n xa}>Q&x = Y,aen x<* f o r s o m e x ^ e ^ ^ x e £ a 6 f i A C ■ Proposition 3.3.5 Let {fi<x\ct G ft} C L(ii). Suppose that L is regular. Then

91 Proof. Now x G (naeQfiay & (n a € n /x a )(x) > 0 O A a € n ^ a ( x ) > 0 => Ma (a:) > 0 V a e f i ^ x G n a € n / i * . ■ Proposition 3.3.6 Let {/ia \ a G fi} C L(i?) 6e suc/i tfm£ A a€ nMa(0) > 0. Suppose that L is regular. Then V (3 G fi, yfy D Xlaeo ^a = {0} if and only if V x G ii, x 7^ 0, ( ^ H E a G ^ /MO*) = °Proof.

1£ Suppose that fi% {0}. Let ax; €G .R, ^ nC Ma # ,xx /^ 00.. Then a€fi YlaeCtp ^a=—{°}-

llp{x) A ( ^ (M/? n ] T Ma)(a:) >)(*) =-: M^W

Ma)(a:) >)(*) =-=00

since by Proposition 3.3.4. that 3.3.4,, /z£ fl lZ^aen/3 (^aGft Ma)* A*a)* = {0}. Conversely, suppose supposex that i%n n = T h e n x A . Then W3(x) (M/3 n EEaGfi/3 a e n 3 A*a)(a:) ^ X * ) - 0 °w( ) A ((Ea E a e€n^oc){x) ^ ^a)(a:) == 00 ssoo either ax £ MM£ £

■

or ° r *x£ $■ ( E ad2a€Q3 € n * / * " ^")*)*■■

Theorem 3.3.7 Let {fia \ a G fi}U{//} C L(/J) be such that AaG^Ma(0) > 0. Suppose that /J, = Ylaen ^^- SuPPose that L is regular. Then /i* = ®aenMa if and on/y if fi = 0aeftMaProof. By Proposition 3.3.4, \i* = X]aGftM<**- Hence the desired result follows from Proposition 3.3.6. ■ Proposition 3.3.8 Let {/j,a \ a G ft} C L(i?). Suppose that // a (0) = a V a G f l . T/ien ^/ v Ma* C ( ^ M a ) * . a€H

Proof.

aGfi

Let x G Xlaefi Ma*- Then x = Xlaefiu<* ^OI (Eaeft Ma) (*) (Eoten Ma)(a>)

some w

c* E Ma*- Thus

X— £ V{A(^a ee rn2 M = V{A /xaa (x aj i ^ aa )) |1x = a G f i X**} Z^aGfi = da = V{A^GfiMa(^a) | 0o= ==Eo J^aen aGfi /x a (v a ) 1 =: V{A, Gf2 Vvaa}} n/*a)(0). = ( E a e€r t M a X O ) . (Ea

Hence x G ( E a G n M a ) * . B Proposition 3.3.9 Let {fjba \ a G fi} C L(i?) fee 5^c/i £fta£ /z a (0) = a V a G fi. 7/ £/iere exists b G L, 6 ^ a, sucft £/ia£ b > V{ /z a (x) | x ^ /xa*} V a G fi, tfien SaGfi^ a * = (Z/aG^Ma)*-

92

Let x G E f t € n ^ ) * " T h e n v ( A a € f i M a ( ^ ) | x = T,aenx<*} = ( X ^ Q M C * ) ^ ) = a. Thus the supremum is attained when x — J2aen%a for some xa G /i a *. ■

Proof.

Corollary 3.3.10 Let L be a chain. Let {^a\& € ft} Q L(R) be a finite collection such that Ma(0) = a V a G ft. / / Ma is finite-valued V a G ft, #&en Eo <£Q Ha* = (£„t€ nMa)*' ■ Example 3.1.22 shows the need for the assumption of a chain in Corollary 3.3.10. Proposition 3.3.11 Let {^a | a G ft} be a collection of L-subrings of R such that AaGoMa(0) > 0. Suppose that L is regular. If ^cten ^ = ^aenHa, then Z^aGfi Ha* = ©aefiMa** Proof. The desired result follows from Proposition 3.3.6 and the fact that Hoc* Q (fla)*. ■ Proposition 3.3.12 Let {/j,a \ a G 0 } be a collection of L-subrings of R. Then fip J2*en V* = E ^ n / W * and ( E a € f i Ma) M/3 = E a e t t ^ ^ - ■ Proposition 3.3.13 Let fi, v G L(R) be such that /i(0) = v(0) = a. If X £ ^ //ix ,. } ^ ,!/„}) < a, (V{/i(x) 1| x j )) V V (V{i/(x) ( V W x ) || xX $ (V{MW M)

tfien (/xi/)* = M*^*Proof.

Clearly, (/xi/)(0) = a. Now x G (M^)* <^> (H")(X) z

z

^(^)) | x = J27=i Vi ii neN} = a^x Zi e v* (i = 1,..., n) o x e M*^*- •

= «^

= J27=i Vi i f°

r some

V

{A?=I(M(2/0A

V%

e

M* a n d

Theorem 3.3.14 Suppose that R has an identity. Let {Ma|^ £ ft} Q L(R). Suppose that ft is finite and 3 a G £\{0} such that let /jba(0) = a V a G ft. Suppose that aR = J2aen V* andV x G R, x ± 0, (fj,aHp)(x) = 0 V a, /? G ft, a ^ /3. TTiera V a 6 O, /i Q z's on L-ideal of R. If L is regular, then an = ©^e^MaProof. Let x,y G R. Then fxa(x-y) /xa is an L-subring of R. Now

> fia(x) Afj,a(y) and MaMa Q Ma since

: ORMa = ORMa = ((J^M/3)Ma ^ M/3)Ma == ^2^ Pen pen

M^M« M^Ma = = MaMa HOLHOLC Q/ i Ha a

by Proposition 3.3.12. Thus // a G L/(iJ). Now for a, /? G ft, a ^ /?, let 2/fc e /x*, z* G /x£ for A; = 1 , . . . , n. Then (MaM/3)( E L i 2/fc^) > ° s i n c e L

93

is regular. Thus Y2=\ Vkzk = 0 and so /z*/z£ = {0}. By Proposition 3.3.4, R = (aRy = E a e n A C Hence by [261, Theorem 30, p.175], R = © a e n /x*. Thus CLR = © a € ^/z a by Theorem 3.3.7. ■ Let Ra be a subring of R and let fi G LI{R). Let n'a denote the restriction of p{0)Ra/i to Ra. Then fi'a G LI(Ra). Theorem 3.3.15 Suppose that R has an identity. Let {Ra\a G ft} be a finite collection of ideals of R such that R = ®aenRa- Let /i G LI(R). Suppose that L is regular. Then there exists a collection, {fia | a G l l } , of L-ideals of R such that ji = 0 a 6 ^ / i a , and in fact fia = /i(0)^ a /i V a G i ] . If p(0) > V{/i(x) | x £ /z*}, then fi'a is an L-ideal of Ra and R/fi ~ (BaenRa/PaProof. By Theorem 3.3.7, fi(0)R = © ^ M O W Now /i(0)*/x - ( £ a G f i 4 MOJO/- = SaGfi^(°)-R«M by Proposition 3.3.12. Now/x(0)jR/i = /i and by Theorem 3.1.34, fi(0)Rafi is an L-ideal of i?. Since p(0)R — (Ba£np{fyRa and p(0)Ra/j, C p(0)Ra V a 6 O, we have that iz = ©aGn^(0)i?aM- Let /xa = /i(0)Rap V a E l ) . Now suppose that /i(0) > V{zi(x)|x ^ //*}. Let x G /i*\/xa*Then" n

Ma (a:) = V i A j L ^ / i ^ ) ^ ^ ) Ap(zi))

\ x = J ^ y ^ n G N}.

By Proposition 3.3.13, /xa* = (^(OJ^/z)* = (p(0)Ra)*p* = ^aM*- Since x ^ zi^ = Ra/i*, either there exists yi £ Ra or zi £ /i*. If yi £ Ra, then p(0)Ra(yi) = 0. If 2?i ^ /i*, then /i(z;) is bounded away from /i(0). Hence 6 > V{/i a (x) | x ^ /i a *} V a G f i , where 6 = V{/i(x) | x ^ /i*}. Thus by Propo­ sitions 3.3.9 and 3.3.11, /i* = ©aGftMc**- Hence by [261, Theorem 30, p.175] and Theorem 3.2.4, we have that R/fl

~ iJ//X* ~ 0 a G « ^ a / ^ a * -

©aGft^W/4

since /za* = (/4J*. ■ Proposition 3.3.16 Let {/ia | a G 11} U {/i} C L(.R). Suppose that L is regular. If \x = © a € fi/i a , then V x G / i * , x = SaGf* x « for uniQue x<* € Pa and fi(x) = A{/i a (x a ) | x = ] P x a , x a G /i* }. Proof. By Theorem 3.3.7, /x* = ®aefi/
94 Proof. Now x G (/xi/)* <£> (/ii/)(ar) > 0 <S> V{Af=1(^(7/i) A i/fe)) | x = n E * L i 2 / ^ > G N} > 0 ^ a: = ^ = 1 ^ ^ f o r s o m e Vi € ^*, ^ G i/* <^> x G Let /x and v be L-subsets of a nonempty set X. The Cartesian crossproduct ii x v is usually defined by V x, y G X, (/J, X i/)(a;, j/) = /JL(X) A I/(T/).

Definition 3.3.18 Let {Xa \ a G Q} be a collection of nonempty sets and let fia G LXa V a E f l . Define the Cartesian cross-product of the /aa byW xa G Xa, ( x a€fiMa)(^) = A a€ nMa(^a) 5 where x = (xa) and (xa) denotes an element of the Cartesian cross-product x a e ^ X a . Proposition 3.3.19 Let {Ra \ a G Q,} be a collection of commutative rings and let / i a be an L-subring (L-ideal) of RQ V a G ft. Then xaenfi>a is an Lsubring (L-ideal) of x a € ^ i 2 a , where by xaenRa we mean external direct sum of the Ra. Proof.

Let x — (xa), y = (ya) G xaenRa. (xct€Q/iCk)(x-y)

Then

=

A a G f i / ^ a ( ^ a - Vet)

> = =

haen(fia(xa) A fia(ya)) (AaGfi/ia^a)) A (A a € Q/X a (y a )) ( x a 6 f i / i a ) W A (xa€QfjLa)(y).

Also (XaeWa)(xy)

AaGfiMa(^a2/a)

>

>

>

f A a G fi(^ a (rr a ) A A*a(3/a)) if A*a i s a*1 L-subring of i? a ; A aG n((/x a (x Q ) V fia(ya)) if ^a is an L-ideal of Ra; f (Aa€fi/^a(^a)) A (AaenfJ,a(y*)) if /xa is an L-subring of Ra; (Aaefi^a(^a)) V (A a€ n/i 0 (2/ a )) if /xa is an L-ideal of Ra; f (x a efi^a)(x) A (x a € n/z a )(j/) if ^ a is an L-subring of i? a ; ( x a € n ^ a ) ( x ) V (xaecifjLa)(y) if // a is an L-ideal of Ra. ■

Let {Ra\ct G fi} be a collection of commutative rings. Let / i a be an Lsubring (L-ideal) of Ra V a G Q. Assume ^ a ( 0 ) = fJ
95 Proposition 3.3.21 Let {jda \ a G 0 } C L(R). Assume /i a (0) = fip(0) Va, /? G ft. Suppose that L is regular. IfJ^aen I1* ~ ®aenMa, then ^2a€Q fJ>a — XcxenV* on (£ aGn Mc*)*Proof. The desired result follows from Proposition 3.3.16 and the fact that since x = Xlaefi Xa ^ or U I U C l u e xa € Ma*5 we can think of x = (xa), where all but a finite number of the xa are 0. I Proposition 3.3.22 Let {i? a |o; G Q.} be a collection of commutative rings. Let fia be an L-subring (L-ideal) of Ra V a G fi. Assume na(0) = fJ>p(0) = a Va,/3 G fi. Suppose that L is regular. Then in x a € Q.R a , ®/3en x Q 6 n/i(f' =Proof.

x

ae^Ma-

Now

(©/36ft XaGfi f^a^ix)

(/3) )(z/3) || xX — = = A{(x A{(xaenc*€ftMa flJW)(xp) = 2^/3efi Y,fenxfaxxPi f*xP == ((XXPl) Pl) >> x/3 = 0 if /? 7^ 7, x^ G Ay, 7 G ft} = 0 if p ^ 7, x^ 77 ^/?77 = eH.nen} = A{A a enH^ixpa) 1 a:= I2penxP>xP = (*/3a)} A{^ a a(x = A{)U (xaa) | x = = (x ( s aa ) .,,xxaa G #jR ,a GG ft}, ft}, a ,aa

where the first equality holds from Proposition 3.3.16 and the last equality holds since for a ^ /?, /1Qt (xpa) = a since x^ = 0. ■ In Proposition 3.3.22 with ft = {1, 2}, we have that in the external direct sum Ri xi?2, the (internal) L-direct sum (/xi x a U ) ® (a{o> x ^2) equals /ii x ^2, where a = ^i(O) = /i2(0). This motivates the following definition. Definition 3.3.23 Let {Ra\ct G ft} be a collection of commutative rings and let fia be an L-subring of Ra V a G ft. Assume /xa(0) = VpiO) Va,/3 G ft. T/ien t/ie Cartesian cross-product XaenVa is called the complete L-direct sum of the fia. Definition 3.3.24 Let {Ra\ct G ft} be a collection of commutative rings and let fia be an L-subring of Ra V a G ft. Assume fia(0) = ^/?(0) Va,/? Gft.Le£ SaGfi ^ Q denote the weak direct sum of the Ra. Define the L-subset Y^en Ma °f!2%nRoc by\f x = (xa) G T,®eQ R*, ( E a e n ^ l W = Xaefi^^a). T^n a 5 SaGfi ^ ^ ca/Zed £/ie (external) weak L-direct sum of the fia. Proposition 3.3.25 Let {Ra\a Eft} be a collection of commutative rings and let fia be an L-subring of Ra V a G ft. Assume ^ a (0) = /^(O) = a Va,/3 G ft. V a, /3 G ft, Ze£ i/4 = ^ a if (3 ^ a and z/a = a/Q} if P = a-

96 (l)Vx (1)VX:

= (x (zaQ)) e xaen a €R n ai,? a , ifx^O,

then V 0/? €e fi,

) /3) ({xxaeQnrf? i aennvW)(x) xQ€nl/(f))(a;)= ; =00; € n Mrix /ien ((2)Vx 2 j V l : = {x a) e (*<*> G £E® a ee n Re, #<*> ifx^O, */ * + °. ithen V aa e6 fi,

(£/#>" £><«)(*) = o. Proof.

(1)

(Xaefi/if nn x a^e n ^ }X) (xx)) (x,

= = = =

)

((x. X ^»en/4f Q X X ^ )(a:)A(> X X ) A ( 'xa ^e ^n i^ ^X) *^)) ^ ( x a ) ) A ( A a e n ^ a ^ H ^ *)) 0

since xa ^ 0 for some x a and for those X& ^ 0, v&\xa)

= 0 if /? = a and

}

^ (*a)=0if/3^a. (2) The desired result follows by an argument similar to that used in (1). H

3.4

Maximal L-Ideals and Irreducible L-Ideals

Definition 3.4.1 Let xx G LI(R). Then fj, is called a maximal L-ideal of R, if /j, is a maximal element in the set of all non-constant L-ideals of R under pointwise partial ordering. Definition 3.4.2 a G L\{1} is called a maximal exist c G L\{1} such that a < c < 1.

element

if there does not

Theorem 3.4.3 Let /x G LR. Then \i is a maximal L-ideal of R if and only if xx can be expressed as /x = 1 ^ Ua#, where xx* 25 a maximal ideal of R and a is a maximal element of L\{1}. Proof. Let xx = 1 ^ UOR, where xx* is a maximal ideal of R and a is a maximal element of L\{1}. Clearly, xx is non-constant. Let v be a non-constant L-ideal of R such that /x C v. Then /x* C z/* and v* ^ R. Since /x* is a maximal ideal of i?, /x* = v*. Now V x $. xx*, a = xx(x) < v{ x) < 1 and hence u(x) — a V x £ xx*. Thus n = v. Hence xx is a maximal L-ideal of R. Conversely, suppose xx is a maximal L-ideal of R. First we show that /x(0) = 1. Let v G LI(R) be defined by i/(x) = 1 if x G xx* and i/(x) = xx(x) if x ^ xx*. Then xx C XA Thus either xx = v or z/ is constant. In either case, /x(0) = 1. We now show that |/x(i?)| = 2. Clearly, |/x(i?)| > 2. Let x, xy G i? be such that \i{x) < 1 and xx(xy) < 1. Let xx(x) = a and XX(T/) = b. Let z/ G L H be defined by v(z) = xx(z) Va\f z € R. Now

97 i/(0) = /i(0) V a = /z(0) and v(x) = /J,(X) V a = /J>(x)- Hence v is non-constant. Since L is distributive, v G LI(R). Clearly, /i C v. Since /i is a maximal L-ideal and z/ is nonconstant, fx = u. Hence V>(y) = v(y) = !*(y) V a>a = fi(x). Similarly, fi(x) > fi(y). Thus /J,(X) = n(y). Hence \n(R)\ = 2. Let /x(i?) = {1, a}, where a G L\{1}. Then zz = 1 ^ U a^. Clearly, zz* ^ R. Suppose 3 an ideal B of R such that fi* C B C R. Then z/ = 1 B U CLR G LI(R), V is non-constant and /x C v C 1#, which is a contradiction. Hence zi* is a maximal ideal of R. Finally, if a is not a maximal element of L\{1}, then there exists c G L such that a < c < 1. Then 1 ^ U CR G L7(.R) and /z < 1 ^ U CR < 1^, a contradiction. I By Theorem 3.4.3, the existence of a maximal L-ideal of R requires the existence of a maximal ideal of R and the existence of a maximal element of L\{1}. However, there exist complete Hey ting algebras L that do not satisfy this requirement. For example, the lattice [0,1] is such an algebra. Definition 3.4.4 Let fi G LI(R). Then /z is called a generalized maximal L-ideal of R if for any L-ideal v of' R, if/z C */, then either fi* = v* or v — 1R. Theorem 3.4.5 Let /z G L(R). Then /i is a generalized maximal L-ideal of R if and only if fi can be expressed as fi = l^m U a ^ , where /z* is a maximal ideal of R and a G L\{1}. Proof. Let ji = 1 ^ U OR, where /i* is a maximal ideal of R and a € L\{1}. Then clearly, \x is a generalized maximal L-ideal of R. Conversely, let /z be a generalized maximal L-ideal of R. Then n* is a maximal L-ideal of R, |/x(i2)| > 2 and as in the proof of Theorem 3.4.3, fj,(0) — 1. Let x, y £ /z*. Let fi(x) = c and fi(y) = d. Then /z* C \ic and /z* C /z
98 Theorem 3.4.8 Assume that R is with identity and contains at least two elements and L is a chain. Let /i be a proper ideal of R. Then there exists a generalized maximal ideal v of R such that /JL C V. Proof. Let M = {a | a G L, fia ^ R}. Clearly, 1 G M and thus M ^ U . Let A = UaGM^a- Then A is a proper ideal of R and 1 G R\A. Thus there exists a maximal ideal B of R such that AC. B. Now let y G R\B and b = /J>(y). Then b £ M and thus b < 1. Since b £ M, lib = R and hence /x(:r) > b V x e R. Accordingly, /J,(X) = b V x € R\B since 5 is maximal. Let ^ = l^U 6R. Then v is a generalized maximal L-ideal of R such that / i C i / . | Definition 3.4.9 ^4n L-ideal ^ of R is said to be reducible if there exist fi, v G LI(R) such that fi D £, v D £ and ^ fl v = £; £ zs 5aid £0 6e irreducible otherwise. By definition, maximal L-ideals are irreducible. If L is a chain, then gen­ eralized maximal L-ideals are also irreducible. Definition 3.4.10 Let c G £\{1}. Then c is called irreducible L,a > c and b>c=>aAb^c.

if\fa,b

G

Theorem 3.4.11 Assume R is with identity. Let £ = 1Q U CR, where C is an irreducible proper ideal of R and c is a irreducible element of L. Then £ is an irreducible L-ideal of R. Proof. Clearly, £ G LI(R). Suppose there exist /x, v G LI(R) such that /x D £, v D £, and / i f l i / = f. Now 3 y, z e R such that /i(?/) > £(?/) and 1/(2) > £(z). Thus y, z £ C, and hence £(2/) = £(z) = c. Let A = (CU{y}) and £ = (C U {*}). Then C C A and C C B. Let r e An B. Then r = u + w/ = w + xz for some u, w £ C and x,v E R. Thus vy — xz = w — u E C. Hence £(i>2/ — xz) = £(w — iz) = 1. Now n{vy — xz) > £(vy — xz) = 1. Hence u(yy — xz) = 1. Thus u(vy) = fi(vy — xz + xz) > fi(vy — xz) A fi(xz) = /J,(XZ). Similarly, fi(xz) > fi(vy). Hence /i(vy) = fi(xz) and similarly v(vy) — v{xz). Thus \x(xz) = \x{yy) > fi(y) > £(y) = c and v(xz) = v(z) > £(z) = c. Now £{xz) = /i(xz) A v(xz) ^ c since c is irreducible. Thus £(xz) = 1 and hence xz G C. Thus r = w +xz G C. Hence AnB C C. Since C C A f l S , i H 5 = C, which is a contradiction. Thus £ is irreducible. ■ Theorem 3.4.12 Let L be a chain and let £ be an irreducible proper L-ideal of R. Then £* is an irreducible ideal of R and £ = 1^ UCR for some c G £\{1}.

99

Proof. First we show that f(0) = 1. Suppose f(0) < 1. Let /x,i/ e LR be defined as follows: ^

\ f(x)

ifx^*;

*/(*) = f (0) V x G i ? . Then /a, i/ G £ / ( # ) , f = ^ H i/ and f C /x C 1*, f C i/ C 1R. This is a contradiction. Hence £(0) = 1. Clearly, \£(R)\ > 2. Suppose \{(R)\ > 3. Then 3 z,w G R such that 1 > £(z) > l(w) = c. Let £(z) = a. Let \i,v G LR be such that /i(x) = 1 if x G fa, /x(ar) = f(x) if x £ £>a, v{x) = £(x) if x G £ a and i/(x) = a if x £ fa. Then since L is a chain /x,i/G ZJ(iJ). If x G ffl, then /x(x) A i/(s) = 1 A f (x) = f (x) and if x ^ f0, then /x(x) A i/(x) = f (x) A a = £(x). Hence \i n i/ = f. Since jx(z) = 1 > a = f (z) and i/(w) = a > c = f(w), / i D { and ^ D f, which is a contradiction. Hence If (i2)| = 2. Thus f = 1^ u CH for some c G £\{1}- ^ £* i s n o t irreducible, then there exist ideals A and B of R such that A D (*, B D ^ and AnB = £*. Hence I A U C * , l ^ U c * G L/(i?), U U C R D f, l B U c H D f, and ( U U c H ) n ( l B U c H ) = f, which is a contradiction. Hence f * is irreducible. ■ Theorem 3.4.13 Let R be a ring with identity and let L be a chain. Let £ G LR. Then f is an irreducible L-ideal of R if and only if £(0) = 1, f* is an irreducible ideal of R and \£(R)\ = 2. Proof.

The proof follows from Theorems 3.4.11 and 3.4.12. ■

Theorem 3.4.14 Let R be with identity, S a ring, and let f be an epimorphism of R onto S. (1) Iff is an irreducible L-ideal of R and f is f-invariant, then /(£) is an irreducible L-ideal of S. (2) If L is completely distributive and n is an irreducible L-ideal of S, then f~1(r]) is an irreducible L-ideal of R. ■ Theorem 3.4.15 Let R be a Noetherian ring with identity. Let f G LI(R) with f (ii) = {1, c}, where c is an irreducible element. Then f can be expressed as the intersection of finitely many irreducible L-ideals of R. Proof. Clearly, f* ^ R. Since R is a Noetherian ring with identity, there exist finitely many irreducible proper ideals of R, say Ci, C 2 , . . . , C n , such that f* = njLi(?». Let f; = Id U cR for i = 1,2,..., n. Then & is an irreducible L-ideal o f i ? V l < z < n a n d f = n? =1 &. ■

100

3.5

Prime L-Ideals and Semiprime L-Ideals

Definition 3.5.1 Let £ G LI(R). Then £ is called a prime L-ideal of R if £ is non-constant and \i o v C £, //, v G LI(R) => either \i C £ or z/ C £. Theorem 3.5.2 Let £ G LI(R). Then £ is a prime L-ideal of R if and only if £ is nonconstant and [iv C £, JJL, V G LI(R) => either /i C £ or z/ C £. Proof.

The proof follows from Corollary 3.1.19. I

Theorem 3.5.3 If £ is a prime L-ideal of R, then £* is a prime ideal of R. Proof. Let £ be a prime L-ideal of R. Now let A and B be ideals of R and AB C £*. Then 1^, 1 B G LI{R) and 1A ° 1 B £ l£* ^ £• Since £ is a prime L-ideal of R, either 1 A £ £ O I , 1 B S £ > a n d hence either A C £* or JB C £*. ■ Definition 3.5.4 Lei c G L\{1}. 77ien c zs ca/Zed a prime Va, 6 G L , a A 6 < c = > a < c or b < c.

element

of L if

Theorem 3.5.5 Let £ G L^. TTien £ is a prime L-ideal of R if and only if £(0) = 1, £* is a prime ideal of R, £(R) = {l,c}, where c is a prime element in L. Proof. Suppose £ is a prime L-ideal of R. By Theorem 3.5.3, £* is a prime ideal of R. We now show that £(0) = 1. Suppose that £(0) < 1. Since £ is nonconstant, there exists x G R such that £(#) < £(0). Let fi, v G LR be defined by fi(x) = 1 if x G £* and n(x) = 0 if x £ £*; z^(x) = £(0) V x G i?. Then /x, i/ e L/(iJ) and /z o i/ C £. Since /z(0) = 1 > £(0) and i/(x) = £(0) > £(x) for x ^ £*, fi 2 £ a n < i v % £• This is a contradiction since £ is a prime L-ideal. Hence £(0) = 1. We now show that |£(#)| = 2. Since 1 G £(#) and £ is nonconstant, |£(i?)| > 2. Let x, y G i2\£* and let £(x) = c. Then C(x) = (xc) Q £• Clearly, l(a;>, c# G L/(JR) and l< x) g £. Now l (x> o cR C l(x) H c# = c^) C £. Since £ is a prime L-ideal and 1 ^ %. £, CR C £. Thus £(x) = c = CR(?/) < £(T/). Similarly, £(?/) < £(x). Hence £(x) = £(?/) = c. Thus |£(i?)| = 2 and hence £ = 1^ UcR, where £* is a prime ideal of R and c G L\{1}. We now show that c is a prime element of L. Suppose that c is not a prime element of L. Then there exist a, b G L such that a ^ c, b j£ c, and a Ab < c. Thus a^, 6# G LJ(i2), a# £ £, and &# g £. However, aRobR = (aA b)R C £. This contradicts the fact that £ is a prime L-ideal of R. Hence c is a prime element of L. Conversely, suppose £ satisfies the given conditions. Then £ is not constant. Assume that there exist ix, v G LI(R) such that \xo v C £, but \x g £ and ^ 2 £. Then there exist x, y e R such that /i(x) ^ £(x) and v(y) ^ £(7/). Then

101

£0*0 = £(y) = c, x, y G R\£*, fi(x) ^ c, and i/(y) £ c. Since x j G i?\£* and £* is a prime ideal of i?, xy £ £*. Then £(XT/) = c. Now /z(x) A v(y) < (fj, o v){xy) < £(xy) = c. Since c is a prime element of L, either /i(x) < c o r v(y) < c, a contradiction Hence £ is a prime L-ideal of R. ■ Theorem 3.5.6 Let £ G LR. Then £ is a prime L-ideal of R if and only if £ is nonconstant and xa oy 6 C £, x, y G i?, a, 6 G L, £/ien either x a C £ or ?/& C £. Proof. Suppose that £ is nonconstant and xaoyh C £, x, y G i2, a, 6 G L. Then either xa C £ or y& C f. Let /z, z/ G LI(R) and ^oi/ C £. Then xaoyh C f, V x a C / i and V 1/5 C i/. Now it is not possible for x a o ?/& C £, ^ c o i;d C £ such that yb £ £ and u c £ £, where x a , uc C JJ, and 2/5, ^ C i / since ucoyh C £. Hence x a C £ V x a C /i or ?/& C £ V 3/5 C ZA Hence either /i C £ or z/ C £. Thus £ is a prime L-ideal of R. The converse is trivial. ■ Theorem 3.5.7 Let R have an identity. Then every maximal L-ideal of R is a prime L-ideal of R. M Theorem 3.5.8 Let £ be a prime L-ideal of R. Then for any x, y G R, either t(xy) = £(x) or £(xy) = £(y). Proof.

The proof follows from Theorem 3.5.5. ■

Theorem 3.5.9 Let f be a homomorphism of R onto a ring S. Let /i be an f-invariant L-ideal of R such that fi has the sup property. If n* is a prime ideal R, then f(f~t*) is a prime ideal of S. Proof. By Lemma 3.2.5(3), /(^*) = /(^)*- Let u, v G S be such that uv G /(/i*) = /(M)*- There exist x,y G R such that /(x) = u and f{y) = v. Then f(xy) = uv G /(^)*- Hence

/(/i)(0) = f(Mf{xy))

= V{fi(z) I /(*) = /(*»)} = MO).

Thus there exists w G R such that fi(w) — /J(0) and f(w) = f{xy). But then n(xy) = fi(w) since /i is /-invariant. Hence xy G [i*. Thus either x G //* or y G /x*. This implies that either / ( x ) G /(/x*) or /(?/) G / ( M * ) - Hence /(//*) is a prime ideal of 5. ■ Corollary 3.5.10 Let f be a homomorphism of R onto a ring S. Let /x be an f-invariant L-ideal of R such that n(R) is finite. If /i* is a prime ideal of R, then /(/z*) = /(/1)*. ■

102

Theorem 3.5.11 Let f be a homomorphism of R onto a ring S. Let fi be an f-invariant L-ideal of R. If fi is a prime L-ideal of R, then f(ji) is a prime L-ideal of S. Proof. By Theorem 3.5.5, ^(0) = 1, fi(R) = {1, a}, where 0 < a < 1, and /z* is a prime ideal of R. By Lemma 3.2.5(3) and Corollary 3.5.10, f (H)(0) = 1 and /(//)* is a prime ideal of 5. Thus it suffices to show that /J,(R) = f(/j.)(S). Let x G R be such that y,(x) — a. Then

/(/*)(/(*)) = vM») I /(») = /(*)} = o since /x is /-invariant. Hence fi(R) = f(fi)(S).

I

We now give an example to show that if fi is not /-invariant, then Theorem 3.5.11 need not be true. Example 3.5.12 Let F be any field. Let R = F[x,y], where x and y are algebraically independent over F and S = F[x,y]/Q, where Q = (y2) • Let P = (x). Then P is a prime ideal of R. Let W = (P + Q)/Q. Since (y + Q)2 G W and y + Q £W,W is not a prime ideal of S. Let f : R —► S be the canonical homomorphism. Let fi be an L-subset of R such that for all x G R, ji(x) = 1 if x G P and ^x(x) = 0 if x £ P. Then \i is a prime L-ideal of R. Since /(//)* = f(fJ>*) = f(P) — W and W is not a prime ideal of 5, it follows that f(/j) is not a prime L-ideal of S. Note that fi is not f-invariant. For, let a — x and b = x -\- y2. Then a + Q = b + Q, i.e., f(a) = f(b). But /i(a) = 1 and fi(b) = 0. Proposition 3.5.13 Let f be a homomorphism of R onto a ring S. Let v G LI(S). Then (1) if v* is a prime ideal of S, then f~1(u)^ is a prime ideal of R; (2) if v is a prime L-ideal of 5, then f~x(v) is a prime L-ideal of R. Proof. (1) By Theorem 3.2.6(2), / " V ) * = / " V * ) - Let x,y G R be such that xy G f~l(v*). Then f(x)f(y) = f(xy) G i/*. Hence either f(x) G i/* or f(y) G v*. Thus either x G f~x(v*) or y G f~x(v*). Hence f~x(v)* is a prime ideal of R. (2) Now / _ 1 ( ^ ) ( 0 ) = v(f(0)) = 1 and / _ 1 ( ^ ) * is a prime ideal of R by (1). Now i/(S) = {1, a}, where a G L\{1} is a prime element. Let y e S be such that v(y) = a. Then there exists x G R such that f(x) = y and V such y and x, / - V X * ) = "(/(*)) = a- T h u s / -1 (")(#) = KS). H e n c e / ~ V ) is a prime L-ideal of i?. ■ Definition 3.5.14 Let v G LI(R). Then v is called semiprime constant and if ji2 C v, \x G LI(R), then fj, C v.

if v is non-

103

By definition, a prime L-ideal is semiprime. Theorem 3.5.15 Let v G LI(R). Then v is semiprime if and only if v is nonconstant and /i/iCi/, fi e LI(R) => /x C v. ■ Theorem 3.5.16 Let v be a semiprime L-ideal of R. Then (1) /i(n) Ciy, fie LI(R), n G N ^ / i C i / , (2) [in C i/, fi e LI(R), ne N => xx C v. Proof. (1) If n = 1 or n = 2, the clearly xx /J, C v. Suppose that V k < n, ^ C i/, /x G L/(i?) => xx C */. Let xx(2): Let x G # and a G L be such that (x a ) 2 C v. Then v (#a) ^ - Thus (:ra) o (x a ) = (a; 2 ) C XA Hence (xa) C xv and so x a C XA (2)=>(3): Suppose that fi e LR and zx2 C XA Let x e R. Then zx(x) = /x(x) A/J,(x) < /J?(x2) < v(x2). Hence (xa)2 = (x2)a C i/, where a = /x(x). Thus x a C i / and so ji{x) C z/(:r). Hence / i C i / . (3)=»(1): Follows by definition. ■ Theorem 3.5.19 Let v be a semiprime L-ideal of R. Then v(x2) — v(x) V xeR. Proof. Let x G R and a = i/(x 2 ). Then (a; a ) 2 = (x2)a Q v. Hence xa C v and so v(x) > a. Thus v(x2) — a < v(x) < v(x2). Hence v(x2) = v(x). ■ Theorem 3.5.20 Let v be a proper L-ideal of R. Then for each a G ^(L), either va = R or va is a semiprime ideal of R if and only if v is a semiprime L-ideal of R. ■ Theorem 3.5.21 Let v G LR. Suppose that v = \Um U &#, where v* is a semiprime ideal of R and 6 ^ 1 . Then v is a semiprime L-ideal of R.

104

Proof. Clearly, v is nonconstant. Let fi G LI(R) be such that ji2 C v. If x G ^*, then /i(x) < i/(x). When x G i?\^*, since z/* is a semiprime ideal of i?, it follows that xx ^ z/*. Thus we have fj,(x) A ^x(x) < /J?(x2) < v(x2) = b. Hence fi(x) < b = v(x). This shows that \i C v. Therefore, v is a semiprime L-ideal of R. ■ Example 3.5.22 Suppose C is a prime ideal of R and L = {l,a, 6,0}, w/iere a and b are not comparable. Then l c G LR is a semiprime L-ideal of R, but not a prime L-ideal of R. The reason is that although C is a prime ideal of i?, the zero element of L is not a prime element. Example 3.5.23 Suppose that R is the ring of integers. Let L = [0,1] and a, 6 G L\{1} be such that a > b. Define £ G LR as follows: 1 £(#) = ^ a b

ifx = 0 if x is a nonzero even integer otherwise

\fx G R. Then £ is a semiprime L-ideal of R. Hence we see that the cardinality of the image of a semiprime L-ideal of R may be greater than 2.

3.6

Radicals of L-Ideals

Let £ G LI(R) and let V^ be the family of all prime L-ideals fi of R such that £ C ji and £* C /x*. Definition 3.6.1 Let £ G LI{R). fined by

^ = (

The L-radical

H{HMG^}

1R

of £, denoted by y/£, is de­

ifPt* ifPt = (l>-

Theorem 3.6.2 Let £ be a constant L-ideal of R. Then y/% = 1R. Proof. Let /x G V$. Then £* C ^ . Since £ is constant, £* = R. Hence fi* = R. Thus p£ =
(1) VT* c (v^)*; (2) (V?)(*) = 1 V x e (v?)*; W 4 c v^.

105

Proof. (1) Let x G y/&. Then x G P for all prime ideals P of R such that £* C P. Let fieV^. Then x G /x* and thus /x(x) = 1. Hence (\/£)(z) = 1. Thus x G (V?)*- Hence v ^ C (y^)*. (2) and (3) are immediate. ■ The following example shows that in general, v^T ^ (V£)*. Example 3.6.4 Let R = Z 6e t/ie nnp o/ integers and let L = [0,1]. Let P n = (2 n ) , t/ie idea/ generated by 2 n , and Zet P = flP n . Let £ € LR be such that

t(x) ^ '

=

f

X

^

\ 1 — J"

G P

if x £ P, and ix is the smallest i such that x £ Pi.

Then £ G LI(R) and y/^ now show that if/j,e P^, of P , ^(0) = 1 and fi(R) a < 1 — ;£. Suppose that Then

= {0} is a prime ideal of R. Thus y/^ = £*. We then /z* = (2). Let fi eV^. Then ^* is a prime ideal = {1, a}, w/iere 0 < a < 1. Let n e R be such that /z* = {0} or /i* = (p) , p an odd prime. Let x = 2n.

H(x) = a < 1 - - < 1 - — — = £(x), n n -\-1 a contradiction. Hence /J,* = (2) . Thus (V?)*

ffence v ^

= (H{AX|/X G P J ) *

= n{^*

I Me PJ

=

(2).

(V£)*.

In Example 3.6.4, we note that V{£(x) | x £ ^ }

= £(0).

Theorem 3.6.5 Let £ be a nonconstant L-ideal of R and let V{£(x)|x £ y/^} —a. If a < £(0) and a is prime, then yft^ — (Vf)*. Proof. Let P ^ R be a prime ideal of R such that £* C P. Then v ^ C P. Let \i G LR be such that fi(x) = 1 if x 6 P and /z(:r) = a if a: ^ P. Then / I E P ^ . Let x 6 (>/?)*• Then ( v ^ W = 1 a n ( i t n u s M^) = 1- Hence x G /x* = P Thus for any prime ideal P such that £* C P, x G P Hence x G V?* • Thus

v/£ =(>/?)..■ Corollary 3.6.6 Let fi be a nonconstant L-ideal of R. If {£(x) \ x £ y/£^} is finite, then y^T = (\/£)*- " The following example shows that the condition a < £(0) in Theorem 3.6.5 is not necessary for y/£^ = (\/£)*.

106

Example 3.6.7 Let L = [0,1]. Let R be a ring which has an infinite strictly descending chain of prime ideals Pi D Pi D P3 D Then C\Pn = P is a prime ideal of R. Let £ G LR be such that

c(x) = f ^

'

X

1 1— j -

ifxeP if x £ P, and ix is the smallest i such that x £ Pi.

Then £ G LI(R) and £* = P is a prime ideal of R. Thus y/T~^ = £*. Let fin G LR be such that fin(x) = 1 if x G Pn and jin{x) = 1 — ^ if x £ pn. Then jin G V^ and /in5te = Pn. Now y/% C fl/i n and (fl/i n )* = Dfin^ = £*. #ence V ^ = (\/£)*Theorem 3.6.8 Let \i,v G LJ(iJ) and Ze£ /x(0) = 1 = v(0). Then x/fTTVu =

Proof. Now /i D fiDu and /x* D /i* D v* = (/i fl ^)*. Thus y//7 D y/jTTTP. Similarly, y ^ 2 y/iTTTu. Hence y ^ ^ \ / ^ 2 yfjTTTv. Let A G P^ni/- Then /i n v C A. Hence z i o ^ C / x n ^ C A . Thus either /x C A or v C A, say, xx C A. Since /x(0) = 1, zx* C A*. Thus A 6 ? M and so 7 ^ C 7 ^ U 7V Thus y/Jj^TP 2 v ^ ^ V^- Hence y/jTrTv = y/JL n v ^ . B Theorem 3.6.9 Le£ /x,r/ G LI(R) y^xfl v^-

and let xx(0) = 1 = i/(0). T/ien y/JSI/ =

Proof. Clearly, (xxz/)(0) = 1 = (/x D ^)(0). Hence (/xi/)* C (/x fl v)* since /j,v C fi n v. Thus ^//x^ C y/JTrTP. Let A be a prime L-ideal of i? such that fiv C A and (/i^)* C A*. Then either /x C A or z/ C A. Thus \x fl ^ C A. Since (/x H i/)(0) = 1, (/x fl i/)# C A*. Thus vT^T^ C yfiw. ■ Corollary 3.6.10 Let £ G L/(ii) ana7 f (0) = 1. Then V ^ " = V? Vn G N. ■ Theorem 3.6.11 £e* £ G L/(i?). T/ien v V f = \/lProof. If £ is constant, then clearly V v f = V€- Let £ be nonconstant. Let /x G Vt:. Then V^ Q V» Let a; G (v£)*. Then (>/£)(*) = 1. Now 1 = (V£)0*0 < M x )- H e n c e /*(ar) = 1 = /i(0). Thus (V?)* C zx*. Hence \i G P / j . Thus V<: C P /^. Clearly, P /£ C 7^. Hence ^ v f = \fl- ■ Definition 3.6.12 Suppose that R has an identity. Let {/xa | a G fi} 6e a collection of finite-valued L-ideals of R such that /xa(0) = ^ ( 0 ) Va,/3 G fi. 7%en tfie / i a are said to 6e pairwise comaximal if /xa ^ a/?, a = /i(0), V a E f l and /xa + ftp = CLR Va, / 3 € f i , a ^ / 3 .

107

Lemma 3.6.13 Let L be a chain. Suppose that R has an identity. Let {/j,a | a G H} be a collection of finite-valued L-ideals of R. Then the /j,a are pairwise comaximal if and only if the Ma* are pairwise comaximal and = M/3(0)Va , / ? € Mc»(0) =

fi.

Proof. Suppose that ji^ are pairwise comaximal. Then Ma(0) = fJ>p(0) VOJ, (3 G ft by definition. Also, V x G R, a = (iia + fj,p)(x) = W{fia(xa) Afip(xp)\x = x ct -f ^/3}> where a = Ma(0). Since Ma and M/3 are finite-valued, there exists ^OL € /xa*, x/3 G M/3* such that x = xa + xp and so fia(xa) = a = fip(xp). Thus # = Ma* + M/3*- Since Ma 7^ a i? 7^ M/3 5 Ma* ^ R ^ M/3*- Conversely, suppose that the //«* are pairwise comaximal. Then i? = M«* + M/3* a n d so V a; G i?, x = xa + xp for some x a G Ma*, #/3 € M/3*- Thus (fia + Hp){x) = a = aR(x). Hence Ma + M£ = a i*- Since Ma* ¥" Ry^ M/3*> V R = y^MoT + yffxpl O R = (V7^)* + (y/I1?)* b y Corollary 3.6.6 <£> aR = y/jjfc + y/jlp. (2) Now aR = v + fiaVaeft. Thus R ==J^[ {y + Ma) C z/ + n Y\ ^° Q a . IK" =Y[n°*= a

aR ==

' + f^oc) C 2/ +

a€fi a6fi

aa€H €H

POLC OjfJ.R

a€£7 a€ft

Hence a H = ^ + FLen ^ a * Since flaefi Ma Q H a€ nMa, aj* = ^ + HaGnMa(3) By Theorem 3.1.37, Ma H M/3 HM/3

= = = = Q =

c

HM/3) (Ma ++ M/3)(Ma M/3)(MaHM/3) Ma (Ma: flM/3)+M/3(Ma Ma(Ma H M/3) + M/?(Ma HM/3) H M/?) M/3Ma MaM/3 + HpV>cx MaM/3-

By Theorem 3.1.36, MaM/3 Q Ma H M/3- Thus MaM/3 = Ma H M/3- The remainder of the proof follows as in [261, Theorem 31, p. 177]. ■

108

3.7

Primary L-Ideals

Definition 3.7.1 Let £ E LI(R). Then £ is called a primary L-ideal of R if £ is nonconstant and for any //, v E LI(R), / i o i / C ^ implies that \x C £ or Theorem 3.7.2 Le£ £ E L^ 6e snc/i £/ia£ £(0) = 1, £* is a primary ideal of R, and £(R) = { l , a } , u>/iere a E £\{1} and a is a prime element of L. Then £ is a primary L-ideal of R. Proof. Since £* is an ideal of i?, £ E LI(R). Clearly, £ is nonconstant. Let fi, v E LI(R) and let /zoi/ C £. Suppose that /x g £ and v <£. >/£. Then 3 u, v E i£ such that fi(u) jt £(u) and u(v) ^ >/?(*>) • Since £(0) = 1 = \/£(0), u ^ £* and v £ (V?)*- Since yC" C (Vt)*> v ^ >/£*• Hence m; ^ £*. Thus f (tzv) = a. Clearly, //(it) ^ a and v{y) j£ a and hence since a is prime, /z(iz) A u(v) j£ a. Also a = £(iw) > (fio v)(uv) > /JL(U) A ^(v), which is a contradiction. Hence £ is a primary L-ideal of R. ■ Corollary 3.7.3 If Q is a primary ideal of R and L is regular, then \Q is a primary L-ideal of R. I T h e o r e m 3.7.4 Let £ be a primary L-ideal of R such that £(R) = { l , a } , where a E L\{1} and a is prime. Then £* is a primary ideal of R. Proof. Let S and T be ideals of R such that ST C £*. Let /i = ls and v = IT- Then / i o i / C ( . Since £ is primary, either \i C £ or z/ C ^/J. If /x C £, then 5 C £*. Let v C V£. Now V£T = (\/|)*- Thus T C (V£)* = VET- Hence £* is a primary ideal of R. ■ Corollary 3.7.5 Let Q be an ideal of R such that 1Q is a primary L-ideal of R and let L be regular. Then Q is a primary ideal of R. ■ R e m a r k 3.7.6 (1) If £ is a constant L-ideal of R, then ^ = 1R. Thus such an L-ideal is always primary. (2) Let £ be a nonconstant L-ideal of R. If \/£ = 1#, then £ is primary. In this case £(i?) may be finite or infinite and £(0) may or may not be equal to 1. However, if R has identity and L is a dense chain, then we will prove that £(0) = 1, £(i?) = {1, a}, where a is a prime element a < 1, and £* is a primary ideal of R. (3) If R is a ring with identity, then every proper ideal of R is contained in a maximal ideal of R. Further, since R is with identity, every maximal ideal of R is prime. Thus every proper ideal A of R is contained in a prime ideal P of R such that P ^ R.

109 L e m m a 3 . 7 . 7 Let R be with identity and let L be a dense chain. nonconstant L-ideal of R. Then 3 / i G ? ^ .

Let £ be a

Proof. Since £ is not constant, 3 x G R such t h a t £(x) ^ £(0). Let £(x) < a < £(0). T h e n £ a ^ R and £ a is an ideal of R. Now 3 a prime ideal P of R such t h a t £ a C P C P . Let /J, e LR be such t h a t /x(x) = 1 if x G P and zx(x) = a if x ^ P . T h e n /x is a nonconstant prime L-ideal of R. Now £ C /x and £* C P = JJ,*. Hence zx G P$. ■ L e m m a 3 . 7 . 8 Let R be with identity, L a dense chain, and£ be a nonconstant L-ideal of R. Let u G R be such that £(1) < f(ix). Then ( \ / £ ) ( l ) < €(u). Proof. Let £(iz) = b a n d £(1) = a. Let a < m < b. T h e n £ m is a proper ideal of R. Let P be a prime ideal of R such t h a t £ m C P and P ^ R. Let fi e LR be such t h a t /x(x) = 1 if a: G P and xx(x) = m if x £ P. T h e n /x G P$. Now (x/£)(l) < M(l) = m < b = f(tx). ■ T h e o r e m 3 . 7 . 9 Le£ R be with identity, L be a dense chain, and £ be a nonconstant primary L-ideal of R. Then £(0) = 1 and | £ ( P ) | = 2. Proof. Suppose t h a t £(0) = b < 1. Let £(1) = a < £(0). T h e n ( V £ ) ( l ) = n < b. Let 6 < TO < 1. T h e n a < n < f c < m < l . Let /x, i/ G LR be such t h a t xx(x) = ra if x G £*, xx(x) = a if x £ £* and v = 6#. T h e n ^ , v G LI(R). Let x G P . If x G £*, t h e n £(x) = 6 and (JXOJ/)(X) < 6 since v(x) = b. Now if x ^ £*, then since /x(x) = a, (/i o z/)(x) = a = £(1) < £(x). Hence /x o v C £. Since This M (0) - m > 6 = £(0), AX g f. Also i/(l) = b > n = ( v £ ) ( l ) . Hence i / ^ ^ is a contradiction since £ is primary. Hence £(0) = 1. Since £ is not constant | £ ( P ) | > 2. Suppose t h a t |£(R)\ > 3. Let £(1) = a. T h e n 3 6 G £(R) such t h a t a < b < 1. Let it G P be such t h a t £(?x) = b. T h e n (./£)(1) < f (zz) by L e m m a 3.7.8. Let /^, i/ G L ^ be such t h a t /i(x) = 1 if x G ft, fJ>(x) = a if x ^ £5, and i/ = bR. T h e n /i, i/ G LI(R) and /x o i/ C £. Now /x(tx) = 1 > 6 = f (TX). T h u s /x £ £. Also i/(l) = 6 = £(u) > ( \ / £ ) ( l ) . Hence i / g ^ T h u s neither /x £ £ nor i/ 2 \/?5 which is a contradiction. Hence |£ ( P ) | = 2 . 1 T h e next result follows from Theorems 3.7.2, 3.7.4, a n d 3.7.9. T h e o r e m 3 . 7 . 1 0 Let R be with identity, L be a dense chain, and £ G LI(R). Then £ is a primary L-ideal if and only if £(0) = 1, £* is a primary ideal of R and \£(R)\ = 2. M T h e o r e m 3 . 7 . 1 1 Let R be with identity, L be a dense primary L-ideal of R. Then y/£ is prime L-ideal of R.

chain,

and £ be a

110

Proof. By Theorem 3.7.10, f (0) = 1, £* is a primary ideal of R, and £(R) = {l, a } , where a G £\{1}. Now (>/£)* = V?* i s a prime ideal of i? and (V%)(x) = 1 V x G ^ . Let /i G L f l be such that /x(x) = 1 if x G V ^ and /x(x) = a if x £ y/£. Then fj, G ^ and ^ = V^T = (>/£)*■ Let x £ V ^ • Then a = £(x) < (V^)(x) < /i(x) = a. Thus (>/£)(iJ) = {l,a}, (V£)(0) - 1, and (V?)* i s a prime ideal of R. Hence y ^ is a prime L-ideal of # . ■ Theorem 3.7.12 Let R be with identity 1 and L a dense chain. Let £ be a prime L-ideal of R. Then V i > 1, £W 2S a primary L-ideal of R if and only if ( £ ^ ) * ^s a primary ideal of R. Proof. Let £(i2) = { l , a } , where a G £\{1} is a prime element. First we show that £(R) = ^(R) V i > 1. Clearly, the result is true for i = 1. Suppose f (#) = £(*)(#), » > 1. Let x G i t Then f (*+1)(x) = V{Aj =1 (^W(u i ) A f (ty)) | x = Y^j=iu3vji Uj'>v3 G i?, j = 1,2,... , n } . Hence £( 2+1 )(x) = 1 or a. Thus by induction f( i+1 )(JJ) = { l , a } . Hence by induction f(ij) = £ ( i ) (#) V i > 1. Now ^W(O) = £(0) = 1 V i > 1 by Theorem 3.1.13(2), and (£«),„ = (£„)* V z > 1 by Theorem 3.1.16. Now the result is an easy consequence of Theorem 3.7.10. ■ Theorem 3.7.13 Suppose that R has an identity and that L is regular. Let {Ra | a G f£} be a finite collection of ideals of R such that R = ® a e fii? a . Let fi G LI(R) be such that /x(0) = 1 > V{/x(x) | x ^ /z*}. Le£ /xa = li?azx V a G f i . Then \x is a generalized maximal or prime or primary L-ideal if and only if all but one of the fi'a is 1 on Ra and the remaining /j,'a is a generalized maximal or prime or primary L-ideal in i ? a , where fj,'a is the restriction of fia to Ra and where we assume L is a dense chain for the primary L-ideal case. Proof. By Proposition 3.3.16, every x G //* has a unique expression x = Eaea x <*' w h e r e x<* € /x* and fi(x) = A{/x a (x a ) | x = £ a G Q Z a , x a e *£}• By Theorem 3.3.15 and Theorem 3.2.4, RIli* ~ Rlii ~ ®aenR<xln'oc - ® a € n # a / 0 4 ) * By [261, Theorem 30, p. 175], /z* is maximal or prime or primary if and only if (/xJJ* = # a for all but one (/^J* and t n e remaining (/x«)* is respectively maximal or prime or primary. If x G /x*, then /i(x) = /x a (x). Hence Im(/x'a) C Im(/x). Suppose that Im(/i) = {1, a}, where a < 1. For those /x'a such that (/x'J* = Ray Im(/x'J = {1}. Suppose (j^)* ^ i?^. Then I m ( ^ ) = {1, a}. Conversely, suppose Im(/x^) = {1, a}. By Proposition 3.3.17 and [261, Theorem 30, p.175], /x£ = (1RPH)* = i^/3M* and i^/x* C i?^. Then Im(/x) = {1, a} since if (fi'p)* = i?/?, i.e., a > 0, then /x* = 0a€ftxx* = 0 a e n # a = R by Theorem 3.3.7. Hence the desired result follows from Theorems 3.4.3, 3.5.5, and 3.7.10. ■

111

3.8

TZ-Radicals of L-Ideals

Definition 3.8.1 Let fi G LR be given. An L-subset Ufa) G LR, defined by Tlfa){x) = V nGNA i(x n ) V x G R, is called the 7^-radical of xx. Theorem 3.8.2 Let \i, v G LI(R).

Then the following assertions hold:

(2) fi c nfa) c ,/ji. (3)n(lZfa))=1Zfa). (4) y/VZ Q Kfa)a V aeL. (5) //V{/x(x)|x i^}=t< /i(0), then Ufa)* = Jfc. (6) fiCv =* ft(/x) C ft(i/). f7j/i + i / C % ) + R ( i / ) . (8) Ufa) fl 7i(i/) = Ufa o i/) = ft(/zi/) = ft(xx fl i/). Proof. (1) ft(/x)(0) = V{xx(On)|n G N} = /x(0). (2) Clearly, /i C Ufa). Let f G 7^. Then f(U) = { l , a } , where a G L\{1} is a prime element. Let x G R and let n G N. Let £(x n ) = 1, then x n G £*. Hence x G £* since £* is a prime ideal. Thus £(x) = 1 = £(x n ). If £(x n ) = a, then £(x) = a. Thus £(x n ) = £(x) V n G N. Hence f (x) = £(x n ) > xx(xn) V n G N. Thus £ D ft M V ( € ? M . Hence ft(/x) C ^//J. (3) and (4) Straightforward. (5) By (4), y/jr; C ^(xx)*. Let x G ft(xx)*. Then /x(0) = Ufa){0) = Tlfa)(x) = V{/x(x n )|n G N}. Thus 3 n G N such that /x(xn) = xx(0). Hence 3 n G N such that x n G /i* or x G yT^- Thus Ufa)* C y ^ . (6) and (7) Straightforward. (8) Let x G ii. Then

nft(i/))(x)

= = < < = < < <

(V{xx(xm)|ra G N}) A (fa(xn)\n G N}) V{xx(xm) A v{xn) | m,n G N} V{(/i o zy)(xm+n) | m, n G N})

y{faoJ){xk) \keN} nfaov){x) TZfau)(x) TLfaC\v)(x)

(nfa)nnfa))(x).

Hence Ufa) n11fa) = ft(xx o i/) = ft(/zi/) = Ufa fl i/). ■ Theorem 3.8.3 Let i? fee with identity and \x G LI(R).

Then Ufa) G LI(R).

112

Proof.

Let x,y G R. Clearly, 71(/J,)(-X) /

n

(x + y ) m + n = x

m

m

.

= 7J(/x)(x). Let m, n G N. Now m+n

\

/

{ YY-'y' + y" £\

^ (

2=0 ^

where

'

i

\

^

/

fm +

2=n+l

n\+n_iyi_n^

/m + n\ [(m + n)!] i ) ~" [(m -f + nn -— i ) !i)!z!] z!]

for 0 < i < m + n. Hence

I < J nn

m-fn m+n

i=0

2=n+l

+ n m+n_ /x((x+j/r ")) > /i(* m n n n ~V)A/i(y n n((x + y)m+n E(T> E ((mm+t")^ V-nn)) > /*(a;m EC t )*n"V)AM(yn 2=n+l E )^m+""V2=0

/x(x m )A/x(j/ n ). /x(x m )A M (2/ n ).

> >

Thus

ftM(x + l/) = V{fi((x + y)k) n{ii){x + y)

= > > > > = = = =

1l(ii)(xy) K(fj)(xy)

= —. = > > = =

\keN} k V{fi((x y) )\keN} V{fM((x ++ y)m+n ) | m, n e N} m+n V{^((x V { ( ^ mm+ ) A?/) / j ( f )n)) | m, | mn, nGGN} N} m ) A fi(y )) | m, n G N}n V{(^(x ( V { ^ m) 1 m € N}) A (V{Ai(j/ ) | n e N}) (V{^(x ) |ATZ(^(y). m G N}) A (V{^(7/n) | n G N}) 1Z(fi)(x) n(fj)(x)An(fj)(y).

Also,

y{fi((xy)n) n) | n G N} V{/i((x?/) V{^(x n y n ) | n G N} V{/x(x") V{/x(xn) V /x(y") fi(yn) | n 6G N} :N} n (V{/x(x (V{//(x ) | n G N}) V V (V{Ax(j/ (V{^(2/n) | n GN}) G N}) n(fj)(x)\/n(tj)(y). n(ii)(x)VK(n)(y).

Hence ft(/z) G L/(iJ). ■ Theorem 3.8.4 Let \i G LI{R). ^ / ^ } V x G #.

Then n(fi)(x)

= V{a | a G //(#) and x G

Proof. Let x G i? and Tl(fi)(x) = u = V{fi(xn) | n G N}. Let & = V{a | a G ^(iJ) and a: G y/JJ^}- Then 6 = V{a | a G /z(iJ) and 3n G N such that /x(xn) > a}. Hence u, > 6. Let n G N and a — \x(xn). Then a G /x(i?) and x G y/JI^. Thus u
W W

" I a*

{l}. z/x G V ^ V A W T7 / o* r™i € N G\ N\{1}.

113

Proof. (1) By Theorem 3.8.2, ^/JT^ C K(fj)ai. Let i G N and let x G n(fi)ai. Then ft(/x)(x) > a*. Thus 3 n G N such that fi(xn) > ai or xn G /x0.. Hence * e y/JiZ. Thus (ft(/x)) a . C 0 ^ 7 . Hence 7e(/x)0. = ^//x^". (2) FoUows by (1). ■ Theorem 3.8.6 Let v G LR. K{y)

Proof. then

= =

Then

U{xa | x G i?, a G L, 3n G N such tfia* ( x a ) n C v) U{/x | /i G L H , 3n G N 5tzc/i tfia* /xn C i/}.

Let x E R and let a G L. If there exists n G N such that (xa)n C i/, ft(i/)(x) = V meN z/(x m ) > v{xn) > a

and thus 7Z(is) > xa. Hence 1l{y) 2 U{x a | x G R, a G L, 3n G N such that ( x a ) n C i/}. Let y G ii and let i/(y n ) = an V n G N. Then (yan)n

C i / V n e N . Thus

WO/) = vMvn) \neN} = v{(j/aB)(y)|neN} <

(U{x a I x G R, a G L, 3n G N such that ( x a ) n C i/})(y).

Hence 7l(i/) C U{xa \ x G i?, a G L, 3n G N such that ( x a ) n C v). Thus 7J(i/) = U{x a | x G U, a G L, 3n G N such that ( x a ) n C i/}. Clearly, by the first equality, TUy) C U{/x | xx G L f l , 3n G N such that /xn C i/}. Now, if xx G L*, n G N, and /xn C i/, then xx(x) < xxn(xn) < */(xn) < ft(*/)(x) V x G i? and so xx C 7i(i/) . Thus 7i(i/) = u{/x | xx G L H , 3n G N such that /xn C i/}. ■ Theorem 3.8.7 Suppose that v G LI(R).

Then

n(v) = U{/x | /x G ZJ(-R), 3n G N such that ^ n C i / } .

114

Proof.

By Theorem 3.8.6, n 3n e N such that fi G € LI(R), LI(R), 3neN that /x 1l{y) D U{fj, | /x K{v) 2U{H ^ C i/}. "}•

Let x G fl, a E L, and ( x a ) n C 1/ for some n G N. Then (x a ) Hence fc(y)

= = C c C c

n

= ((x a ) n ) C i/.

n n U{x v) i2, a G L, I/, 3n G N such that (x (xaa) ) CQ "} U{xaa |I xa; G R, U{(x U{(xa0)) \ x e£ R,ae R,a e L,3neN L,3n eN such that that (^a> ( x a ) nn C C i'v) } U{/x € LI(R), £ ! ( # ) , 3neN 3n G N such that /x / inn C /}. Ci "}■ U { / x|| ^/x G

Thus 7i(i/) n{y) = = U{/x | /x G L/(U), LI(R), 3n G N such that fi /xnn C i/}. ■ Theorem 3.8.8 Let S be a commutative ring, f an epimorphism of R onto S, jie LR and v G Ls. Then (1) If fi is f-invariant, then /(7£(/x)) = TZ(f(fi)) . 1

(2)r\n{v)) = n{r ^)).

Proof.

(1) Let y € S. Then f{1l(n)){y)

= = = = =

Hence /(ft(/i)) = ft(/M) (2) Let x G .R. Then

y{ii{n){x) V{ll(ri(x)\xeR,f(x)=y} | x ER,f{x)=y} n V{V ^(x V{Vn€ ( Z " )) |\xeR,f{x) NM ne N x e fi, f(x) = = y)y} V n€N(V{/x(x") (V{^(x") \xeR,f\x \x€R,f(xn)n) --= }) = yynn}) V n 6 N (V{/i(*)|I z€R,f(z)=y * € £ , / ( * ) = » "'"}) }) Vn€N(V{M(2) : Vn€N/(^)(j/") = TZ(f^))(y). V„ 6 N /(/i)(» n )=W(/(A*))(»). .

/-H^(^))(x) /-W«/))(*) = n{u){f{x)) H{v)U(x)) = = = =

V Vnn66NNll/((/(x))») /((/(x))") n V„ V n e€NN^(/(x i/(/(x n)) )) 1 V „e N6 (/- » ( x "n )))) V„ N ( / (i')(a:

n{f-\u)){x)

ft(/-»)(x)

V x € U. Hence / _ 1 ( t t ( i / ) ) = tt(/-1(i>)). ■ Theorem 3.8.9 Suppose that S is a commutative ring and f is an epimorphism of R onto S. Let \x € LI(R) and u 6 LI(S). (1) If n is f-invariant, then 1Z(f(n)) 6 LI(S). (2)U{f-\u))eLI{R). Proof.

This is a direct consequence of Theorems 3.8.8, 3.2.7, and 3.2.6.

■

115

3.9

7^-Primary L-Ideals and 7£-Semiprimary LIdeals

Throughout the present section, it is always assumed that R is with identity. Definition 3.9.1 Let £ 6 LI(R). Then £ is called K-primary if £ is nonconstant and ^ o i / C ^ / i , i/G LI(R) => either fi C £ or v C 7£(£). T h e o r e m 3.9.2 Let v, £ £ LI{R) thenll(y) C £ = ft(0-

with v C f. If £ is a semiprime L-ideal,

Proof. Prom Theorem 3.8.2, it follows that U{y) C K{£). If £ is a semiprime ideal of i?, then ft(f) = £ by Theorem 3.5.19. Therefore, n{y) C f. ■ T h e o r e m 3.9.3 Let £ e LR. Then £ is a prime L-ideal of R if and only if £ is a semiprime and IZ-primary L-ideal of R. H T h e o r e m 3.9.4 Let £ e LR. Then£ is an IZ-primary L-ideal of R if and only #£(0) = I? £* i>s a primary ideal of R and £(i?) = {1, c}, where c 6 £\{1} and c is a prime element in L. Proof. Suppose £ is an 7£-primary L-ideal of R. First we show that £ is two valued. Since £ is not constant, |£(1?)| > 2. Let x £ £* and let £(x) = c. Then £(1) < c < 1. Let £(0) = a, [i = a^x) and v = cR. Then //, v e LI{R), Hov = a^x) OCR = C(x) C £, and /J, £ £. Thus i/ C 7£(£). Then £(x) — c = v(\) < ft(f)(l) = £(1) < c. Hence £(:r) = £(1). Thus £ is two valued. Let £(#) = {a, c}, where a > c. We now show that £(0) = 1. Suppose a = £(0) < 1. Let fi = CLR and */ = 1 ^ . Let x G £*. Then (/x o ^)(x) < //(a;) = a = £(x). Let a: ^ £*. Then V u,v e R such that x = uv, u £ £* and t; ^ £*. Hence (/x o i/)(a;) = V{/i(u) A z/(v) | u , v 6 i ? , a ; = m>} = V{a A 0} = 0 < £(x). Hence / i o i / C f . Now ^(1) = a > c = £(1) and i/(0) = 1 > a = £(0) = K(£)(0). Thus neither it g ^ nor v £ 7£(£), which is a contradiction. Hence £(0) = 1. Now, let A and B be ideals of R such that Ai? C £*. Then 1A ° 1 B = IAB Q 1^ C £. Thus either 1A C £ or 1 B C V?. If 1 A C £, then A C £*. Let 1 B C ft(f). Then B C ft(f)*. Since £ is finite-valued, V ^ = ft(0*- Hence B C y ^ . Thus £* is a primary ideal of R. Finally, let a,b e L be such that a A 6 < c. Then &* ° a* Q C,R C £. Thus 6R C £ or aR C 71(0. I f ^ £ f. t h e n 6 = M 1 ) < £(1) = c. Let a/* C ft(f). Then a = OR(1) < 71(0(1) = f (1) = c. Hence c is a prime element. Conversely, suppose that £ satisfies the given conditions. Let fi, v e LI{R) and let /z o i/ C f. Suppose that /x g £ and i/ g 7£(£). Then there exist x, y e R such that fi(x) £ £(#) and i/(y) ^ ^(0(2/)- Hence x ^ £* and 2/ ^ ^ ( 0 * = V£*' Then since £* is primary xy £ £* and hence £{xy) = c. Now

116

li(x) A v(y) < ( / z o v)(xy) < £(xy) - c. Since ji{x) £ £(x) = c and c is prime, v{y) < c- Since y £ v ^ , y n ^ * V n € N and hence f (y n ) = c V n G N. Hence 7£(y) = c > 1/(2/) which is a contradiction. Thus either /x C £ or ^ C 7£(£). Hence £ is a primary L-ideal of R. M Theorem 3.9.5 Let £ be a nonconstant L-ideal of R such that V x,y G i?, V a, 6 G L, 2/ xaoyh C f, tfien ez£/ier x a C £ or 3 n G N stic/i £/ia£ (?/6)n C f. TTien £(0) = 1, £* zs a primary ideal of R and £(i2) = {1, c}, w/iere c G £\{1} is a prime element. Proof. Suppose that £(0) = a < 1. Since £ is not constant, 3 x G R such that £(x) < £(0). Now x a o 0i = (x0) a = 0 a C f. But x a g £ and V n G N, (0i) n = 0i ^ f • This is a contradiction to the hypothesis. Hence £(0) — 1. Now, let x,y G R and let XT/ G £*. Then £(xy) = 1. Suppose that x £ £*. Then £(x) < 1 and hence x\ % £. Now x\oyi = (xy)\ C f. Then 3 n G N such that (yi) n = (2/i)n C f. Hence 3 n G N such that £(yn) = 1 or yn G £*. Thus £* is primary. We now show that |f(i?)| = 2. Since £ is not constant \£(R)\ > 2. Let x ^ £*. Then £(1) = c < £(x) = a. Now xi o l a = x a C £. Since xi £ £, 3 n G N such that l a = ( l a ) n C £. Thus £(1) > a = £(x). Hence |£(#)| = 2 and £(i£) = {l,c}, where c G £\{1}. Finally, suppose 3 a, 6 G L such that a Ab < c. Then l a o 1& C l c C £. Suppose l a g £. Then 3 n G N such that 1 6 = ( l 6 ) n C f. Hence c = £(1) > 6. Thus c is prime. ■ Theorem 3.9.6 Let £ G LI(R) be nonconstant. Then £ is IZ-primary if and only if V x, y G .R, V a, b G I/, if xa ° yb ^ £> £/ien either xa C £ or 3 n G N swc/i £/ia£ (2/6 ) n £ £• Proof. Suppose £ is 7£-primary. Then £(0) = 1, £* is a primary ideal of R and £(#) = {l,c}, where c G £\{1} is a prime element. Let x,y e R and a, 0 G L be such that xaoyh (x a o yb){xy) > a Ab. Since c is prime, either a < c or 0 < a Let a < c. Then £(x) = c > a = x a (x) or £ > x a , which is a contradiction. Suppose 6 < c. Let n G N. Then (yb)n(yn) = (yn)b(yn) = b < c = £(yn) or (yb)n Q f, which is again a contradiction. Hence either x a C £ or 3 n G N such that (yb)n C £. The converse follows from Theorems 3.9.4 and 3.9.5. ■ Theorem 3.9.7 Suppose that £ is an IZ-primary L-ideal of R. Then 7£(£) is a prime L-ideal of R. Proof. By Theorem 3.9.4, £ = 1^ U CR, where £* is a primary ideal of R and c is a prime element of L. Then 7£(£) = 1 /T-UCR. Since £* is a primary

117

ideal of i?, y/£^ is a prime ideal of R. Hence by Theorem 3.5.5, 72(0 is a prime L-ideal of R. ■ Theorem 3.9.8 Let L be a dense chain and let £ G LI(R). if and only if £ is IZ-primary. Proof.

Then £ is primary

Follows from Theorems 3.7.10 and 3.9.5. ■

Lemma 3.9.9 Let £ be a finite-valued L-ideal of R, x G R and let a G L. Then xa C 72(0 if and only if 3 n G N such that (xa)n C £. Proof. Now xa C 72(0 <^> 72(0 (z) > « <£> V{£(x n ) | n G N } > a ^ 3 n G N such that £(xn) > a <=> 3 n G N such that (xa)n C £. Theorem 3.9.10 Le£ L be a dense chain. Let £ 6e primary

(IZ-primary).

Then^ = n(0. Proof. By Theorem 3.8.2, 72(£) C vT- Let x, 2/ G i i and a, 6 G L be such that xaoyb C 72(£). Then 3 z G N such that (x a o yb)1 C £, by Lemma 3.9.9. Thus 3 z G N such that (x a ) 2 o (yfe)2 C £. Since £ is 72-primary 3 j G N such that either {xa)i+j C £ or (y&) i+j C £. Thus either x a C 72(£) or j / 6 C 72(0, bY Lemma 3.9.9. Hence 72(0 *s prime L-ideal. Since £ C 72(£) and £* C 72(0*, 72(0 € Vz. Hence V? = 72(£). ■ Theorem 3.9.11 Suppose that R is an Artinian ring with identity and L is a chain. Assume that £ is an irreducible proper L-ideal of R. Then £ is a primary L-ideal of R, and hence 72(0 ™ a V^me L-ideal of R. Proof. Prom Theorem 3.4.12, it follows that £* is an irreducible proper ideal of R and ( = 1^ U CR for some c G L\{1}. As is well known, an irreducible proper ideal of a commutative Artinian ring with identity must be a primary ideal. Consequently, by Theorem 3.9.4, £ is a primary L-ideal of i?, and thus, by Theorem 3.9.7, 72(0 is a prime L-ideal of R. ■ Theorem 3.9.12 Let S be a commutative ring with identity and f be an epimorphism of R onto S. (1) If £ is a primary L-ideal of R and f -invariant, then / ( 0 is a primary L-ideal of S. (2) If r] is a primary L-ideal of S, then f"1^) is a primary L-ideal of R. H Definition 3.9.13 Let £ G LI(R). Then £ is called H-semiprimary if £ is nonconstant and if [iov C £, /z, v G LI(R), then either fi C 72(0 or v C 72(0-

118

Clearly, a primary L-ideal is a ft-semiprimary L-ideal. Theorem 3.9.14 Let £ G LI(R) and let ft(0 be a prime L-ideal of R. Then £ is an 7Z-semiprimary L-ideal of R. M Theorem 3.9.15 Let £ G LR be such that £ = 1^ UCR, where £* is a semipri­ mary ideal of R and c is a prime element of L. Then £ is an ft-semiprimary L-ideal of R. Proof.

Now ft(£) = 1 rr U CR. Since £* is a semiprimary ideal of R,

y/^

is a prime ideal of i?, and hence ft(£) is a prime L-ideal of R. The result now follows by Theorem 3.9.14. ■ Theorem 3.9.16 Let £ be an IZ-semiprimary L-ideal of R and let £(0) = 1. (1) £(1) is a prime element in L. (2) /jf ft(£)* = y/^, then £* is a semiprimary ideal of R. Proof. (1) Suppose 3a,b e L such that a <£ £(1), b j£ £(1) and a A 6 < £(1). Let xx = 1^ UaR and v = 1^ U6*. Then xx, v G LI{R), /x % ft(£) and v % ft(£). But /x o v C 1^ U (a A O)R C £ which is a contradiction. Hence £(1) is a prime element in L. (2) Let x, y e Rbe such that xy G £* and let x £ ^/^ . Let /x = 1 ^ and i/ = l ^ ) . Then xx, */ G LI(R) and fio v = \(xy) C 1 ^ C £. Since xx(x) = 1 > ft(0(0) > ft(0(#)> M 2 ft(0- Since £ is ft-semiprirnary, v C ft(£) and hence (y) C ft(0* = >/£■ Thus y G v ^ . Therefore, £* is a semiprimary ideal of R.M Theorem 3.9.17 Let £ be an ft-semiprimary L-ideal of R and let £(0) = 1. / / f t ( 0 * — \/£*> then ft(0 ^ a prime L-ideal of R. Proof. By Theorem 3.9.16, £* is a semiprimary ideal of R and £(1) is a prime element in L. Now ft(0* = \/£* *s a prime ideal of R. We now show that ft(OCR) = {li c}, where c = £(1). Now ft(0(0) = 1 and ft(0(l) = £(1) = c. Hence l , c G ft(£)(i?). Let x G i?. If £(x n ) = c V n G N, then ft(£)(x) = c. Suppose there exists n G N such that £(z n ) ^ c. Then £(a;n) > c. Let a = £(:r n ), xx = 1^ U aR, and i/ = 1^ U c#. Then xx, i/ G LJ(i2) and /x o ^ C £. Since xx(l) = a > c = ft(£)(l), xx g ft(0- Hence i/ C ft(£). Thus 1 = v(xn) < ft(£)(a:n), i.e., xn G ft(£)* = v^T- Hence x G \A7fT = V ^ = ft(0* w n i c h implies that ft (00*0 = 1. Thus ft(£)(i?) = {1, c}, where c = £(1). By Theorem 3.5.5, ft(0 is a prime L-ideal

offt.B

119

Theorem 3.9.18 Let S be a commutative ring with identity and f an epimorphism of R onto S. Then, (1) If £ is an IZ-semiprimary L-ideal of R, ?£(£)* = y/^, and £ is finvariant, then /(£) is an IZ-semiprimary L-ideal of S. (2) If rj is an IZ-semiprimary L-ideal of S and (y/rj)* = y/rj^, then / _ 1 ( ^ ) is an IZ-semiprimary L-ideal of R. ■

3.10

Characterization of Artinian Rings

In this section, we assume that L is a chain. Theorem 3.10.1 Either R is Noetherian or L does not contain a strictly de­ creasing sequence if and only if for every L-ideal \x of R, /J>(R) is well-ordered under < . Proof. Suppose that /i is an L-ideal of R such that fi(R) is not well-ordered under < . Then there exists a G n(R) which does not have a successor under < . Since fi(R) can not be finite it follows that there exists a^ G fi(R) such that a{ > ai+\ > a, i = 1,2, Then /xai C Mai+i ,2 = 1,2, Hence R is not Noetherian. Conversely, suppose that R is not Noetherian and L contains a strictly descending sequence. Then there exists a strictly ascending chain of ideals in i?, say I\ C h C . . . C In C Let ai G L be such that ai > a;+i, 2 = 1,2, Define the L-subset /x of R by fi(x) = 0 if x £ Ug^ii and fi{x) — ai, where i is the minimum i such that x G /;, i = 1,2, Then fi is an L-ideal of R such that /i(i2) is not well-ordered under < . ■ Theorem 3.10.2 Either R is Artinian or L does not contain a strictly as­ cending sequence if and only if for every L-ideal fj, of R, fJ>(R) is well-ordered under > . Proof. Suppose that \x is an L-ideal of R such that fi(R) is not well-ordered under > . Then there exists a G fJ>(R) which does not have a successor under > . Since fi(R) can not be finite, it follows that there exists a{ G n{R) such that ai < ai+i < a, i = 1,2, Then jiai D fiai+1, i = 1,2, Hence R is not Artinian. Conversely, suppose that R is not Artinian and L contains a strictly ascending sequence. Then there exists a strictly descending chain of ideals in i?, say, i i D I2 D . . . D In ^> Let Oj G L be such that ai < ai+i, 2 = 1,2, Define the L-subset \i of R by /x(x) = 1 if x G flg^/i and /z(x) = ai, where 2 is the maximum 2 such that x G Ii, i; = 1,2, Then )U is an L-ideal of R such that fi(R) is not well-ordered under > . ■ Corollary 3.10.3 R is Artinian and Noetherian if and only if every L-ideal of R is finite-valued.

120

Proof. Every L-ideal of R is finite-valued if and only if for every L-ideal p of R) p(R) is well-ordered under > and < . I Corollary 3.10.4 Suppose that R has an identity. Then R is Artinian if and only if every L-ideal of R is finite-valued. Proof.

Since R is Artinian, R is Noetherian. I

Theorem 3.10.5 Suppose that R has an identity. Let S be a commutative ring with identity. Then R(BS is Artinian if and only if R and S are Artinian. Proof. Suppose that R © S is Artinian. Let / be an ideal of R. Then / 0 S is an ideal of R © S. Hence a strictly descending chain of ideals or R induces a strictly descending chain of ideals of R 0 S of the same length. Thus we see that R must be Artinian. Similarly, S is Artinian. Conversely, suppose that R and S are Artinian. Let p be an L-ideal of R®S. Define the L-subsets p and a of R and 5, respectively, as follows: p(x) — p(x, 0) Mx G R and o~(y) = p(0,y) My G S. Then p and a are L-ideals of R and 5, respectively. Now p(x, 0) = /x((l, 0)(ar,y)) > p(x, y) V(x, y) G R®S and similarly, /z(0,2/) > /x(x, y) V(x, 2/) G iJ©5. Hence V(x, y) G i i © 5, /x(x, 2/) < p{x) A cr(y). Also, /*(*, V) = M((^, 0) + (0,2/)) > /x(x, 0) A /z(0,2/) = p(x) A a(2/) V(x,y) G ^ ® 5 . Thus V(x,y) G .R © 5, /x(x,y) = p(x) A cr(y). Now p and a are finite-valued by Corollary 3.10.4. Hence p, is finite-valued. Thus R © 5 is Artinian. ■ Theorem 3.10.6 Suppose that R has an identity. Let I be an ideal of R. Then R is Artinian if and only if the quotient ring R/I is Artinian and p(I) is finite for every L-ideal p of R. Proof. Suppose that R is Artinian. Then clearly R/I is Artinian. By Corollary 3.10.4, p is finite-valued. Thus p/I is finite-valued. Conversely, suppose that R/I is Artinian and p(I) is finite for every L-ideal of R. Let p be an L-ideal of R. Since p(I) is finite, we have that {p(x) | x G i?, p(x) = p(y) for some y G 1} is finite. Since R/I is Artinian, p/I is finite-valued. Now (p/I)(x + i") = M{p(x + 2/) I y G / } , where x e R. Suppose that /z(x) 7^ ^(2/)Then p(x + y) = p(x) A p(y). We need only consider the case, p{x) ^ /i(2/) My G / . In this case, (p/I)(x + I) = \/{p(x) A p(y) \ y e 1} = p(x) since ^(0) > p(x). Thus {/x(x) I x G i?, //(x) ^ p{y) My G / } is finite since p/I is finite-valued. Hence R is Artinian by Corollary 3.10.4. ■

121

3.11

L- Generators

In this section, L is a dense chain. Let p be an L-subring of R. Let F(p) denote the set of all L-subsets £ of R such that £ C p. Let v G fip). Recall from Definition 3.2.11 that v is an L-ideal in p if v C p and Vx,y G i?, v(x K * --y)> ?;) > v(x) y(x) A v*%) (y) and v{xy) v(xy) > > (u(x) p{y)) V V (u(y) (1/(2/) Ap(x)). A p(x)). (i/(x) A p{y)) Definition 3.11.1 Let £ G ^ ( p ) . (1) Let (£) denote the intersection of all L-ideals \ of R such that £ C x fj^ Let -< £ >- denote the intersection of all x^P °f R, where x is an L-ideal of R such that £ C x(3) Let <^C £ ^> denote the intersection of all L-ideals x in P such that £ C X£ is said to generate (£) (or -< £ >- or - or
v{(£ti( r ik(*ik + E7=i^(%)^)(*) Pin) >>**>£te) bi,Z(xi) *(*) === v{(£ti(^k(*;k £7=i ";(»)«,)(*) 1I Pfa) > a>- en,


£(3/j) >Uj,ri,Xi,yj £(%) > Uj,ri,Xi,yj fc,m € A;,m G N}. N}.

€ /J,rij #,rij ee Z = ll ,, .. .. .. ,,ffcc;;jj = G Z ,, ti = = ll,,......,,rma ;;

Tnen cr 25 an L-subring of R such that £ C cr C /?. Proof.

Now k

k

m

jiVoJCj 2i = = 1l

j=l i=i

C£2

riXi

i=l

m

yyj)d c p, j=l

where c mc.m . bkAAa ax . . .AAdfca^A A ci c\A A ..... .A. A d = 61 A .... . AA6fc i AA. ..

Thus cr C p. Let x G £* and let a = £(z). Then x a = 0p(O)Xa + lxa Q &Hence £ C cr. Let u, v e R. Then cr(u) is a supremum of numbers of the form

122

(J2i=irixi + Ej=i n j%) 0 and o~(v) > 0. Then there exists dq = (A{bi A ai | i = 1 , . . . , fc; fe G N}) A (Ajc^ | j = 1 , . . . , m; m G N}) (g = 1,2,...) such that dg —> cr(w). There exists d p (p = 1,2,...) defined similarly such that dp —► &{v). Now iz G ( x i , . . . , Xfc,yi, • - • ,y m ) and v G ( x i , . . . , x'g, j / i , . . . , y£) imphes that u + v, uv G ( x i , . . . , xk, 2/1,..., ym, x[,...,

x'g, y j , . . . , y'h) .

Thus
> = =

V{d 9 Ad p | g,p = 1,2,...} (V{d,|(Z = l , 2 > . . . } ) A ( V { 4 | p = l , 2 , . . . } ) o~(u) Acr(v).

Clearly, a(u) = a(—u). ■ Let p be an L-ideal of R such that p C p. Suppose that r& C p and x a C p. Then p(rx) > p(r) V p(x) > p{r) A p(x) > b A a. Thus r&xa C p. T h e o r e m 3.11.3 Le£ £ G ^"(p). Let cr be the L-subset of R as defined in Lemma 3.11.2. If A{p(x) | x G R} > V{£(x) | x G i2}, thenu is an L-ideal of R and

* = (0 =-< € >-=< £ > • Proof. To show that cr D (£), it suffices to show that cr is an L-ideal of R such that £ C a C p. Clearly, £ C cr C p. By Lemma 3.11.2, it only remains to be shown that if w, v G i£, then cr(m;) > a{u) V cr(f )• Suppose that cr(u) > 0. Then there exists a representation of u of the form w = ^2i=1 rixi + E j = i n i % > where x;, % G £*. Now p(v) > V{£(x) | x G i?}. Thus wv =

Y!t=i(vri)xi + T?=i(njv)yj and °(uv)

> ( E i U vp(v)(ri)bi(xi)ai + E ^ i K ' ^ p M ^ i J c J ^ t ; ) =

6x A . . . A 6^ A a x A . . . A afc A Ci A . . . A c m

= (EiUfak (**)«< + Ejli ^(yjJcJH, where p(r^) > &;, £(xi) > a* for z = 1 , . . . , fc and £(yj) > Cj for j = 1 , . . . , m. Hence a(uv) > cr(u). Similarly, o~(uv) > o~(v) when a(v) > 0. Thus cr is an L-ideal of R. Hence (£) C a. Thus {x | X ls a n L-ideal of i?, £ C x Q p} ^ 0.

123

In Lemma 3.11.2, (xi) 0 ., (%) c . C £ C (£) for i = 1 , . . . , fc; j = 1 , . . . ,m. Hence a C ( 0 since ( £ ^ 1 ^ + £™ = 1 *&)* Q (0 • Thus a = (£). Now {x | x is an L-ideal of i?, £ C x C /?} C {x H /9 | x is an L-ideal of R, £Q x} Q {x I X is an L-ideal in p, ^ C x}- Thus a -=

(0 2^ £ ^2< ^ > 2
where the latter inclusion holds since (YH=I rixi + E j = i njVj)d Q < £ > • ■ ' If p = l f l , then A{p(x) | x G R} = 1 > V{£(x) \ x e R}. Corollary 3.11.4 Let f G .F(p). Suppose that A{p(x) \ x G R} > V{£(x) | x G R}. (1) If R has an identity, then V x G R, a(x) (%) IfBly

==

v

{ ( E ? = i ( r 0 6 i ( ^ ) a J ( ^ ) 1 p(ri) > bi,€(xi) > au ri, Xi G , . . . ,,k\k k\ k G N}; ri,Xi e R, R,i i = 1I,... e N};

e R such that £(y) > 0, tfien V x G i?,

a(x) > 6,£(y) b, £(y) >> a, a, c)(x) |\ r G Z(y)>c,neN}; (3) If R has an identity and 3! y G R such that £(y) > 0, then V x G R, r > b,£(y) > a,n V{(rby62/a )(x) \| rir GeR a(x) === V{(r R,p(r) ■ a)(x) >t(y) >a ,n GG N}. ■ :b,

An L-ideal (i of i? is said to be finitely generated if 3 a finite set of L-singletons 5 such that ji = (S) or equivalently if 3 an L-subset £ of R such that £* is finite and \i = (£*). Theorem 3.11.5 Let R have an identity. Every L-ideal of R is finitely generated if and only if R is Artinian. Proof. Suppose that R is not Artinian. Then 3 a strictly descending chain of ideals R = Io D h D . . . D J n 3 Define the L-subset // of R by fi(x) = ^~- if x G J n \ J n + i for n = 0 , 1 , . . . and fi(x) = 1 if x G ng 0 ^n- Then fi is an L-ideal of R. Let ( x i ) a i , . . . , (xk)ak Q P- Let x G 7 n \ J n + i , where n is such that ^ £ { f l l ,... ,a f c }. Then ( ( * i ) a i l . . . , (xk)ak) (x) = V{( £ * = 1 fa) i (*<)«,)(*) I Ti G i?, i = 1 , . . . , k\ k G N} = a; (or 0) for some i = 1 , . . . , fc. However, /x(x) = ~^- 7^ a*. Thus [i is not finitely generated. Conversely, suppose that R is Artinian. Then |Im(/x)| < oo by Corollary 3.10.4. Let Im(/x) = { a i , . . . ,afc}, where ai > . . . > a*. Let x i j , . . . , x n j J G i? generate //aj. for j = 1 , . . . , A:. Let

124

x G M a . A ^ - ! , j = 1, • • •, k, where /xao = 0. Then x = YAU e /i 0j -Wj-i

for

s o m e

*•

T x

i ih

where X

H

T n u s

( ^ ( ^ ) i ( ^ ) ^ ) ( x ) = a i = M*)> where 6i = /d(xij), i = 1,... ,rij. Hence ({(*#)<*,• \i = l, • •., ™?; jf = l, • •., fc}) = MThus /z is finitely generated. I We show that every L-ideal with the sup property has a minimal set of generators if R is a principal ideal ring. We also show that every L-ideal of R is finitely generated if and only if R has a composition series. Hence if R has an identity and every L-ideal of R is finitely generated, then we will see in Chapter 6 that every L-ideal /z of R such that /i(0) = 1 has an L-primary representation. We also show that if R has a composition series of length n, then every L-ideal of R has a generating system with no more than n 2 — 1 elements. Whenever we write ^2ieI r^ri G i£, for some index set / , we mean that all but a finite number of the n = 0. If S is a set of L-singletons of R such that x a , Xb G S implies a = 6, then we set (S) = (£), where f is the L-subset of R defined by £(x) = a if xa e S and £(#) = 0 otherwise. Let zi be an L-ideal of R such that /i C p. Suppose that r& and # a C zz. Then /z(rx) > /z(r) V /z(x) > bAa. Thus r&xa C /i. Lemma 3.11.6 Suppose that R is a principal ideal ring. Let \i be an L-ideal of R such that /i C p. For zx* = (x) , x G R, let ru e R, nu € Z be such that the level ideal p,u = (rux + nux) V u G Im(fi). Then (1) p(rux + n w x) = w, (2) IfA{p(r) \r e R} >\/{u\u € I}, then V> = {{{rux + nux)u where I =

| u G /}) ,

lm(fi)\{0}.

Proof. (1) p(rux + nux) > u. Let z G zz* be such that p(z) = u. Then z G pu and so 3 s G i?, n G Z such that z = s{rux + n u x) + n(rux + n w :r). Thus iz = zz(z) > /z(rua; + nux). Hence TZ = /i(r u x + nux). (2) Clearly, ({(r u x + nux)u \ u e I}} C p.

125

Let z G /z*. Then by Theorem 3.11.3, ({(rux + nux)u \ u e I}) (z) = V { ( £ u € / (su)vu(rux + nux)u + ^2ueImu(rux + nux)u){z) \ p(su) Z}. Hence if p(z) = u, then z G /xu and so z = s(r u x + nux) + n(rux + n u x) for some s G i£, n G Z. Clearly, then p(z)

^ ( { ( r n ^ + ^n^)u

\uE

I})

(z).

■

Let 5 be a set of L-singletons of R. Then S is called a minimal L-set (or L-system) of generators for p if (S) U 0^(o) = p and there does not exist xa G S such that (<S\{xa}) U 0^(0) = pTheorem 3.11.7 Suppose that R is a principal ideal ring. Let p be an L-ideal of R such that p C p. For p* = (x), x G ii, let ru G i?, nu G Z 6e s?/c/i that pu = (r u x + n u x) V w G Im(p). Suppose that p has the sup property. If A{p(r) | r G R} > W{u \ u e I}, where I = lm(p)\{0}, then {(rux + nux)u \ u G 1} is a minimal L-set of generators of p. Proof. By Lemma 3.11.6, {(rux + nux)u of p. Let v E I and v =

({(ruX + nux)u

\ u G 1} is an L-set of generators

| u G 7}\{(r v x + nvx)v})

.

Then i/ is a L-ideal of R and i/ C p. Suppose that 3 u G / such that u < v. Then v = p(rvx + n v x) and z/(rvx -f- n v x) = V{izi A . . . A un \ rvx -f nvx = z^uei\{v} su\rux -+- Tiu^) + Z^ue/\{v} rnu{ruX + nux), su G it, m u G Z, 5U ^ 0, mu ^ 0, for u G { u i , . . . ,un} C J, n G N} = V{w | r v x + n v x = s u ( r u x -f nux) + mu(rux + n u x), u < i ; , i 4 E l } < v , where = and < occur since p has the sup property. Hence v C p. Now suppose there does not exist u G I such that u < v. Suppose that v(rvx + n v x) > 0. Then k

rvx + nvx — ^2sui(rUix Z=l

k

-\- nUix) + ^ m U i ( r U i x + nUix) 2=1

for some sUi, rUi G i2, rani, nUi G Z (iti > v), z = 1 , . . . , k. Hence TyX ~\~ n
— Pii

for some u > v, a contradiction, since pu C /xv and rvx + m v :r generates pv. Thus i/(rva: -f n v x) = 0. Hence v C p. M Theorem 3.11.8 Suppose that p\,..., pn, and p are L-ideals of R such that f_i = px 0 . . . 0 pn, where pi(0) = p(0), i = 1 , . . . , n. If Gi is a minimal set of generators for pi, i = 1 , . . . , n, then G = G\ U . . . U Gn is a minimal set of generators for p.

126

Proof. By induction, it suffices to prove the result for n = 2. Let z G \x\. Now /x* = /z| © [i\ by Theorem 3.3.7. Hence (Mi®M2)(*)

= = =

V{/xi(x) Afi(y) A/x(2/) | 2f = x + 2/,xG^J,2/6/x5} jziO) A// 2 (0) fii(z) Mi(2)>

where the latter equality holds since /ii(0) = /^(O). That is, \i\ = ji on /zj. Let x 0 G Gi. Then UG (GiU G 22)) === (Gx) ( G i ) © () (Gi and so

{G1\{x aa}}U
({G ((Gi\{x 1 \K»©(G 2 ))(x) 0 }) © (G 2)){x) {Gi\{x }) a})(x) (x) (G1a\{x (Gi)(x) (Gi){x) « G i ) © »(z) (G1(G UG1UG 2)(x). 2)(x).

A similar argument shows that UG 2\{y b})(y)<(G 1 2)(y)U G2) (y) (Gi (G1uG 2\{y b})(y)<{G 1UG for yt, E G 2 . Hence Gi U G 2 is a minimal set of generators of /i. ■ Recall that R has a composition series if and only if R satisfies the ascend­ ing and the descending chain conditions for ideals. Also, if R has a composition series of length n, then every composition series is of length n and every strictly ascending or descending chain of ideals has at most n -j-1 elements. Theorem 3.11.9 The following conditions are equivalent. (1) R has a composition series. (2) Every L-ideal of R is finite-valued. (3) Every L-ideal of R is finitely generated. Proof. (1) «=> (2): Suppose that \x is an L-ideal of R which is not finitevalued. Then Im(/x) contains either an infinite increasing sequence or a de­ creasing infinite sequence. Hence the set of all level ideals of fi contain either a strictly decreasing sequence or a strictly ascending sequence. Thus R does not have a composition series. Conversely, suppose that R does not have a composition series. Then R has either an infinite strictly decreasing sequence or a strictly ascending sequence of ideals. In either case, the sequence can be used to define an L-ideal of R which has this sequence of ideals as its level ideals. Such an L-ideal is not finitely generated.

127

(2) => (3): Since R has a composition series, every ideal of R is finitely generated. Let Im(^) = { a i , . . . , a f c } , where ax > . . . > ak. Let xXj,..., znjj 6 Ji generate /xaj. for j = 1 , . . . , fc. Let x G A S A / S - I , j = 1 , . . . , fc, where Ma0 = 0- Then z = J ^ nxi6 + XT=i 3Wj, where the r{ e R, q{ e Z, x^- G AsAMaj-! f° r s o m e * w i t n either r» ^ 0 or qi =£ 0. Thus (^( r *)i(zij)&i + ^ft(z»i)&i)(*) i=l

= a

J

=

^0*0 >

i=l

where bi = /x(x^), i = 1 , . . . , n^. Hence ({(xij)aj

\i = 1, • • • ,71,;.; = 1 , . . . , k}) = p.

Thus // is finitely generated. (3) => (1): Suppose that R does not have a composition series. Then R has either an infinite strictly descending chain or a strictly ascending chain of ideals. Suppose that R has an infinite strictly descending chain of ideals R = J 0 D 11 D . . . D In D . . . . Define the L-subset \i of R by u

W

J

A

^w-\ 1

i f x € J n \ J „ + i f o r n = Oll>...

ifarengo/n.

Then /x is an L-ideal of R. Take any ( x i ) a i , . . . , (xk)ak Q V- Let x G In \ J n + i , where n is such that ^ £ { a i , . . . ,ak}. Then ( ( x i ) a i , . . . , (xk)ak) (x) = V { ( E t i ( ^ ) i ( ^ ) a , + £ t i ^ ( ^ ) a j ( x ) I r< G JJ, i = l,...,fc; A; G N} = ai for some i = 1 , . . . , A;. However, /x(x) = ^ - j - 7^ a*. Thus \x is not finitely generated. A similar argument shows that if R has a strictly ascending chain of ideals, then R has an L-ideal which is not finitely generated. ■ Corollary 3.11.10 R has a composition series of length n if and only if V{|Jra(^)| I fj, is an L-ideal of R} = n + 1. Proof. The result here follows easily from Theorem 3.11.9 and arguments involving level ideals of L-ideals. ■ The next result, which is the same as Theorem 3.11.5, gives a characteriza­ tion of R as an Artinian ring in terms of finitely generated L-ideals while the results in Section 10 give the characterization in terms of finite-valued L-ideals. The characterization here gives the immediate result that if every L-ideal of R is finitely generated, then every L-ideal \x of R such that ^(0) = 1 has an L-primary representation, as we will see in Chapter 6.

128

Corollary 3.11.11 Let R have an identity. generated if and only if R is Artinian. ■

Every L-ideal of R is finitely

We now apply our results to several algebraic situations. Theorem 3.11.12 Suppose that R has a composition series of length n. Then the following assertions hold, where I is a nonzero proper ideal of R. (l)IfI = Qin.. .nQm is an irredundant primary decomposition of I, then m < n — 1. (2) If{xi,..., xm} is a minimal system of generators for I, then m < n — 1. (3) Every L-ideal fj, of R has a system of generators with no more than n 2 — 1 elements. Proof.

(1) Let Ik Qi for k = 1l ,,......,,mm.. If Ik4 = h+i, Ik === fil n?=1=:lQz 4 + i , then — ^2=1, nm 7I — i^k+lQi

contradicting the fact that I = Q\ f ) . . . fl Q m is irredundant. Thus D( (0) 2? D Jl D .m. . D J m = IJ D flDJiD...Dl 0). Hence m -f 1 < n. (2) Let Jk =: (XU. . .,Xfc), fc = 1 , . . . ,m. If Jfc = Jfc+i, then (2) Let Jk = ( x i , . . . ,Xk), k = 1 , . . . ,ra. If Jk = Jfc+i, then 7I == (( {{ xx ii ,, .. .. .. ,, xx m }} \\ {{ xx f c + ii }} )) ,, m

fc+

contradicting the minimahty of { # 1 , . . . , x m } . Thus iJ Z> JiJl DD . ...... D Jm R D Jm D (0) (0).. Hence m -f- 1 < n. (3) Suppose that Im(/z) = { a i , . . . , a m } , where ai > . . . > a m . By (2), a minimal system of generators for fiaj has < n — 1 elements, j = 1 , . . . ,m. Let { x i j , . . . , Xfcj} be a minimal system of generators for fiaj, j = 1 , . . . , m. If a: G fjLaj\^aj-i » then s = 2 i = i rixij + J2i=i n&ij, where some Xij G ^ a j A M ^ - I with either r* ^ 0 or n2- ^ 0. Thus fc

k =

i)l(^ij)aj i=l

2=1

,)(*) =

a

j

= /^(x).

Hence S = {(xij)aj | i = 1 , . . . , n — 1; j = 1 , . . . , m} is a system of L-generators for [i. Since m < n-fl by (2) and Corollary 3.11.10, we have that |£| < n 2 —1. ■ Suppose that R has an identity and R = f?i © . . . © i ^ , where ft; is a subring of it, i = 1 , . . . ,n. Then an L-ideal /x of f? is said to be compatible with the above decomposition of R if fi = Yl7=i H-Ri- We recall that Ri is an ideal of R and /x|fii is an L-ideal of i?, z = 1,..., n.

129 Theorem 3.11.13 Suppose that R has an identity and R = F\ 0 . . . 0 Fn, where Fi is a field, i = 1 , . . . ,n. Iffi is a compatible L-ideal of R with respect to the above decomposition of R, then /x has a generating system with no more than n elements. Proof. Since fi = YM^IIAF^ A* = ®£=iHFi by Theorem 3.3.7. Let e; be the multiplicative identity of Fi, i = 1 , . . . ,n. Then |Im(/x|/r.)| < 3 and 0 G Im(^|jr.) if n > 1. Thus either {(e;)^( ei )} is a minimal generating set of / i ^ or yu(ei) = 0, i = 1 , . . . ,n, as in Theorem 3.11.7. Thus /x is generated by a subset of {(ei)M(c.) | z = 1 , . . . ,n} by Theorem 3.1.30. ■ Let S/K be a finite dimensional separable field extension and let F/K be an arbitrary field extension. If d, i = 1 , . . . ,n, are the inequivalent field composites of S and F over K, then S®K F = C\ 0 . . . 0 C n , [67, p. 87]. Hence Theorem 3.11.13 has an immediate apphcation here. Other apphcations along these fines can be found in [67].

3.12

EXERCISES

Let R be a Dedekind domain. Let L = [0,1]. 1. (112) Show that every prime fuzzy ideal and irreducible fuzzy ideal is (generalized) maximal. 2. (112) Let v\,..., i/n, and /x be fuzzy ideals of R such that /x = v\... vn and ^i(O) = . . . = vn(0). Prove that /z(0) = 1/^(0), i = 1 , . . . , n. Prove also that if i/i,..., i/n are finite-valued, then fia = (^i)a---(^n)a Va G [0,/i(0)]. 3. (112) Let ii, v, and A be finite-valued fuzzy ideals of R such that ji(0) = u(0) = A(0) and Va G [0,/x(0)],/xa 7^ (0). Prove that if /xi/ = /xA, then 1/ = A. 4. (112) Let 1/ be a fuzzy ideal of R such that Im(i/) = {a$,ai,... ,an}, where ao < a\ < . . . < an. Prove that there exists fuzzy ideals 6 and fi of R such that v = <5/i, /x is a fuzzy maximal ideal of i2 with ao 6 Im(/x), Im(£) = {ao, a\,..., a n } if i^ai is not a maximal ideal of R and Im(5) = {a\,..., an} if i/ai is a maximal ideal of R. 5. (112) Let v be a finite-valued fuzzy ideal of i£ such that i/(0) = 1. Prove that 1/ is a product of a finite number of fuzzy maximal ideals. Prove also that this factorization is unique except for the order of the factors. Let R be a commutative ring with identity and let L = [0,1]. A fuzzy ideal of R is called fuzzy principal if it has a minimal generating set S such that if xa and yb G S with a > b, then x = ray for some ra G R.

130

6. (15) Suppose that \i is a fuzzy ideal of R with the sup property. Prove that \x is fuzzy principal if and only if every level ideal of \x is principal. 7. (15) Prove that R is a principal ideal ring if and only if every fuzzy ideal of R with the sup property is fuzzy principal. 8. (15) Let R be Artinian. Prove that R is a principal ideal ring if and only if every fuzzy (generalized) maximal ideal of R is fuzzy principal. 9. (61) Suppose that R is a noncommutative ring. Use the techniques of Section 1.9 to show that every fuzzy left ideal of R is a fuzzy two-sided ideal if and only if every crisp left ideal is a two-sided ideal. 10. (61) Suppose that R is a noncommutative ring. Use the techniques of Section 1.9 to show that R is regular if and only if \x A v = [iv holds for every fuzzy right ideal fi of R and every fuzzy left ideal v of R.

131

Chapter 4

L-SUBMODULES In this chapter, we introduce the reader to some ideas of fuzzy submodules. The results are taken primarily from [187, 235, 250, 258, 259]. A left module or module over a commutative ring R with identity is an additive group M together with an operation • from Rx M into M such that (1) ( r + s) • x = r • x -f s • x, (2) r • (x + y) = r • x + r • y, (3) r • (s • x) = (rs)- x) (4) 1 • x = x, Vr, s G -R, x, y € M. We write rx for r • x. If i? is a field and M is a module over R, then M is called a linear space or vector space over R. Throughout this chapter, unless otherwise stated, R is a commutative ring with identity 1 and 1 / 0, M is a module over R, and 0 denotes the zero element of M. Also / denotes a nonempty index set.

4.1

Basic Concepts

In this section, we first consider several operations of L-subsets of a module induced by the addition and scalar multiplication. We then introduce the notion of an L-submodule. Definition 4.1.1 Let fi, v G LM. Define fj, + z/, — /J, G L

M

as follows:

(fi + u){x) = V{/x(2/) A v{z) | y, z € Af, y + z = x}, (-/x)(x) =

fi(-x)

Mx G M. Then fi + v is called the sum of fi and v, and —fi the negative

of fi.

132

Let fj,{ G L M , 1 < i < n and n G N. Since + is associative and commutative, we can consider /xi -f /X2 + • • • + ^n and write it as Y^i=i M*Let \ii G LM for each i e I,\ I \> I. Then £ i € / /x; G L M is defined by

($3/^)0*0 = V{Aie//Xi(x») I Xi G M,i G 7, ^Txi = x} iei

V a: G M, where Ylxi — ^2ieiXi anc ^ there a r e a ^ most finitely many x^s not equal to 0. J2iei & 1S c a u e d ^ n e weak sum of the /x^'s. Clearly, when J = { 1 , 2 , . . . , n} and n > 2, X ^ e / ^ = ZlILi M»Definition 4.1.2 Let r E R and /x G LM. Define r/x G LM as follows: (r/x)(x) = V{/x(y) I y G M, ry = x} V x £ M. Then rfi is called the product of r and /x. Theorem 4.1.3 Let r, 5 G R and /x, z/, £, /x^ G L M , z G / . TTien the following assertions hold: (1) 1/x = /x, (-l)/x = -/x;

f2jn w = i w ; (3)

/J>

Q v => rfi C rv\

(4) K5^) = (rs)w

(5) r(/x + v) = r/x +rv\ (6) r(Uieifii)

=

Uieir/j,i;

(7) (r/x)(rx) > /x(x) \f x e M; (8) £(rx) > /x(x) V x G M <£> r/x C £; ^ (r/x + 51/) (rx + 5y) > /x(x) A v(y) \/ x, y E M; (10) £(rx + sy) > u(x) A v(y) V x, 1/ G M 4=> r/x +5Z/ C f. Proof. Parts (1) through (6) are straightforward. (7) Now (r/x)(rx) = V{/x(y) | 2/ G M,r?/ = rx} > /x(x) \/ x e M. (8) If £(rx) > /x(x) \/ x e M, then (r/x)(x) = V{/x(y) | 3/ G M, ry = x} < V{£(n/) I 2/ G M,r2/ = x} = £(x) V x e M, i.e., r/x C f. If r/x C £, then £(rx) > rfi{rx) > /x(x) V a: G M by (7). (9) By (7) and the definition of +, it follows that (r/x + sv)(rx + sy) > (r/x)(rx) A (sv)(sy) > /x(x) A v(y) V x, y G M. (10) Suppose that £(rx + 57/) > /x(x) A ^(y) V x, 2/ G M. Then (r/x + 51/) (z)

— V{(r/x)(ix) A (5z/)(v) | w , v G M , u + v = 2} = V{(V{/x(x) I x G M, rx = ix}) A (V{I/(T/) | 2/ € M, sy = v}) I ?x, v G M, w + v = z} = V{/x(x) A v(y) I a;,2/€M,ra; + si/ = 2}

< a*)

133

V z G M. Hence r/x + sv C f. Conversely, suppose that r/x + sv C £. Then £(rx + sy)

> > >

(r/x + sv) (rx + sy) (r/x)(rx) A (s^)(sy) u(x)Av(y) (by (7))

V x, 2/ G M. ■ Theorem 4.1.4 Let r, s € R and fi e LM. Then (1) r\i C /x <£> /x(rx) > /x(x) V x e M, (2) r/x + s/x C /x <£> /x(rx + sy) > /x(x) A fi(y) Proof.

VxjeM.

The proof follows from (8) and (10) of Theorem 4.1.3.

■

Theorem 4.1.5 Suppose that N is also a module over R and f is a homomorphism of M into N. Let r, 5 G R and /x, v G LM. Then (1) /(/x + v) = /(/x) + /(,/), (*; / M = r/(/x), ^ /(r/x + sv) = r/(/x) 4- */(i/). Proof. (1) This assertion follows easily. (2) We have that /Mfo)

= = = =

=

V{(r/x)(x) | x G M, / ( x ) = y} V{V{/x(u) | w G M, rtx = x} | x G M, / ( x ) = y} V{/x(u) | u € Af, f(ru) = y} V{p(u)\ueM,r(f(u))=y}

rf(n)(y)

V y G N. Hence /(r/x) = r/(/x). (3) This is an immediate consequence of (1) and (2). ■ Definition 4.1.6 Let( G LR and /x G L M . Define (• /x, CO M G £ M as follows: (C • /x)(x) = V{C(r) A /X(T/) | r G iJ, y G M, ry = x}, (C0/x)(x) = V{Aj l =1 (C(ri)A/x(x i )) | r2- G R,Xi G M, 1 < i < n, n £ N,X^ = 1 r;Xi = x}, Theorem 4.1.7 Let /x G L M . Tften fij /or all r e R, l{ r } • /x = r/x; (^) /or allr eR, x e M, n

(l{r}QfJL)(x)

= V{A?=1/x(a;i) | x2- G M, 1 < z < n , n G N,r ]Tx z - = x}. ■ 2=1

134

Definition 4.1.8 An L-submodule

of M is an L-subset /J, G LM such that

(Ml) M) = 1, (M2) /x(rx) > fj,(x) V r G R and x G M, (MS) fi(x + y) > /x(x) A /x(y) V x,y e M. When L = [0,1], an L-submodule is called a fuzzy submodule. We denote the set of all L-submodules of M by L(M). Since R is a module over itself, it follows from Definition 4.1.8 that \x is an L-submodule of the module R if and only if /x is an L-ideal of the ring R. Since — \x — — x V x G M, condition (M2) implies that n{—x) > ji{x) V x G M. Hence \i G L(M) if and only if fj, is an L-subgroup of the additive group of M and satisfies condition (M2). Theorem 4.1.9 Let /x G LM. Then \i G L(M) if and only if /i satisfies con­ dition (Ml) and the following condition: (M4) \x{rx + sy) > fi(x) A /i(y) V r, s G R and x, y G M. Proof. Suppose fi G L{M). By the definition, fi satisfies condition (Ml). Since /x also satisfies conditions (M2) and (M3), it follows that fi(rx + sy) > fi(rx) A fi(sy) > ji{x) A fi(y) V r, s G R and x, y e M. Thus /x also satisfies condition (M4). Conversely, assume that \x satisfies conditions (Ml) and (M4). Then n(rx) = fi(rx + r6) > fi(x) A /i(0) = ji{x) V r G i? and x e M and /x(ar + 2/) = /i(lx + ly) > xx(x) A fi(y) V x, y e M. Thus /x satisfies conditions (M2) and (M3). Hence /x G L(M). ■ Theorem 4.1.10 Let ji G L M . Then /x G L(M) i/ and only if fi is an Lsubgroup of the additive group of M and satisfies the following condition: (M2)' rfjL C /x V r G R. ■ Theorem 4.1.11 Let \i G LM. following conditions: (Ml)' 1{0} C n, (MB/ rfj,C /j,Vr eR, (MS/ \i + fJ. C fi. ■

Then /x G L(M) z/ and only if \i satisfies the

135

T h e o r e m 4.1.12 Let \i G LM. Then \i G L(M) if and only if fi satisfies the following conditions: (Ml/ l w C /x, (M4/ r/i + s / i C / z V r , s G i J . ■ We now discuss some basic properties of L-submodules. If /x, v G L M , then [i + v is defined as in Definition 1.2.1, i.e., Mx G M, (/x + i/)(x) = V{/x(?/) A v(z) \y,z G M, x = y + z}. T h e o r e m 4.1.13 Let \i,v G L(M). Tften /x + v G L(M). Proof. By Theorem 1.2.9, /x + z/ is an L-subgroup of the additive group of M. By Theorem 4.1.3(5) and Theorem 4.1.10, r(fi + v) = rfi + rv C /j, + v V r G R. Hence /x + i/ G L(M). ■ The following theorem follows from the preceding two theorems. T h e o r e m 4.1.14 Let r{ e R and ^i G L(M), 1 < i < n, n G N. T/ien

Z7=i^i

e L(M). M

The following theorem is a generalization of Theorem 4.1.13. T h e o r e m 4.1.15 Let \ii G L(M) for each i G / , where \I\ > 1. Xften ^ i G / /x* G L(Af). ■ T h e o r e m 4.1.16 Let C G L/(/J) and /x G L(M). 27ien C 0 /z G L(M). Proof.

Clearly, (C 0 /i)(<9) = 1. Now

(C©^)(™0

= >

VJA^CC^^A/x^)) | 5 i G i ? , ^ e M , l < i < n , n G N , E?=i **t = rx} V{AjL1(C(rri) A/x(xi)) | r» G # , x* G M, 1 < i < n,n G N,

>

V{A?=1(C(ri) A/i(xi)) | r< 6iJ,Xi G M , 1 < i < n , n e N ,

=

(C©M)(*)

EILi (r^Jx* = r 4

V r G .R, x G M. Since A is infinite V-distributive, it follows that (C 0 n)(x + y) > (C 0 /*)(*) A (C 0 /i)(y) Vx,y G M. Hence C© M £ £ ( M ) . ■ Suppose M = R. Then by Theorem 4.1.16, if C, M € £ / ( # ) , then £ 0 M G U(i?).

136

4.2

L-Submodules of Quotient Modules

In this section, we introduce two methods of constructing L-submodules of a quotient module. Theorem 4.2.1 Let v G L(M) and let A be a submodule of M. Define £ G LMIA as follows: £([*]) = V M u ) | u G [x]} (4.1) Vx G M, where M/A denotes the quotient module of M with respect to A and [x] represents the coset x + A. Then £ G L(M/A). Proof. As in Theorem 1.3.12, £ is an L-subgroup of the additive group of quotient module M/A. Now t(r[x])

= = = > = > =

£([rx]) \Z{v(v) | v G [rx]} V{v{rx + y) | y G A} \/{i/(rx + rz) \ z € A} \/{v(r(x + z)) | z G A} V{i/(x + z ) | z e A } V{i/(u) I u G [x]}

=

£([*])

V r G iJ and x G M. Hence £ G L(M/A). ■ We now consider a special case of Theorem 4.2.1, i.e., the case that L is a chain and v G L(M). Let a E L and let ^4 = va. Then A is a submodule of M. Thus by Theorem 4.2.1, £ G L(M/A), where £ is defined by equality (4.1). Let x e M and consider £([x]). If [x] = 4 , then £([x]) = 1. If [x] ^ A, then x ^ A and so v{x) < a. Thus for any y G [x], there exists z e i such that y = x + 2. Hence

^(y) = K x + z) > z/ ( x ) A K*) — K^OSimilarly, z/(x) = v(y + (-z)) > v(y) A ^ ( - 2 ) = 1/(2/). Consequently, v{x) = v(y). Therefore, £([x]) can be given by

«N) = { I (x)

if v(x) > a otherwise

V x e M. For the case in which A = i/[a], a similar conclusion can also be obtained. We now investigate mapping properties of the L-submodule £ denned in Theorem 4.2.1. Let //, v G L(M) be such that /1 C 1/ and assume L is regular. It is known that both /J,* and 1/* are submodules of M. Clearly, fi* C 1/*. Thus /x* is a

137

submodule of v*. Moreover, it is also clear that v\v* G L{y*). Therefore, it follows from Theorem 4.2.1 that if we define £ € W'l** as follows: e(M)=V{i/(z) | *€[*]} V x G z/% where [x] denotes the coset x + z/*, then £ G L(z/*//x*). The Lsubmodule £ is called the quotient of v with respect to /x and written as vj\x. Let ^ G L(M) and assume that N is also a module over R and / is a homomorphism of M into JV. As in Theorem 1.3.13, /(/x) G L(xV). Definition 4.2.2 Le£ M and N be R-modules and let /x G L(M) and v G L(N). (1) A homomorphism f of M onto N is called a weak homomorphism of ji into v if /(/x) C v. If f is a weak homomorphism of fi into z/, then we say that /i is weakly homomorphic to v and we write /J, ~ v, or simply /i ~ v. (2) An isomorphism f of M onto N is called a weak isomorphism of /x into v if /(/x) C v. If f is a weak isomorphism of /x into u, then we say that [i is weakly isomorphic to v and we write ji — v, or simply /i ~ v. (3) A homomorphism f of M onto N is called a homomorphism of /x onto v if f(n) = v. If f is a homomorphism of /x onto v, then we say that /x is homomorphic to v and we write fi & v, or simply /i « v. (4) An isomorphism f of M onto N is called an isomorphism of \i onto v if /(/x) = v. If f is an isomorphism of /x onto v, then we say that /x zs / isomorphic to v and we write \x — v, or simply /x = v. The proof of the next four results are similar to the those of Theorems 1.4.10-1.4.13. Theorem 4.2.3 Let /x, v G L(M) regular. Then v\v* « z///x. ■

be such that /x C v. Assume that L is

Theorem 4.2.4 Let v G L{M). Suppose that N is also a module over R and £ G L(N) is such that v w £. Suppose that L is regular. Then there exists fi G L(M) such that [i C v and vj\i = £|$*. I Theorem 4.2.5 Let /x, v G L(M). Assume that L is regular. Then i//(/x fl v)

- (/* + ^)/M- ■ Theorem 4.2.6 Let /x, z/, £ G L(M) be such that /x C v C £. Suppose that L is regular. Then (£//x)/(z///x) = £/v. ■

138

The next theorem suggests another method of constructing L-submodules of a quotient module. Theorem 4.2.7 Suppose that R is an integral domain, M is a divisible module on R (i.e., r G R\{0} and x£M=>3y£M such that ry = x), and A is a prime submodule of M (i.e., A is a submodule of M, and rx G A, r G R, x G M => either r = 0 or x G A). Let v G L(M). Define rj G LM
r,([x]) = l \

, , v u

L ^ue[x] \ ) V x G M. Then n G

if =A

^

otherwise

(4.2) v

J

L(M/A).

Proof. Clearly, 77(^4) = 1. Let r G R, x G M and consider rj(r[x]). If r[x] = A, then 77(7-[#]) = 1 > T/([X]). If r[x] ^ A, then r / 0 and [x] ^ A. Thus since M is a divisible module and A is a prime submodule of M, it follows that A = {rz I z G A}. Hence n{r[x\) = rj([rx]) = Ave[rx]i/(v) = AyeAi>(rx + y) = AzeAv(rx + rz) = AzeAv(r(x + z)) > Az^Av{x + z) = Au€[a.]i/(ix) = v([x]). Now let a:, 2/ G M and consider rj([x] +[2/]). If [x] = A or [7/] = A or [x] -f- [?/] = .A, then clearly

^(W + M)>^(W)Ar;([2/]). Suppose [x] 7^ A and [2/] ^ ^4 and [x] + [y] ^ A. Then */([*]+ [2/])

= = > > = =

Awelx]+[v]"(w) Au>tV>eAv((x + u') + (y + v')) Au>iV'eA(v{x + u') A v{y + v')) (Au>eAv{x + u')) A (Av>eAi/(y + 1/)) (AueW^M) A (A^ji/fi;)) *KM)AT7([y]).

Hence 77 G L(M/A). ■ As a consequence of Theorem 4.2.7, we have the following result. Theorem 4.2.8 Let v G L(M) and A be a submodule of M. Assume that R is a field. Then the L-submodule rj defined by equality (4.2) is an L-submodule of the quotient module M/A. ■ Theorem 4.2.9 Let v G a divisible module over R, Tor(M) — {x I x G M,3r rj defined by equality (4.2)

L(M). Assume that R is an integral domain, M is and A is the torsion submodule of M (that is, A = G R\{0} such that rx = 9}. Then the L-submodule is an L-submodule of the quotient module M/A.

Proof. By our assumption, it is well known that A is a prime submodule of M. Consequently, the desired result follows immediately from Theorem 4.2.7. ■

139

4.3

L-Submodules Generated by L-Subsets

Theorem 4.3.1 Let /j,i G L(M) for each i e I, where \I\ > 1. Then fli€//Xi G L(M). M Let n G LM. By Theorem 4.3.1, n{i/ | /x C v, v G £(Af)} is an L-submodule of M, called the L-submodule generated by the L-subset /x. We denote this L-submodule by (xx) , i.e., (fi) =n{v

| /iCi/,i/eL(M)}.

Let £ G L(M). If £ = (/x) for some /x G L M , then ji is called a generating L-subset of £. Let /x, z/ G LM. The following can be easily verified. (1) xx G L ( M ) *> (AX) = xx. (2) /x C i/ =» (xx) C .

(^(/xj^C^UGLCTV), where iV is a submodule of M, and /X|JV a ^ d (xx) |JV are the restrictions of xx and (/x) to xV, respectively. Moreover, (1A) = 1(A) > where A is a nonempty subset of M and (A) is the submodule generated by A. Theorem 4.3.2 Let fii G L(M) for each i G / , \I\ > 1. Then (UieIfj,i)

=

Proof. Now Yliei Mi 1S s e e n t o be the smallest L-subgroup of the additive group of M that contains all the xxi's. Thus it follows from Theorem 4.1.15 that Y^ieiVt e L(M)- Hence (Uiei^i) = Y,iei fr- B Theorem 4.3.3 Let fi G L M . Define £ G LM as follows: £{x) = 1 ifx = 9, £(x)

=

Mx G M, i.e., i = l{e}

V{A?=1xx(xi) | n G # , a* G M, 1 < i < n, n G N, X^iLi rixi — x} otherwise

^ 3) ^ ' '

U (1R 0 xx). Then (/x) = f.

Proof. Clearly, xx C f. By definition, £(0) = 1. Let r G R and x G M. If rx = 0, then £(rx) = 1 > £(#)• Suppose rx ^ 6. Then x ^ 0 and £(rx)

= > = >

V{A?=1/x(xi) V{A?=1xx(xi) V{Af=1/i(xi) V{A^=1xx(xi)

|n |^ |n |^

G i?, Xi G M, 1 < G i?, x{ G M, 1 < G R, Xi G M, 1 < G # , ^ G M, 1 <

z < n, n i < n, n i < n, n i < n, n

G N, £ ? = 1 r i x * = r x ) G N, YA=\ rrixi = rx) G N, r ( £ ? = 1 ^Xi) = rx} G N, £ " = 1 riXi = x}

140

Hence £(rx) > £0*0 V r G R and x G M. Let x, y e M. If one of x, y and x + y is equal to #, then £(# + y) > £0*0 A £(y). If x, y, and x + y are all not equal to 5, then using the infinite V-distributivity of A we can show that £(# + y)> £0*0 A£(y). Thus £(x + y) > f (a;) A£(y) Vx,yeM. Hence £ G L(M). Finally, let r] G L(M) with \i C 77. By Theorem 4.1.9, it follows that £ C 77. Hence (/x) = £. ■ Theorem 4.3.4 Let x e M and a e L. Then {xa) =

l{e}U(U{(rx)a\reR}).

In particular, when x = 8 or a = 0, (x a ) = 1{0}. Proof.

By equality (4.3), the result is evident.

■

Theorem 4.3.5 Let \i G LM and assume that /z*\{0} = {xi,X2,. ■. ,#n}> where n G N. Ze£ a* = /i(#i) and /i; = (^)a i 5 1 < i < n. T/ien (/x) = l ^ j . U ^, where m

v = U { ^ r^.^. |r^. G i?, 1 < j < m, 1 < ix < i2 < . . . < z m < n). i=i

Proof.

4.4

This is an easy consequence of Theorem 4.3.3.

■

Free L-Submodules

Definition 4.4.1 Let M be a free module over R on the set B with respect to the function f : B —> M. Let (3 be an L-subset of B. Let £ be an L-submodule of M. Then £ is said to be free with respect to (3 if f{/3) = £ on f(B) and V modules Y over R and L-submodules rjofY with g : B —» Y and #(/?) = rj on g(B), there exists a unique homomorphism h of M into Y such that g = h o f and /i(£) C 77. Theorem 4.4.2 Let (3 be an L-subset of a set B. Then there exists a free L-submodule £ of a free module M over R such that the following properties hold: (1) If\P(B)\ is infinite, then |£(M)| = \0(B)\. (2) If\P(B)\ is finite, then |£(M)| < 2 ^ B ) I . (3) If L is a chain, then |£(M)| < \/3(B)\ + 1.

141

Proof. Let M be a free module over R on the set B with respect to the function / : B —> M. Then we may assume that / is injective and f(B) is a basis for M. Now V x e M, x ^ 0, x has a unique representation of the form k

* = 5>
(4.4)

2=1

where 0 ^ n € R, z{ £ B, i = 1, 2 , . . . , fc; /(*<) ^ /(*,■) if i ^ j . Define the L-subset £ of M as follows:

X)

^

f 1 ~ \ A^Pizi)

if x = 0, ifx = Y.U

Ufizi)

+ 6 (as in (4.4)).

(4 5)

'

Then £ is an L-submodule of M. Now V x e M, f(0)(x) = \/{fi(z) | f(z) = x}. Thus by equation (4.5), m(x)

=

l 0

***/(*)

, 46)

Thus /(/?) C £. In fact, /(/?) = £ on / ( £ ) . Let Y be a module over R, g & function of B into Y, and 77 an L-submodule of Y with g : B —► Y such that #(/?) = 77 on #(£). Since M is free, there exists a unique homomorphism hoi M into Y such that hof = g. We now show that ft(0 C 77. Let 7/ G Y If y = 0 or if /i_1(2/) = 0, then clearly h(f)(y) < ^(2/). Suppose that y ^ 0 and h~1(y) ^ 0. Let x G M b e such that fo(x) = y. Then a; has a unique representation as in equation (4.4). Since h o / = g, we have that y = h{x) = Y2i=i ri9(zi)- Since g(fi) C 77, we have that 77 o g D /?. Thus f(») > A t i ( r ? o 5 ) ( ^ ) > A?=i/J(*i) = £(*)• Hence 77 2 MO-

'

Theorem 4.4.3 Let M and M ' 6e free modules over R on sets B and B', respectively. Let (3 and /3f be L-subsets of B and B', respectively. Let fj, and fj! be free L-submodules of M and M' with respect to /3 and /?', respectively. If there exists a one-to-one function g of B onto B' such that g(/3) = /?', then there exists an isomorphism f of M onto M' such that /(/x) = /z'. ■ Theorem 4.4.4 Let \i be an L-submodule of M. Then there exists a free mod­ ule X and a free L-submodule £ of X such that fi is isomorphic to a quotient

oft

142

Proof. Construct X on M and £ with respect to /x as in Theorem 4.4.2. Let g : M —> M be the identity map and / : M -» X. Then £ = /(/x) on / ( M ) and there exists a homomorphism h of X onto M such that ho f = g. There exists an isomorphism i of X/(Ker h) onto M such that i o k = h, where k is the natural homomorphism of X onto X/(Ker /i). Then i _1 (/x) is the desired quotient. ■ Definition 4.4.5 Let B be a subset of M. For x G M, we say that a represe­ ntation of x, x = 2 i = i rizii where ri G R and Zi G B (i = 1, 2 , . . . , k) is irredundant if r\Zi ^ 0, i — 1 , 2 , . . . , k, and Zi =£ Zj if i ^ j . In this case, we k

sometimes write x = £ *r 2 z;. 2=1

Definition 4.4.6 Let B be a subset of M and fi an L-submodule of M. Let a be an L-subset of B such that a C /x. Then a is said to be a B-linearly independent L-subset of /x if (1) B is a linearly independent subset of M and (2)V x G M, if x = 5^i=i rizi is an irredundant representation of x, where r t G R and zi G B (i = 1, 2 , . . . , k), then /JL(X) = AjLjor^). It is clear that if cr is a B-linearly independent L-subset of /x, then a = fi onB. Theorem 4.4.7 Let B be a subset of M and a an L-subset of B. Let £ be the L-subset of M defined as follows: £(x) = 1 if x = 9, n

£(x)

=

y{Af=1a(zi)

| x = ]T *nzi,rieR,ZieB,

i = 1,...,n;

2=1

n G N}

otherwise

Then £ = (cr). Proof. Clearly, £(0) = (a) (0). Let x e M,x ^6. Let /i be any L-submodule of M such that /x D cr. For any irredundant representation x = Y27=i rizi> w e have A? =1 a(zi) < Aj^/xfe) < M E i L i f ^ ) = **(*)• Hei *ce f(x) < fi(x). Thus £(#) < (cr) (x). Hence £ C ( a ) . We now show that £ 2 (^) by showing that £ is an L-submodule of M such that £ D cr. Clearly, £ D a. Let a;, y G M. If x + y = 0, then £(x H- y) = 1 > £(:r) A £(2/). Suppose that x -\- y ^ 0. Let x = n

m

i=i

i=i

Y^ *r{Zi and 2/ = ^

* r j z j - Then x + y has an irredundant representation

*+» = £ * « fc=l

143

which is determined by summing the irredundant representations of x and y followed by combining and simplifying terms for which bi = bf- for some i and j , and discarding terms equal to 9. Hence Z(* + V)> AJU (A?=1 £(#) A£(y). Let r £ R. If rx — 0, then £(rx) > £(#). Suppose that rx ^ 9. Now rx = Y^i=\(rri)zi' ^y discarding terms in this representation of rx which equal 0, we obtain an irredundant representation for rx, rx = ^ *(rrj)zij,

where

i=i

{*ii, ■••,**,} Q {*!,...,2 n }. Thus f(rx) > A ^ c ^ . ) > A j L ^ f e ) . Hence £(rs) > £(*). ■ Definition 4.4.8 Lei /x be an L-submodule of M, B a subset of M, and cr an L-subset of B such that crC.fi. Then cr is said to be a basis for ji if B is a basis of M, cr is B-linearly independent, and (a) = ji. Example 4.4.9 Let R denote the field of real numbers, let V = R 2 , and L = [0,1]. Let /i be the fuzzy subspace l x , where X = {(x,#) | x £ R}. Let B = {(1,1), (1,0)}, B' = {(1,1)}, a W 5 r / = {(1,0), (0,1)}. Let a and a' be Lsubsets B and B', respectively, defined as follows: cr((l, 1)) = 1, cr((l,0)) = 0 and cr'((l, 1)) = 1. Then a is a basis for /i. Even though B1 generates xx* and cr' is a B'-linearly independent L-subset of /x, cr' is not a basis for /x since B' does not generate V. Let (3 be the L-subset of B" such that (3 = /x on B". Then (1,1) = (1,0) -h (0,1) is an irredundant representation of (1,1), but (3 is not a B"-linearly independent L-subset of fi since /3((1,0))A/J((0,1)) = 0 ? U = M(1,1))Theorem 4.4.10 Let /x be an L-submodule of M, B a subset of M, and cr an L-subset of B. Suppose that a is a B-linearly independent L-subset of /x. Then a is a basis for /x if and only if B generates M. Proof. Suppose that B generates M. Let x £ M, x / 0. Since B is a basis for M, x has an irredundant representation, x = Y^i=z\rizi^ where zi £ B, i = 1 , . . . ,n. By (2) of Definition 4.4.6, fi(x) = Ai=1a(zi). Hence (cr) (x) > /i(x) by Theorem 4.4.7. Thus (cr) D /x. Clearly, (cr) C xx. Hence (cr) = xx. Thus cr is a basis for \x. The converse is immediate. ■ Theorem 4.4.11 Let xx be an L-submodule of M. Then xx has a basis if and only if ji is free.

144

Proof. Suppose that ji is free. Then we may assume that M is free over R on a set B and that [i is free with respect to an L-subset /? of B. Let M' be a free module over R on B and construct a free L-submodule / / of M' using (3 as in Theorem 4.4.2. It follows by Theorem 4.4.3 that there exists an isomorphism f of M onto Mf such that / ( ^ ) = \il'. Now y! has a basis as can be seen by equation (4.5). It thus follows that /i has a basis. Conversely, suppose that /i has a basis a. Then there exists a subset B of M such that a is an L-subset of B, a is jB-linearly independent, and B is a basis for M. We let / of Definition 4.4.1 be the identity map. Since a is a basis for /x, f(o) = fj, on B. Now let Y be any module over R on B with g : i? —> Y. Let 77 be any L-submodule of Y such that g(a) = 77 on #(£). Let x G M and let a: = XliLi r ^ 2 D e a n irredundant representation of x, zi G B, i = 1,..., n. Define /i from M into Y by /i(x) = ]T^ =1 rig(zi). Then /i is a homomorphism of M into y. Now 77(/i(x)) = 77(^X1 r ^ ^ ) ) > A ^ ^ ^ C ^ ) ) = A? =1 cr(^) = /x(x). Hence /i(/z) C 77. Clearly, ho f = g. Let /i7 be any homomorphism of M into y such that h' o / = g. Let x 6 M have the irredundant representation x = YH=I rizi- T n e n h'(x) = YA=I rih'(zi) = S r = i r ^ ( ^ ) = Hx)' Hence h is unique. ■ Theorem 4.4.12 Suppose that L is a chain. Let R be a field. Then every finite-valued L-subspace of V is free. M Theorem 4.4.12 can be proved using techniques of this section. However, we do not prove it here since we consider the situation where L is a chain in Chapter 8. Theorem 4.4.13 Suppose that R is a field and that M has dimension n > 2. Then there exists an L-subspace of M which is not free. Proof. Let q = 2n — 1, and choose 1-dimensional subspaces S i , . . . , Sq of M such that Si nSj = {0}V i ^ j . For example, if {zx,..., zn} is a basis for M, we may take Sj to be the subspace generated by YA=\ rizii where r* e {0,1} and the sequence rxr2 .. .rn represents j as a binary number. The sets {9}, M\ Uqk=zl Sfc, Sk\{0}(k = 1 , . . . , q) partition X, and we can define ( {l,...,q} li(x) = < {k}, [ 0

iix = e if 1 < k < q and x 6 if x € M\ U£ =1 5fc

Sk\{6}

from M into the power set of { 1 , . . . , q}. It is easily seen that fi is an L-subspace of M. However, JJL cannot be free by Theorem 4.4.2 since |/x(M)| > 2 n . ■

145

4.5

Residual Quotients

For a subset X of M, recall that (X) denotes the submodule of M generated by X. For any x G M, we let (x) denote the submodule of M generated by {x}. For any submodule N of M, we have N : M = {r \ r G R, rM C N} and y/N: M = {r \ r G R, 3m G N such that rmM C iV} . In this section, we introduce and study the notion of residual quotients of L-subsets in preparation for the next three sections. Definition 4.5.1 For \i,v G LM and £ G LR, define the residual fj,: v G LR and ji : £ G LM as follows: \x\ v = U {77 I 77 G L H , T)- v C fi} /x:C = U { £ | £ € L M ,

C-fC/i}.

M

Theorem 4.5.2 Lei \±,v e L and C G l A T/ien (%) /i : z/ = U {r a | r G i?, a G L, r a • z/ C /i} ; (%) fi : C = U {xa I x G M, aeL, ( • x a C fi} . Proof.

(1) Clearly, by Definition 4.5.1, U {r a I r G i?, a G L, ra • z/ C /x} C fi : v

Let 77 G 1/^ and rj-iyC.fi. Let r £ R and 77(7*) = a. Then (r a • i') (a:)

=

— < <

V {ra(s) A 1/(2/) \seR, v

A

y G M, 57/ = x} r

{^OO ^(2/) 12/ ^ -W> z/ = x) (r)'v){x) u(x)

\/x G M. Thus r a • v C /x and so /i : 1/ C U {r a I r G i?, a € L , r a • z/ C /i} . Therefore, fi: 1/ = U {ra I r G i?, 0 6 I , r a • z/ C /i} . (2) The proof is similar to (1). ■ We now give some basic properties of residual quotients. Theorem 4.5.3 Let fi,v G LM and C G LR. Then (1) (fi : v) - v C ji, (2) ( . ( / i : ( ) C / i , (3) C • 2/ c fi <^ C £ M : v <& v Q v '• C-

quotients

146 Proof.

(1) By Definition 4.5.1,

[(/x : v) • v\{x)

= = = < <

V {(/x : i/)(r) A i/(y) | r 6 U, 2/ € M , r y = x} V{(V {r,(r) | 77 6 L * , 17 ■ u C /x}) A 1/(2/) | r G i?, y € M, ry = x} V{r7(r) A 1/(2/) | 77 6 LR, r e R, y G M , 77 • */ C xx, rt/ = x } V{(77 • i/)(n/) I 77 € L * , r G i?, y G M , 77 • v C /x, ry = x} V {/x(x) I 77 G L * , r G R, y G M , 77 • 1/ C xx, r y = x }

= M*) VxeM. (2) T h e proof is similar t o (1). (3) This is an immediate consequence of (1) a n d (2). ■ T h e o r e m 4 . 5 . 4 If ^ (i e J), 1/ € LM (1) (Diez/Xi) : v = nieI(fii : 1/);

and C G LR,

then

f«;(n i€ /^):C = n ie /(^:C). Proof.

(1) By Definition 4.5.1, (Hiez/Xi) : 1/

= = = = C

U (77 | 77 G L * , U {77 I 77 G L * , U (77 I 77 G L * , U {77 I 77 G L * , rit G /(/Xi : v).

77 • 1/ C nieI^i} 77 • v C xx; Vz G J | 77 C fii : v Vz G / } 77 C rii € /(/Xi : v ) }

=

V{(n i G / (xxi : i/))(r) A u(y) \ r G R, y ry = #} V{Ai€/((AXi : i/)(r) A i/fo)) | r G iJ, y ry = x} V{A i 6 / ((xxi : v) • i/)(ry) | r G iJ, y G ry = x} V {Ai G //Xi(ry) | r € iJ, y G M , r y = (Hiej/XiXx)

By T h e o r e m 4.5.3, ((n i € /(Ait : *)) ' y){x)

< < < = Vz G M . T h u s

(nte/(/4» : ^ ) ) • ^ ^ n i G / / X i ,

and so n» € /(/ii : 1/) C (Hie/XXi) : v.

G M, G M, Af, x}

147

Hence nieI(fii

: v) = (nieIfii)

: v.

(2) The proof is similar to (1). ■ T h e o r e m 4.5.5 Let /x, */ G LM and ( G LR. (1) If fie L(M), then /x: v = U {77 | 77 G LI(R), 77 • 1/ C xx} . (2)If
(a ( r ) ■ 1/)(a:)

= = < < <

= = C

(l{o}Ua ( r > )-i/ 1{0} • v U a • 1/ l{^Ua(r)-i/,

V {a< r) (s) A 1/(3/) \ s e R, y G M, 33/ = x} V {a A 1/(2/) I s G (r) , y G M, sy = x} V {(r a • v)(ry) \ t G JJ, y G M, i(ry) = *} V {/x(ry) I t G # , 2/ € Af, t(ry) = x} V M * ( n / ) ) \teR, yeM, t(ry) = x)

= Kx)

Vx G M. Thus a( r ) • ^ C l ^ j . U /x = /x and so U {77 I 77 G £J(-R), 77 • v C xi} D =

U {r a | r G # , a G L, r a • v C xx} xx : XA

Hence xx : 1/ = U {77 | 77 G LI(R), 77 • */ C xx} . (2) Clearly, U{£ | £ G L(Af), C ^ C / i } C /i : (. Let a; G M, a G L, and C * xa Q V, Let £ = ( x 0 ) . By Theorem 4.3.4, C-(x a )

(C • (rx)a)(y)

= = =

C-(lwU(U{(rx)a|reii})) C-lWU(U{C-(rz)a|rGi*})) lWU(U{C-Mfl|r€/J})),

= V {C(«) A (rx)a(z) \ s G R,z e M, sz = y} = V {((s) Aa\ s eR, srx = y} < V {C(t) Aa\teR, tx = y} =

(C ■*«)(*)

Vr e R and y G M. Thus £ • (rx) a C £ • x a Vr G i? and so U{(-(4

|rGi?}CC-xaC/x, C * C /X.

148

Since (xa) G L(M), it follows from Theorem 4.5.2 that U { £ | £ G L ( M ) , C-f C M } D / i : C .

Hence / i : C = U U | f GL(M),

C-^/i}.B

M

T h e o r e m 4.5.6 Let xx G L(M), v G L , and C 6 U ( i 2 ) . T/ien xx : zy G LI(R) and xx: C G L(M). Proof. Since l{o} • ^ C 1^} C /x, it follows that 1{0} C xx : v. By Theorem 4.5.5, it is easy to verify that (xx : i/)(n) A (xx : i/)(r 2 )

=

< =

(V { ^ ( n ) | 771 G L/(iJ), 771 • 1/ C /x }) A (V {772(7*2) I r/2 € L/(i?), 772 ■ v C /x }) V{77i(ri) Ar72(r2) | 771,772 G U ( i J ) , r/i • v C /x, 772 • 1/ C xx} V{(77i +r&)(ri) A (771 +77 2 )(r 2 ) | 77!,772 G L/(iZ), 771 • 1/ C xx, 772 • 1/ C xx} V{(T7I +77 2 )(ri - r 2 ) | 771,772 G L/(iJ), (771 + 772) • z/ C 771 • v + 772 • 1/ C xx + xx C xx} V {77(ri - r 2 ) I 77 G L / ( # ) , 77 • 1/ C xx } (xx: i/)(n - r 2 )

= >

V {77(sr) | 77 G L/(i?), 77 • v C xx } V {77(7-) I 77 G U ( i i ) , 77 • 1/ C xx }

= < <

V n , r 2 G # . Now (/x : u){sr)

= 0* = ")(r). Vr, s G i ? . Hence xx : v G LI(R). Similarly, xx : C G ZJ(Af). ■ Let xx, z/ G L(M) and £ G LI(R). /x : f is called the residual q u o t i e n t L-submodule of xx and £ and /x: z/ is called the residual quotient L-ideal of xx and v respectively. T h e o r e m 4.5.7 Let /x G LM. (1) If Vi eLM,ie 7, then xx : (Uiem) = n iG /(xx: z/;). (%) 7/C< €LR,ie 7, tficn xx : (U ie /Ci) = n iG/ (xx : &). Proof.

(1) By Definition 4.5.1, xx : (Uiem)

= = C

U {77 I 77 G L H , 77 • (U i€ /i/i) C /x } U {77 I 77 G L H , U i e j (77 • Vi) C /x } U {77 I 77 G L H , 77- i/i C xx }

149

Vz e / . Hence \i: (U i G /^) C niei(ii: (n i€ /(/x : Vi)) • ( U j € / ^ )

*/;). By Theorem 4.5.3, =

UjeI((nieI(fi

C C

U i € /((/x : ^ ) ■ i/,-) U j G //i

: i/*)) • Vj)

=

M-

Thus (n»€/(/x : i/i)) C fj,: (UieIUi). Hence \i: (U i€ ji/i) = (n i€ /(/x : vi)). (2) The proof is similar to (1). ■ Theorem 4.5.8 Let c e L and N be a submodule of M. Then (l;v U cM) : 1 M =

1N-.M

U CR.

Proof. If r £ N : M, then r M £ iV and so there exists y e M such that rj/ £ iV. Thus [(ljv U c M ) : lAf](r)

= = < =

V {^(r) | r/ G £,/(#), 77 • 1 M C 1N U c M } V {
Hence ( 1 ^ U cM) '• 1 M £ 1JV:M U CR. If x ^ iV, then [(liV:M U c*) • l M ] ( i )

= =

V {(liv U c M )(r) A lM(z) c.

Thus (1JV:M U c#) • 1 M Q IJV UCM and so l;v:M (IN U CM) : 1 M = 1JV:M U C^. ■

4.6

\ r e R, z G M, rz = x}

U C ^ C (1N UCM)

:

1M-

Hence

Primary L-Submodules

Definition 4.6.1 An L-submodule ji of M is called primary following condition: C ' v £ M, C € LI(R),

v e L(M)

if it satisfies the

=> eztfier £ C 7£(/z : 1 M ) or u C /j,.

Theorem 4.6.2 Suppose M = R. Then an L-submodule is a primary submodule if and only if it is a primary L-ideal of R.

L-

150

Proof.

Let /x be an L-submodule of M. By Theorem 4.5.5, K(n:lM)

= =

n(fi:lR) TZ(V{r,\r]eLI(R),

77 • 1R C /x })

= ^(v{771 ^ei/CH),

= no*).

vQv})

Hence the desired result follows immediately from Definitions 3.7.1 and 4 . 6 . 1 . 1 Since a primary L-submodule is also a 7^-primary L-ideal when M = R, properties of 7^-primary L-submodules also apply to primary L-ideals. Theorem 4.6.3 Let fj, G L M . Then fi is a primary L-submodule of M if and only if H = 1/x* U c M , where \i* = {x | x G M, fi(x) = 1} is a primary submodule of M and c is a prime element in L.

Proof. Suppose that \± is a primary L-submodule of M. Now ^ is a submodule of M. Thus 1 G fi(M) and \/JL(M)\ >2.Letx,ye M\/x* and /J,(X) = c. Since ^(rx) > fi(x) = c V r G R, {rx)c QjiMreR. Thus by Theorem 4.3.4, (xc) = 1{0} U (U {{rx)c

\reR})Cfi.

Now let i/ = l ( x ) and C = l{o} U cR. Then C e LI(R) and v G L(M). Clearly, 1{0} ' l<x> = 1(0} and c# • 1<X> = c<x). Hence (1{0} U CH) • i/

= = =

C

1 { 0 } • l< x) U cR • l(a;> 1

WUc<^> (xc)

/*.

But since // is a primary L-submodule of M and clearly 1<X) £ / / , £ = l r 0 j u Ci?C ft(/x: 1 M ). Thus C(r)

= < = = =

c n(fi:lM)(r) VneN(fM:lM)(rn) V n e N (V {n(r") | r; G L/(/J), 77 • 1 M C fi }) V {n(ri)(r) I 77 G LJ(iJ), 77 • 1 M C /i }

151 Vr E i ? \ { 0 } , a n d in particular, fi(x)

=

c

=

C(c)

< = = <

V { ^ ) ( e ) | 77 E L / ( i ? ) , 7 7 - l M C / x } V {77(e) | 77 e L J ( i i ) , r / . l M C M } V{v(e)AlM(v)\veLI(R),Ti'lMCiJL} V{(r/-lAf)(y) | 77 E L J ( i ? ) , 77 • 1 M C xx }

< My)Similarly, /x(y) < /x(x). Hence /x(x) = /X(T/) = c. T h u s |/x(M)| = 2 a n d so ^ = 1 ^ U CM- Next we prove t h a t /x* is a primary submodule of M. Let r e R, x E M , a n d rx e n*. T h e n l< x ) 6 L(Af), l< r ) 6 £ ! ( # ) , (r) (x) C /x„, and l(x> • l(r) = 1<x)(r) £ 1 ^ C /X. T h u s either l ( r ) C 7£(/x : 1 M ) or l( x ) C )tx. Note t h a t 1^) C \x implies t h a t (x) C ix* and hence x E /x*. If l ( r ) C 7£(/x : 1 M ) , t h e n by Corollary 3.8.5 and Theorem 4.5.8, we have l(r> C 7 e ( l ^ : M U Cfl) = ly/^M

U C

*'

Hence (r) C y/JI^TM a n d so there exists m E N such t h a t r m E JLX* : M , i.e., m M C /x*. Therefore, /x* is a primary submodule of M. Next we show t h a t c r is a prime element in L. Suppose t h a t c is not a prime element in L. T h e n there exist a,b £ L such t h a t a j£ c and 6 ^ c, and a Ab < c. T h u s l ^ j . U 6 M ? M and ( 1 { 0 } U a * ) ( l ) = a g c = ft(/x: 1 M ) ( 1 ) and so 1{0} U CLR g 7£(/x : 1 M ) - B u t (1{0} U a ^ ) • (1{0> U &M)

= C C

1(0} ' 1(0} U l{o> • bM U a # • 1^} U a f l - 6 M 1{0} U c M /x.

This contradicts t h e fact t h a t /x is a primary L-submodule of M . Therefore, c is a prime element in L. Conversely, suppose t h a t ji = 1 ^ U C M , where /i* is a primary submodule of M and c is a prime element in L. Let £ E L / ( i ? ) a n d 1/ E L(M), a n d ( - ^ C / x . u C Suppose t h a t 1/ g /x a n d £ 2 ^ ( ^ : 1 M ) = 1 f~^ R- T h e n 3 r E R and x G M such t h a t £(r) £ (1 r ] j U c j i ) ( r ) a n d i/(x) g /x(x). T h u s r £ v > * : Af and x ^ /x*. Since /x* is a primary submodule of M , r ^ y/JI^T~M and x ^ \i* implies t h a t r x ^ /i* and so /x(rx) = c. Hence C(r) A v{x) < (C • ^ ) ( r x ) < fi(rx)

= c.

152

This contradicts that fact that c is a prime element in L. Thus either £ C IZfa: 1M) OI v C p. Hence /i is a primary L-submodule of M. ■ Theorem 4.6.4 Let /i G L{M). Then'fi is a primary L-submodule of M if and only if it satisfies the following conditions: (1) /z* is a primary submodule of M, (2) fa: 1M)(1) is a prime element in L, (3) ra • Xb C /x, r G JR, X G M, and a,b e L => either ra C 7£(/x : 1 M ) 0?" #6 C ^.

Proof. Suppose that /i is a primary L-submodule of M. By Theorems 4.6.3 and 4.5.8, it follows that fi satisfies (1) and (2), and fi = 1 ^ U CMNow suppose that 3 r G R, x G M, and a, 6 G L such that r a • x& C /x, but ^a 2 ^ ( M : 1 M ) = 1 /—7M U C# and x 6 £ M- Then r ^ ^/Jt^TM and a; ^ //*. Since /z* is a primary submodule of M, it follows that rx ^ /i*. Thus on the one hand since ra • X}> u-*, we have that a Ab < ji{rx) = c and on the other hand, since ra % Ufa : 1 M ) and Xf, <£. fi, we have that a ^ (Jlfa : 1 M ) ) ( r ) = c and b j£ //(#) = c. This contradicts condition (2). Hence fi must satisfy condition (3). Conversely, suppose that /z satisfies (1), (2), and (3). Let c = fa : M ) ( 1 ) be a prime element in L and a = 1. Let x G M\ii* and 6 = JLA(X). Then C /i. Clearly, xa g M- Hence it follows from (3) that 1& C Ufa : 1 M ) , and so b = U(l) =

(M:1M)(1)

= = = < =

c V{77(1) I 77 eLI(R), ry-lMC/i} V {77(1) A lM(x) I 77 G LI(R), fi(x) b.

V'IMQI*}

Thus fi(x) = fa : 1M)(1) = c and so fi = 1 ^ U CM. Hence ^ is a primary L-submodule of M by Theorem 4.6.3. ■ Note that condition (3) of Theorem 4.6.4 can be replaced by the following condition: (3)' C-iyC^(eLR1iyeLM=^ either C £ Ufa : lM) or v C /x. We now present some properties of primary L-submodules. Lemma 4.6.5 If N is a primary submodule of M, then N : M is a primary ideal and hence y/NiM is a prime ideal of R.

153

Proof. Assume that AT is a primary submodule of M. Let s, t G R be such that steN:M. Then stM C N.Ut £ N : M, then there exists x e M such that s(tx) G N but tx £ N. Thus there exists m G N such that s m G N : M. Hence N : M is a primary ideal and so y/N : M is a prime ideal of R. ■ Theorem 4.6.6 If ji is a primary L-submodule of M, then fi : 1 M is a pri­ mary L-ideal of R and hence 7Z(fj,: 1 M ) is a prime L-ideal of R. Proof. Suppose that /x is a primary L-submodule of M. Then by Theorem 4.6.3, /x = 1 ^ U CM, where /x* is a primary submodule of M and c is a prime element in L. By Lemma 4.6.5, /x* : M is a primary ideal and so ^//x* : M is a prime ideal of i?. Thus /x : 1 M = l/z«.:M U CR is a primary L-ideal of R by Theorem 3.7.2 and hence 1Z(fi: 1M) = 1 /—r^j U cR is a prime L-ideal of R by Theorem 3.9.7. ■ Theorem 4.6.7 Let v G LM and fi be a primary L-submodule of M. Then (1) v C /x =» /i : v = 1R; (2)i/£n=> K(p : v) = U(fi : 1 M )Proof.

(1) If 1/ C xx, then r)'vClR.vClR-/j,

= ij,\/rie LI(R).

Thus

/ I : I / = U{T7 |TJ e Z J ( i i ) , 77 • 1/ C /x} = 1*. (2) Suppose that v % /x. If r\-v C xx for some 77 G LI(R), then 77 C 7£(/x : 1M) by Definition 4.6.1. Thus xx : v = U {77 |r? G LI(R), r? ■ 1/ C fi} C ft(/x : 1 M ). Also /x : 1 M Q M * ^ and so 7£(xx : 1 M ) C ^ ( M : v). Hence TZ(fi : ^) = lZ(1Z(fi : lM))=%:i/).i Theorem 4.6.8 Lei xx G L(M), C £ LI{R), and \i be a primary L-submodule ofM. Then (1) C g ft(/i : 1 M ) =» M • C = M5 (%) C Q A* • 1 M => A*: C = 1MProof. (1) Suppose that C £ ^ 0 * : 1M)- If C • ^ £ A* for some v G L(M), then v C /i by Definition 4.6.1. Thus by Theorem 4.5.5, ix: C = U {v I 1/ G £ / ( # ) , C • v Q tA Q I*Clearly, ( • /x C 1^ • /x = /x and so /x C /x : £. Hence /x : C = A4(2) Assume that C C /x: 1 M - Then C - 1 M £ ( / Z : 1 M ) ■ 1 M C /X by Theorem 4.5.3. Thus C ' ^ C ' I M C XX VI/ G L(M). Hence xx: C = 1M- ■

154

Theorem 4.6.9 Let N be a module over R and f an epimorphism of M onto N. (1) If ii is a primary L-submodule of M and is f-invariant, that is, f(x) = f(y) => ^(x) = jj,(y), Vx,2/ G M, then /(//) is a primary L-submodule of N. (2) If v is a primary L-submodule of N, then / _ 1 ( ^ ) is a primary Lsubmodule of M. Proof. (1) Suppose that \± is a primary L-submodule of M and is / invariant. By Theorem 4.6.3, we have /z = 1 ^ U CM, where /z* is a primary submodule of M and c is a prime element in L. It is easy to see that /(M) = !/(/*.)

Uc

^-

We now show that /(^*) is a primary submodule of N. Let r E R,y e N and ry = r/(a?) G /(/i*) for some a; G M. Then f(rx) = r / ( # ) G /(/i*). Since /i is /-invariant, it follows that rx £ /J,*. Also since /z* is a primary submodule of M, it follows that either x G /i* or 3m G N such that rm G zz* : M. Hence either G /(/z*) or 3m G N such that rmN = rrnf{M) = f(rmM) G /(/**). y = f(x) Thus /(/i*) is a primary submodule of N. Hence by Theorem 4.6.3, /(//) is a primary L-submodule of N. (2) Suppose that v is a primary L-submodule of N. By Theorem 4.6.3, we have v = lVm U c/v, where z/* is a primary submodule of iV and c is a prime element in L. It is easy to see that f~l(v)

= lf-i{l/m)

UcM.

We now show that / - 1 ( ^ * ) is a primary submodule of M. Let r G R,x G M and ra; G / - 1 ( ^ * ) . Then r/(a;) = /(ra;) G z/*. Since v* is a primary submodule of JV, it follows that either /(a;) G ^* or 3m G N such that rmN C */*. Hence or either x G f'1^*) 3m G N such that rmAT = rmf(M) = f(rmM) C */* m -1 l and so r M C Z ^ * ) . Thus f~ {v*) is a primary submodule of M. Hence / _ 1 ( ^ ) is a primary L-submodule of M by Theorem 4.6.3. ■

4.7

7r-Primary L-Submodules

Let /i be a primary L-submodule of M. Then n = 7l(fj,: 1 M ) is a prime L-ideal of R by Theorem 4.6.6. We call /i a 7r-primary L-submodule of M and 7r the prime L-ideal associated with the primary L-submodule JJL. Theorem 4.7.1 Let ji be a ^-primary L-submodule of M. (1) If r G i?, x G M, a, 6 G L, and r a • #6 C ^, #ien ra C n or x^ C /j,. (2) IfC G LJ(i?), i/ G L(M), andC-vCfj,, then ( C IT or v C fi.

155

Proof. The desired result can be obtained from Theorem 4.6.4 and Defini­ tion 4.6.1. ■ Theorem 4.7.2 Let \x be a TT-primary L-submodule of M and v G L(M). If [i\v^ 1#, then \i\v is a ir-primary L-ideal of R. Proof. Assume that \i\v^\R. Then v g /i and 1Z(/J, : v) = 1Z(/J, : 1 M ) = ^ by Theorem 4.6.7. Let 77, £ G LI(R) be such that 77 • C Q M : v- Then by Theorem 4.5.3, 77 • (C • ^) = (77 • C) • ^ C (fi : v) • 1/ C p. Since /x is a 7r-primary L-submodule of M, 77 C 7£(/x : 1 M ) = 7£(^ • v) or £ • ^ Q V>- Thus 77 C ir or ( C |U : 1/. Therefore, /x : 1/ is a 7r-primary L-ideal of i2. ■ Theorem 4.7.3 Let fx G L(M) and £ G LI{R). If n is a TT-primary Lsubmodule of M and fi : £ 7^ 1M, *ften 11 : Q is also a TT-primary L-submodule ofM. Proof.

If 77 G U ( i i ) , 1/ G L(M), and 77 • z/ C /x: £, then by Theorem 4.5.3, 77-(C^)=C-(r7^)CC-(^:C)C/x.

Since /z is a 7r-primary L-submodule of M, £ • 1/ C ^ or 77 C 7£(^z : 1 M ) = 7r. Thus v C ^ : C or 77 C 7£(/z : 1M) = TTNow we prove that 7£((^ : C) : 1M) = 7^(M : 1M) = TT- First we have (fi:C):lM

= =

UJ77 I 7 7 G L H , 7 7 . 1 M C / X : C } U|77|77GLi?,C-(77-lM)C^|

=

U{77 | 7 7 G L H , 7 7 . ( C - 1 M ) C / X }

=

n : (C : 1 M ) .

Next since \x : v ^ 1 M , it follows that £ • 1 M 2 M- Finally by Theorem 4.6.7, we have K((n : C) : 1 M ) = Ufa : (C : 1 M ) ) = n(fi:

1 M ) = TT.

Therefore, /1: £ is a 7r-primary L-submodule of M. ■ Theorem 4.7.4 Le£ AT also be a module over R and f an epimorphism of M onto N. (1) Iffi is a TT-primary L-submodule of M and f-invariant, then f(fi) is a TT-primary L-submodule of N. (2) If v is a 7r-primary L-submodule of N, then f~l{y) is a ix-primary L-submodule of M.

156

Proof. (1) Assume that fi is a 7r-primary L-submodule of M and / invariant. Then f(fi) is a primary L-submodule of N by Theorem 4.6.9 and /(M)

= = 1

/(^)

U C

^-

Thus f{fl)

I 1N

WW:1^)

= 1/(/X«.):1JV

= 1

UC

> /7w^

#,

Uc/l

"

Now we prove that yjf{fi*) : AT = v7^"T~M. Let r G A/ZO^*) '• N. Then 3m G N such that r m G /(/i*) : AT and hence rmAT C /(/i*), that is, rmf(M) = f(rmM) C /(/x*). Since yu is /-invariant, it follows that r m M C //* and hence r G yJJj^TM- Thus yff(fi*) : N C y T ^ T M . Similarly, V / / ^ T M C y/f{iZ)7N. Thus y/f(iI*)7N = y/fi*:M and so 7r = 7£(^ : 1 M ) = 7 £ ( / ( A 0 : liv). Therefore, /(/i) = l/(At„) U c/v is a 7r-primary L-submodule of AT. (2) The proof is similar to (1). ■

4.8

Primary Decompositions

In this section, we study primary decompositions of L-submodules. Theorem 4.8.1 The intersection of a finite number ofTr-primary L-submodules of M is a 7r-primary L-submodule of M. Proof. It suffices to prove this for two 7r-primary L-submodules of M. Let fi, v be two 7r-primary L-submodules of M. Then by Theorem 4.6.3, we have fi — 1M„ U aM a n d v = lUm\J

bM,

where /i*, v* are primary submodules of M, a, b are prime elements in L, and

lZ(fi:lM)

= =

=

ly^Ua^ 7T

7l(i/:1M).

Thus a = b and y/Jt^TM = \Jv* : M is a prime ideal of i?. Clearly, \i fl ^ =2/£*n^* u a M- Since /i*, i/* are primary submodules of M with \Jfi* : M = \Jv* : M as associated prime ideal, it follows that /x* fl v* is also a primary submodule of M with y/fi* : M = \Jv* : M as associated prime ideal.

157

On the other hand, by Theorem 4.5.4, we have (/xni/):lM

= =

Ou : 1 M ) H (1/ : 1 M ) (l/i.:M U a H ) fl ( 1 ^ : M U aR)

—

l(/i*:M)n(i/.:M) U a i*-

Thus ^((^n^:lM)

= =

l^:M)nK:M)Ua, 1

/ v

^

U a

=

Tl(fjL: 1 M )

=

7T.

*

Therefore, fidis is also a 7r-primary L-submodule of M. ■ Theorem 4.8.2 If \ii (1 < 2 < n) are TTi-primary L-submodules of M, t/ien

ft((n?=1^): i M ) = n j L ^ ^ : i M ) = n ^ . Proof. Assume that ^ (1 < 2 < n) are 7ivprimary L-submodules of M. Then by Theorem 4.5.4, we have

(n? = 1 ^):l M = n? =1 (/x i :l M ). Since /^ (1 < z < n) are ^-primary L-submodules of M, it follows by Theorems 4.5.8 and 4.6.3 that Pi : 1M = 1M*:M u (ci)R € LI(R). Thus by Theorem 3.6.8, we have

ft(n?=1(^: iM)) = n^nim

: iM) = n ^ .

Therefore,

n((n^lfM): iM) = n^mm

: iM) = n?=17ri. ■

Definition 4.8.3 Let /J, 6 L(M). A decomposition of /x as a finite intersection, /i = n^ =1 //i, of primary L-submodules of M is called a primary decompo­ sition of ii and the set of prime L-ideals {TZ(fii : 1M) | i = 1,... , w}, ^ e set of associated prime L-ideals of fjb. A primary decomposition /i = f l ^ / i i is called irredundant if no m contains n j = 1 -^/Xj and an irredundant primary decomposition of \x is called normal if distinct \ii have distinct associated prime L-ideals.

158

Theorem 4.8.4 Let {i G L(M) and fj, = D^fa be an irredundant primary decomposition of /i, where the fa are TTi-primary L-submodules of M. Then a prime L-ideal ir G {7^ | i = 1 , . . . , n} if and only if 3v G L(M) such that 7Z(n : v) = 7r. Hence the set of prime L-ideals {71^ | i = 1 , . . . , n} is independent of the particular irredundant primary decomposition of \x. Proof. Since [i = H^fa, it is clear that for any v G L(M), by Theorem 4.5.4, we have ix : v = (n?=1fa) : v = H^ifa : v). By Theorems 4.6.7 and 4.7.2, fa : v — 1R if v C fa and fa : v is a 7rj-primary L-ideal of R if v £ Hi- Hence fa : v G LI(R). Thus by Theorem 3.6.8, we see that Ufa : v) = 1l(n?=l(fa : 1/)) = n J L ^ O * : v) = n™=17r5., where the intersection is taken over those Sj such that v £ l^Sj • Now suppose that 7Z(fi: v) = 7r is a prime L-ideal of i?. Then by Theorems 3.1.4 and 3.1.32, we can see that 7T = nJL^sj

D 7T3l • . . . • 7TSm

and so n D irSj for some Sj. Since 7TSj 3 7r, it follows that n = irSj. Next, consider any one of the associated prime L-ideals -K\ of fj, = n™ =1 ^. Let v = n^ = 1 • 'i/ij. Then by Theorems 4.5.4 and 4.6.7, we have Liny

=

(njj = l W f c ): ( n - = 1 ) i # ^ )

=

fa:(n^=1}j^j).

By the irredundancy of the set of /i*, we have n™=1 ^ / x , 2 A^- Thus by The­ orem 4.6.7, we see that 1Z(fi: v) = TZ(fa : (pi^=1^iij)) = 7l(fa : lM) = ^i- ■ We see from Theorems 4.8.1 and 4.8.4 that an L-submodule of M that is an intersection of primary L-submodules of M is primary if and only if it has only one associated prime L-ideal. Theorem 4.8.5 Let /i G L(M). If \i has a primary decomposition, then fi has a normal primary decomposition. Proof. Assume that \i has a primary decomposition /i = C^=1fa. If fak G {/xi,..., Lin} are such that

fax,...,

ft(Wi • 1M) = . . . = Tl(fak : 1M) = 7T;, let /ij = Hj^iiJiij • Then \x\ is a ^-primary L-submodule of M and 7£(^ : lM) = TTi by Theorem 4.8.1. Thus fi =

fi,1n...nfi,r]

159 where the /4 have distinct associated prime L-ideals. If /^ 2 ^JLij =£i^j f° r some z, then /x^ is deleted. Therefore, /x has a normal primary decomposition. ■ Theorem 4.8.6 Let fi G L(M). Suppose that /x has a primary decomposition, /x = n ^ / i i . Then /x* = nf=1/x* is a primary decomposition of /x*, and if it is normal, then the decomposition of /x = n^ =1 /ii is normal. ■ In the case of a normal primary decomposition, we have a one-to-one corre­ spondence between the primary L-submodules of the decomposition and their associated prime L-ideals. We shall call a primary L-submodule /x^ in the normal primary decomposition /x = n™=1/Xi isolated if the associated prime L-ideal ^ is minimal in the set of associated prime L-ideals of /x. Theorem 4.8.7 Let \x G L(M). Suppose that /x = n™=1/ii is a normal primary decomposition of \x. Then there exists a finite set {TTI, . . . , 7r m }, m < n, where the TTi are minimal in the set of associated prime L-ideals of \i — flJl^/Xi, such that 71(/J, : 1 M ) = (^iLi^i and /i, : ( U ^ L ^ ) = fi when m > 2. Proof. Suppose that /x = fl-J^/Xi is a normal primary decomposition of /x. Then by Theorem 4.8.2, we have ft(/x : 1M) = ft((n?=1^) : 1 M ) = n ? = 1 f t ( W : 1 M ) = nJLiTTi. Let C be any prime L-ideal of R such that £ 2 7£(/i : 1M)- Then by Theorem 3.1.36, we see that C 2 n"=17Ti DTTi • . . . -7Tn. So £ 2 TTf for some z. Thus £ contains some 7^ that is minimal among 7Ti,..., 7rn. Hence if we select those iti in {7Ti,..., 7rn} that are minimal and reindex, then we have 1Z(/I : 1 M ) = H ^ T T ; . If m > 2, then by Theorem 4.5.7, we see that \i : (U^TTi) = n^i(Ax : 7r»). Since TT; <£ 7i(/x : 1 M ) = H ^ T T ; , it follows from Theorem 4.6.8 that /x : 7Tj — /x Vz G { 1 , 2 , . . . , n}. Therefore, /x : (U^!-^) = /x. ■ Theorem 4.8.8 Let ji = f l ^ / i ; be a normal primary decomposition of fi and Hi be isolated 7T{-primary'L-submodules of M. Then

ViG{l,2,...,n}. Proof.

Since 7Ti • . . . • 7ri_! • 7T i+ i • . . . • 7Tn C

H^j^TTj,

it follows from the minimality of TTi that f l ^ ^ T T j £ 7Tj, and hence n " = 1 -^7^- 2 n^TTj- = ft(/z : 1 M )- Thus by Theorem 4.6.8, \x : (H? =1 ^ T T , ) = / i V i € {l,2,...,n}.B

160

4.9

EXERCISES

Let R be a commutative ring with identity and M a unitary ii-module. 1. (258) Let ^ b e a n L-ideal of R such that /i(0) = 1 and let a,/3 be Lsubmodules of M. Prove that (a : /i) is an L-submodule of M and that a : (3 is an L-ideal of R such that (a : /?)(0) = 1. 2. (258) Let fi, v be L-ideals of R such that /x(0) = 1 = i/(0) and let a, /3 be L-submodules of M. Let r a be an L-singleton of R. Prove the following statements: (a) a C (a : /i), (b) / i 0 ( a : / i ) C a , (c) (a : / ? ) © / ? £ a, (d) r a l # C (a : /?) if ralM

C a,

(e) (a : /x) C (/? : /x) if a C /?, (f) {a : v) C. (a : fi) it fi C. v, (g) (a : fi) = 1 M if /i C (a : 1 M ). 3. (258) Let fix ..., /xn be L-ideals of it! such that /ij(0) = 1, z = 1 , . . . , ra, and let a , / ? i , . . . , / 3 n be L-submodules of M. Prove the following statements: 0 ) <* : E i L i ^ = HjL^a : fj,i), ( b ) a : E ^ i A = n2ssl(a:A). 4. (258) Let /i, */ be L-ideals of R such that /i(0) = 1 = i/(0) and let a, /? be L-submodules of M. Prove the following statements: (a) ((a : /x) : /3) = (a : /x©/3) = (a : ^9) : M, (b) ((a : /i) : i/) = (a : \iv) = (a : v) : /J,.

161

Chapter 5

L-SUBFIELDS In this chapter, we present the essentials of fuzzy field theory for the under­ standing of the current development of commutative ring theory and fuzzy algebraic geometry. Throughout this chapter, F always denotes a field. As usual, 0 denotes the additive identity of F and 1 denotes the multiplicative identity of F. We let P denote the prime subfield of F. The results of this chapter are taken from [158-162, 165, 174].

5.1

L-Subfields and L-Field Extensions

The results in this section hold for fields of arbitrary characteristic. We in­ troduce a concept for L-subfields which ordinary subfields do not possess. We call an L-field extension with this property, "neutral." Definition 5.1.1 By an L-subfield of F, we mean an L-subset n of F such that /i is an L-subgrowp of the additive group of F, fi is an L-subgroup of the multiplicative group of F, and u(l) = 1. The set of all L-subfields of F is denoted by L(F). When L = [0,1], an L-subfield of F is referred to as a fuzzy subfield of F. Suppose that //, v G L(F) are such that \iQv. Then we write v/u and call u/u an L-field extension. Terminology for v/u is used in a manner which is consistent with that immediately above when L = [0,1]. Suppose that vj^i is an L-field extension. Then we let L{y/u) denote the set of all L-subfields £ of F such that // C £ C v. If £ G L(u/u), then we call £ an L-intermediate field of v/u. L(y) denotes the set of all 77 G L(F) such that r\ C v. It follows immediately that a subset A of F is a subfield of F if and only if 1^ is an L-subfield of F.

162

Theorem 5.1.2 Let fi G LF. Then a necessary and sufficient condition for \x G L(F) is that every fia (a G L) is a sub field of F. ■ Theorem 5.1.3 Let //, G L(F). (1) For allxeF, x^ 0, H(0) = /i(l) > /i(x) = fi(-x)

= /x(x _1 ).

(2) n* is a subfield of F, where /i* = {x G F | fJ,(x) = ^(1)}^ If L is a regular, then \i* is a subfield of F. ■ Theorem 5.1.4 The set L(F), together with L-subset inclusion C, constitutes a complete lattice whose meet is L-subset intersection. Its maximal and mini­ mal elements are lp and l p , where P is the prime subfield of F1 respectively. Moreover, L{F) is closed under intersection. ■ Recall that for a ring R, L{R) denotes the set of all L-subrings of R. Definition 5.1.5 Let R be a commutative ring with identity. Let \JL,V€L L(R) be such that \i C v and x £ LR be such that x £ ^- Define fi[x) to be the intersection of all £ G L(R) such that fi U x Q £ £ vDefinition 5.1.6 Let /i, v G L(F) be such that fj, C v and let x € LF be such that x Q v. Define /z(x) to be the intersection of all £ G L{yj\x) such that

In Definition 5.1.5, fi[x] is the smallest L-subring of R which contains /xUxIt is called the L-subring of v generated by x over fi. In Definition 5.1.6, fi{x) is the smallest L-subfield of F which contains /z U x- It is called the L-subfield of v generated by x over \i. Let R be a commutative ring with identity 1 and let P' denote the intersec­ tion of all subrings of R which contain 1. We assume throughout this chapter that every \i G L(R) has the property that fi{l) = 1. Let /x G L{R). Then V x G P ' , /i(x) = 1 since P' — {nl\n G Z}. Unless otherwise specified, (a: a ) 0 is defined to be l i when x G P\{0} and a G L\{0}. Theorem 5.1.7 Let x £ LR and fj, G L(R). Set n = /xUx- Define the L-subset v of R as follows: v(x) = V{AyeAfl(y) \ACR,l<\A\
P'{A}}

163

Proof. It follows immediately that v(0) = 1 = i/(l), 77 C v, and v(—x) = i/(x) V x G ft. Let x, y e P . Now x G P'[A], y G P'[B], A, B C P , 1 < |A|, |B| < 00 imply that x + 2/ and xy are both in P'[A U P ] . Thus V x, y e R, both v{x i/(x + + y) 1/(0:2/) > > y) and v(xy) =

A{T/(;Z) | z\ e zeAuB} A\J B} A{n(z)

(At* € i 4^(^)) AA( A V G(A B 7v€B 7 (v(v)). U)). {AueAV(u>))

Hence V 1, ?/ G i?, both i/(x + 2/), i/(xy)

> = =

B C R, V{(A A (A (AV€veB v(v)) | A, \A,BCR, V{(A BV(V)) u€^(u)) A ueAV{u)) Il <<\A\,\B\ < 00,x e P'[A},y e P'[B}}) < e P'[S]}) |A|,|£|'[A],i (V{A ^77(u) (V{Au€ t7(ti) I A C P , 1\AQR,l<\A\
Thus 1/ G L(P). Now let £ € L(P) be such that f D 77. Let x G P . If x = 0 or 1, then £(x) = 1. Suppose that 0 ^ x ^ 1. Let A be a nonem­ pty subset of P such that x G P'[-A], say A = {xi, x 2 , . . . , x n } . Then a: = E ^ i . - i n ^ i ) * 1 • • • ( * n ) S **!...*» e P 7 , ^ - € { 0 , 1 , . . . , ^ - } , ^ GN, j = 1 , . . . , T I . Now for Nj = { 0 , 1 , . . . , n j } , j = 1 , . . . ,n, £(x)

> > > = > =

A {U^((ffcciill.....
Thus £(*) > V{A r7(2/) \ACR,l<\A\
--=

V{(E(*i1...i„)m1...,I.((*l)a1)<1 •••((Xn)a„)i")(x) | M ^ . - O «ii...i„, fcii...*„ e R,X(XJ) = a.j ifij ^ 0,aj € {X(XJ), 1}

=

,Xj G P , i j €G { 0 , 1 , . . ..,nj},rij ,rij},nj €G N , j = l , . . . , n ; n €G N} ifij = 00,Xj e i?,ij

T/ien cr = /x[%].

164

Proof. that

Since (xj)a C /

;\kii...in

x

C fi[x] and ( ^ . . . i j ^ . . . ^ Q V Q /*[*], it follows

JUj^.. . ^ ( ( ^ l ) - ! ) * 1 -

..{Xn)aJ"Qtix}-

Hence cr C /z[x]. In order to show that a D /JL[X], it suffices to show that a G L(R) and that a D p U x- Now a(0) > jx(0) V x(0) = /x(0) = 1 since 0 = 00. Suppose that z G R, z ^ 0. Then fi(z) A 1 = /x(z) and (T(Z) = o-(l2r) > M(1) A X(*) = 1 A xOO = xOO- T h u s * 2 M U X- We now show that a G L(R). Let x, y e R. If either cr(x) = 0 or cr(y) = 0, then clearly cr(x + y) and cr(x) A <j(y). Suppose that a(x) ^ 0 ^ i . . < » ( * i ) i i - ••(^nj

n

jaj

where a = ( A j ^ ! . . . ^ | ij eNjJ

= 1,.. = l,...,n})A(A{aj l , . . ..,n}). ,n}). 1,.. . ,n}) A(A{a i | j =

Also, ! > . .

••imJvi^.. im

((yiKr- ••((2/m)6 m ) i '"=(E^- ..^(yiY1 ■ ■

■(yn)im)b,

where b

b= A j ^ . . . ^m | Zj . ...,m}). . ,m}). = ((A{vu...i ij 6GM M ij,, j = l,...,m})A(A{i>j l , . . . ,m}) A(A{fy |1j j==1 ,1,.. Now x+t/ is the sum of the above expressions for x and y and so cr(x+y) > aAb. Thus a(x + y) > supremum of all such aAb. Since A is infinitely V-distributive, it follows that er(x + y) > cr(x) A cr(y). Similarly, a(xy) > 0. Then a; = S ^ i . - . ^ ^ i ) * 1 • • • (xn)in for some kilm..in G jx* and x i 5 . . . , xn G x*- Then x G P'[A], where ^ = {fcn...in I ij eNj,j

= l,...,n}\j{xj

| j = l,...,n}.

Thus it follows that a C v. Suppose that v{x) > 0. Then x G P'[A] for some A = {2/1,..., y n } C F such that AyeAV(y) > 0. Now for a suitable ordering of the 2/i, ^yeAViv)

= = <

A^^UxXl/) M?/i) A ^(2/2) A . . . A MO/m-i) A x(2/m) •.. x(Vn-i) A x(y n ) cr(x),

165

where the inequality holds since x G /i*[x*]- Hence , it follows that v C a. The proofs of the following results are entirely similar to the proofs of the previous two results. Theorem 5.1.9 Let x G LF and /i G L(F). Set rj = fiUx- Define the L-subset v of F as follows: u{x) = V{AyeAV(y)

I A C iJ, 1 < |A| < oo,x G P(A)}

V x G F. Then /x(x) = i/. ■ Theorem 5.1.10 Le^x G L F and/i G L(F). Define the L-subset a of F as follows: a(x) = V{((E(*« 1 ...i n )u il ... ln ((a:i)a 1 ) il ■ • • ( ( a n J O S E C V ^ J ^ . . . ^ z/ij ^ 0 and a, G {x(*j), 1} ifij = 0, xj G F, ^ G { 0 , 1 , . . . ,71,,}, v(hqi...qrn) = *V.. V - g m ^ F> xG/j) = 6 i * / # / 0 and bj G {x(%), 1} if qj = 0, %• G F, gj G { 0 , 1 , . . . , m j } , n j , ?rij, n, m G N} V x G F Then ji{x) = cr. ■ Theorem 5.1.11 Le£ x G LR. Then the following conditions hold. (1) Suppose that fi G L(R). Then V a G L, /ia[Xa] £ (Mx])a- //" ^ e images of fj, and x are /im£e and L is a chain, then fJLa[Xa] — (MxDa V a G L. (%) Suppose that /i G L(R). Then /i*[x*] £ (Mx])*« /f £ Z5 regular, then M * [ X 1 = (M[XD*.

Proof. (1) x G /xa[Xa] <£> x = S^n-tnC^i)* 1 • • • ( x n) Z n , where the kilmm,in G /xa and Xj G Xa, j = 1, • • • ,n => x G (Mx])a where the implication holds by Theorem 5.1.8. The implication becomes an equivalence when the images of \i and x are finite and L is a chain by Theorem 5.1.8. (2) x G (/i[xD* =» x = £fc j ' — 1> • • • > n O # G /i*[x*]j where the implication holds by Theorem 5.1.8. Since L is regular, the implication becomes an equivalence by Theorem 5.1.8. ■ Theorem 5.1.12 Let \ G LF. Then the following conditions hold. (1) Suppose that \i G L{F). Then V a G L, fia(Xa) Q (Ai(x))a- //" the images of \i and x are finite and L is a chain, then fia(Xa) = (M(x))a V a £ L. (2) Suppose that fi G L(F). Then /i*(x*) C (M(X))*« /f £ *5 regular, then M*(X*) = (MX))*- ■ Suppose that /J,, v e LR. Then (/i, fl v)a = /i a n i/a Va G L. If L is regular, then (/i D i/)* = /i* fl i / \

166

Definition 5.1.13 Suppose that L is regular. Let v j\x be an L-field extension. Let xa C i/, a > 0. Then xa is said to be algebraic over \x if there exists n G N, there exists ki G /x*, there exist c, bi G £\{0} such that fi(ki) > bi for i = 0 , 1 , . . . , n and such that 0C = Yl7=ofti)bi(xay w ^ n°t a^ ki = 0. (In this case, xa is said to be algebraic with respect to the polynomial p(y) = Y2™=Q kiy1 over fi.) If xa is not algebraic over /z, then xa is called transcendental over \x. v/fj, is said to be algebraic if every xa C v is algebraic over \i. If v/n is not algebraic, then v/fi is called transcendental. Definition 5.1.14 Suppose that L is regular. Let v/fi be an L-field extension. Let xa C */, a > 0. Then xa is called neutral over \x if there exists b G L, 0 < b < a, such that xj C //. / / every such xa is neutral over /z, then vj\x is called neutral. Clearly, vj[i is neutral if and only if u* = v*. If xa C v is neutral over fi and L is regular, then xa + (—Xb) = 0C for 0 < ji(x) = b < a and 0 < c = a A b. That is, xa is algebraic over //. Theorem 5.1.15 Suppose that L is regular. Let vj\x be an L-field extension. Then u/fi is algebraic if and only if v*/u* is algebraic. Proof. If v/fi is algebraic, then it follows easily that v* / a* is algebraic. Suppose that v*'/a* is algebraic. Let xa C i/, a > 0. Then there exists n G N and ki G ^*> h not all 0, i = 0 , 1 , . . . , n, such that Yl7=o ^iX1 = 0. Let bi = /j,(ki), i = 0 , 1 , . . . , n and c = (A?=0bi) A a. Then c > 0 and i X o ( ^ k (*<*)* = 0C- Thus vj\i is algebraic. ■ Theorem 5.1.16 Suppose that L is regular. Let x G LF and v G L(F). Mx] = Mx) */ and onty ^ / M x ) / ^ *5 algebraic.

Then

Proof. By Theorems 5.1.11, 5.1.12 and 5.1.15, u(x) = Mx] => /^*(x*) = M*[x*] «=> M*(x*)/M* is algebraic <^> /i(x)/V is algebraic. That M*(X*) = M*[x1 => /x(x) = A*[x] follows from Theorems 5.1.8 and 5.1.10. ■ Theorem 5.1.17 Suppose that L is regular. Let v/fi be an L-field extension and let £ G L{yj/i). Then vj\i is algebraic if and only if z//£ and £//z are algebraic. Proof. v*/u* is algebraic if and only if v*/£* and £*/u* are algebraic. Hence the desired result follows from Theorem 5.1.15. ■ Definition 5.1.18 Suppose that L is regular. Let v/ji be an L-field extension. Then fj, is said to be algebraically closed in v if V xa C */, a > 0, if xa is algebraic over ^, then xa C /j,. Let £ G L(v/fi). Then £ is called the algebraic closure of fi in v if £//i is algebraic and £ is algebraically closed in v.

167

T h e o r e m 5.1.19 Suppose that L is regular. Let vj^i be an L-field extension. Then /x has a unique algebraic closure in v. Proof. Let C denote the algebraic closure of /x* in v*. Define the L-subset £ of F as follows: £(x) = v{x) V x G C and £(#) = 0 otherwise. Then £ G L{yj\i). Since C C i/*, £* = C. Thus f//z is algebraic by Theorem 5.1.15. Let xa C i/, a > 0, be algebraic over f. Then x is algebraic over £* = C and so x e C. Since £(#) = i/(a;), xa C f. Hence £ is algebraically closed in i/. Suppose that C G L{yj\x) is such that £//z is algebraic and £ is algebraically closed in v. Then C / V 1S algebraic and so £* C C = £*. In fact, £* = C. Since £ = z/ on C, C Q £• Hence £/£ is neutral. Thus since £ is algebraically closed in */, £ = f. Thus the desired unicity holds. ■ Definition 5.1.20 Suppose that L is regular. Let vj/x be an L-field extension. Let x € LF be such that x^v. Then x is said to algebraically independent over fj, z/Vxi,. ,.,xn€ X* ^ hx...in G /x*, V bilmm.in, c G i \ { 0 } , where ^(**i...*») > &n...tn and for ai = x(x*), i = 1 , . . . ,n, /

J
■■((Xn)aj"=0c

implies kilmm.in = 0 /or a// 2i . . . i n . We note that in Definition 5.1.20, if x* = {^} and a = x(x)> then x is algebraically independent over /x if and only if xa is transcendental over /x. T h e o r e m 5.1.21 Suppose L is regular. Let V//J, be an L-field extension. Let X G LF be such that X^vThen x is algebraically independent over \x if and only if Xa is algebraically independent over /xa V a G L\{0}. Proof. Suppose that x is algebraically independent over /x. If Xa is empty, then the result holds trivially. Hence assume Xa / ^ Suppose that 0 = Sfcii.-.tntei)* 1 • • • 0 ? n ) S where kilm.,in G \ia and x1,..., xn G Xa> Then 0 a = £(fct 1 ...ija((ai) a i )* 1 ...((x n ) f l n )» n , where x(z*) = a», z = l , . . . , n . Since x is algebraically independent over /x, the kilm„in = 0. Thus Xa is algebraically independent over /xa. Conversely, suppose that Xa is algebraically independent over /j,aV a £ L\{0}. Suppose that 0c = X)(**i -*» K -«» ((*l)«i )^ • • ' ((*")a„ )'», where c, the bilmm,in and a i , . . . , an are as in Definition 5.1.20. Then with a = c, we have that xi,...,xn GXo and fc^...^ G xxa. Hence the fc^...^ = 0. Thus x is algebraically independent over xx. B T h e o r e m 5.1.22 Suppose that L is regular. Let vj\i be an L-field extension. Let x £ LF be such that x^vThen x is algebraically independent over /x if and only if x* i& algebraically independent over /x*.

168

Proof. Suppose that x is algebraically independent over /x. Assume that 0 = ^2kii-in(xi)il • • • (xn)in, where the feil#..in G /i* and ^ 6 x*, j = 1 , . . . ,n. Then 0C = ^ ( ^ . . . i j ^ . . ^ ( ( Z l k r • • . ( ( * n ) o j S where c

= ( M ^ i - d * ? € NjJ

= 1 , . . . ,n}) A (Aja^lj = 1 , . . . ,n}

and the fc^...^ and a l 5 . . . , a n are as in Definition 5.1.20. Then the fc^...^ = 0. Conversely, suppose that x* is algebraically independent over fi*. Then an argument entirely similar to that given above yields that x is algebraically independent over [i. I Definition 5.1.23 Suppose that L is regular. Let u/fi be an L-field extension. Let x € LF be such that x Q v. Then x is called a transcendence basis of v 1 I'I if X is algebraically independent over fi and vjJLX(X) is algebraic. Theorem 5.1.24 Suppose that L is regular. Let vj \i be an L-field extension. Let x £ LF be such that x^vThen vj \i has a transcendence basis. In fact, X is a transcendence basis for vjyL if and only if x* is a transcendence basis of

Proof. Suppose that x is a transcendence basis of vj\x. Then x* is alge­ braically independent over /x* by Theorem 5.1.22 and ^ * / M * ( X * ) is algebraic by Theorems 5.1.12(2) and 5.1.15 since ^//x(x) is algebraic. Conversely, suppose that x* is a transcendence basis of V*//J,*. Then x is algebraically indepen­ dent over \i by Theorem 5.1.22 and v/fJ<(x) ls algebraic by Theorems 5.1.22 and 5.1.12(2). Thus x is a transcendence basis of vj\i. The existence of a transcendence basis for vj\i follows since v* J\x* has a transcendence basis. ■ By Theorem 5.1.15, we see that the notion of an algebraic L-field extension i///x is characterized in terms of the supports of v and \x. We now introduce a different notion of an algebraic L-field extension vjji which is characterized in terms of the level subfields of v and ji. Definition 5.1.25 Let vj\i be an L-field extension. Let xa C v, a G L\{0}. Then xa is called l-algebraic over [i if there exists n G N, ki G /i*, bi G L\{0}, such that fi(ki) > bi for i — 0 , . . . , n such that n

0a = $ > * ) * , ( * « ) ' z=0

with the hi not all 0. / / every such xa is l-algebraic over //, then u/fi is called l-algebraic.

169

Theorem 5.1.26 Let vj^x be an L-field extension. Then v/fi is l-algebraic if and only if vaj [ia is algebraic Va G A W Proof. Suppose that i///j, is /-algebraic. Let a GL\{0} and x G va. Then xa C v and so there exists n G N , ^ G ^*,6» G L\{0}, ^(fc.) > 6», i = 0 , 1 , . . . , n, such that 0 o =E?=i( fc i)6i(^a)* with the ft* not aU 0. Hence 0 = Y^=0hx\ Since 6; > a, k{ G /i a for z = 0 , 1 , . . . , n. Thus x is algebraic over \ia. Conversely, suppose that vaj[ia is algebraic Va G L\{0}. Let x a C i/, a > 0. Then x G i/a. Thus there exists n G N, h G /i a not all 0, z = 0 , 1 , . . . ,n, such that 0 = SiLo** 2 ^- Hence 0 a =J27=o(^)a(xaY- Thus x a is /-algebraic over fi. Hence v/fi is /-algebraic. ■ Theorem 5.1.27 Let vj\i be an L-field extension. Let £ be an L-intermediate field of' v)LI. Then v j LI is l-algebraic if and only if vj^ and £//i are l-algebraic. Proof. By Theorem 5.1.26, we have that vj \x is /-algebraic 4=>i/a/'/J,a is algebraic Va G L\{0}ova/£a and £ a /^a are algebraic Va G L\{0}«»i//f a n d £/^z are algebraic. ■ Clearly, an L-field extension v/fi which is /-algebraic is also algebraic. How­ ever, the following example shows that the converse is not true. Example 5.1.28 Let L = [0,1]. v j \i is an L-field extension which is algebraic, but not l-algebraic: Let F = P(t)(x), where P is a field, t is transcendental over JP, and x is a root of the polynomial f(y) = y2 + ty + t over P(t). By Eisenstein's criterion, f{y) is irreducible over P{t). Let v = ljr. Define the L-subset [i of F as follows: /x(r) = 1 if r G P, fi(r) = | if r G P(i)\P, and fj,(r) = \ if r G F\P(t). Then \i is an L-subfield of Fand LI C v. Now X3 -f ( - i i ) = 0 i and so X3 is algebraic over LI. In fact, vjLI is neutral and so is algebraic. Also, (X3)2 + £3 (2:3) + £3 = 0 3 . Thus x3 is l-algebraic over fi. However, xa is not l-algebraic over n for a G ( | , 1] for if^^=.Q(ki)bi{xay = 0a, then n(ki) > bi > a > | which is impossible since then hi would be in P, but x is not algebraic over P. If we were define /-transcendence over an L-subfield \i of F, it would be natural to call xa /-transcendental over /J, if xa were not /-algebraic over LI. With this definition, xa in Example 5.1.28 would be /-transcendental over LI for a G (f, 1]. Now xa is neutral and so is algebraic over /x. Let x — xa- Then Xi = {x} is not transcendental (algebraically independent) over P(t) = /i 3 / 4 even though x 1S /-transcendental (/-algebraically independent) over [i. That is, /-transcendence (/-algebraic independence) cannot be characterized in terms of level subfields. These complications deter us from pursuing the notion of /-transcendence at this time.

170

Definition 5.1.29 Let L he regular. Let v//i be an L-field extension. Define the L-suhset fi^of Fas follows: ( n ) ( x ) = f V(X)

^

v

'

1 0

VXE/X*

otherwise.

Then u^ is called the neutral closure of ji in v. If £ G L{v/ji), then £ is called neutrally closed in v if Vxa C v,a > 0, xa neutral over £ implies xa C £ . In Definition 5.1.29, it follows easily that /z(n) G L(u/fi). Also, an Lintermediate field £ of v/fi is neutrally closed in v/fi if and only if £ = v on £*. Hence fi^is neutrally closed in v. Definition 5.1.30 Suppose that L is regular. Let U/JJ, be an L-field extension. Then // is said to be compatible in v or simply v j'fi is compatible if A{v(x) | x G /x*} > V{v(x) | x G F\u*}. fi is said to be strongly compatible in v or simply vj[i is strongly ible if A{/i(x) | x G /x*} > V{u(x) | x G F\fj,*}

compat­

Theorem 5.1.31 Suppose that L is regular. Let v/fi be an L-field extension. If/j, is compatible in v and V//J, is algebraic, then v/fi^ is l-algebraic. Proof. Let xa C v with a > 0. Since i/*///*is algebraic, there exists n G N, hi G //* for i = 0 , 1 , . . . , n such that Y%=o ^X% = 0- Suppose that x £ /i*. Then Y%=o(ki)bi(xa)% = 0 a , where u^n\ki) = b{ > a since /i is compatible in v. Hence xa is /-algebraic over (j,(n\ If x G zx*, then xa + (—xa) = 0 a and

K)c^).i 5.2

Separable and Inseparable Algebraic Ex­ tensions

Throughout this section F is assumed to have characteristic p > 0. Definition 5.2.1 Suppose that L is regular. Let v/fi be an L-field extension. Let xa C v, a > 0. Then xa is said to be separable algebraic over JJ, if there exists n G N, there exists ki G /z*, there exist c, bi G £\{0} wz#i ^(A;;) > &;, i = 0 , 1 , . . . ,n, swc/i that 0C = X^r=o(^)fei(a:a)2j where not all the ki = 0 and

171

the polynomial YH^o^y1 * n V ^s separable over /i*. If xa is algebraic over (JL, but not separable over /i, then xa is said to be inseparable over \i. If vj\i is algebraic and every such xa is separable over /x, then vj\i is said to be separable algebraic; otherwise vj\i is said to be inseparable algebraic. Definition 5.2.2 Suppose that L is regular. Let v/fi be an L-field extension. Let xa C i/, a > 0. Then xa is said to be purely inseparable over ji if there exists n G N U {0}, there exist fci, k2 G /x*, there exist c, &i, 62 G £\{0} with fjb(ki) >bi,i = 1,2, such that (ki)bl(xa)pn + (k2)b2 = 0C, w/iere A^ / 0 ^ A:2. 7/ ever*?/ stxc/i x a is purely inseparable over \±, then v/fi is said to be purely inseparable. The proofs of the following results follow from Theorem 5.1.15 and defini­ tions. Theorem 5.2.3 Suppose that L is regular. Let v/ fi be an L-field extension. Then the following assertions hold. (1) vj[i is separable algebraic if and only ifi/*/fj,* is separable algebraic. (2) vj[i is purely inseparable if and only if v*/fi* is purely inseparable. ■ The following theorem holds by Theorems 5.1.17, 5.2.3, and standard crisp results. Theorem 5.2.4 Suppose that L is regular. Let vj\i be an L-field extension and £ G L(yj\i). Then the following assertions hold. (1) v/fi is separable algebraic if and only if v/£ and £//z are separable algebraic. (2) vja is purely inseparable if and only if v/£ and £//i are purely insepa­ rable. ■ Theorem 5.2.5 Suppose that L is regular. Let v/fi be an L-field extension. Let xa C 1/, a > 0. If xa is algebraic (separable algebraic, purely inseparable) over /JL, then fi(xa)//^ is algebraic (separable algebraic, purely inseparable). Proof. Define the L-subset £ of F as follows: £(z) = v(z) if z G (ix(xa))* and £(z) = 0 otherwise. Then £ G L(i///x). Now £* = (ii(xa))* = /x*((x0)*) (by Theorem 5.1.12(2)) = n*({x}) and /i*({x})//i*, i.e., £*//x* is either algebraic, or separable algebraic, or purely inseparable according as x has these properties over //*, respectively, or equivalently as xa has the corresponding properties over \x. The desired result now follows from Theorem 5.2.4 and the fact that

i 2 Kxa)- ■ Theorem 5.2.6 Suppose that L is regular. Let vj\i be an L-field extension. Then there exist unique maximal L-intermediate fields a andn ofiy/fi such that a/li is separable algebraic and 7r//i is purely inseparable. If u/fi is algebraic, then v/cr is purely inseparable.

172

Proof. That a and /i exist follows from arguments entirely similar to those used in Theorem 5.1.19. Suppose that vj\i is algebraic. Then v*/u* is alge­ braic. Let S be any intermediate field of v*/cr* such that S/fi* is separable algebraic. Define the L-subset £ of F as follows: £(z) = v{z) if z G S and £(z) = 0 otherwise. Then £ G L{y/o) and £* = 5. Thus £/fi is separable alge­ braic. Hence £ C cr by the unicity and maximahty of a and so 5 = £* C cr*. That is, cr* is the maximal separable intermediate field of i/*/u*. Thus i/*/cr* is purely inseparable and so I//<J is purely inseparable by Theorem 5.2.3. ■ As we noted immediately after Example 5.1.28, some complications arise with the notion of /-transcendence for L-field extensions. Never-t he-less, the concept of /-algebraic L-field extensions yields results of some interest. We now introduce these results. Definition 5.2.7 Let vj'/x be an L-field extension. Let xa C z/, a > 0. Then xa is said to be separable l-algebraic over fi if there exists n G N, there exists ki G u*, there exists bi G L\{0} with a{ki) > 6;, i = 0 , 1 , . . . , n, such that 0 a = ]C2=o(^)&i(xa)% where the ki are not all 0 and the polynomial Yl7=o ^iy1 in y is separable over fia> If Xa is l-algebraic over //, but not separable over //, then xa is called inseparable over \x. If vj\i is l-algebraic and every such xa is separable over u, then vj \x is said to be separable l-algebraic; otherwise v/fi is said to be inseparable l-algebraic. Definition 5.2.8 Let vj' ji be an L-field extension. Let x a C i/, a > 0. Then xa is said to be I-purely inseparable over \i if there exists n G N U {0}, fci, &2 G /i*, &i, 62 G L\{0} with fi{ki) > bi, i = 1, 2, s^c/i £/ia£ (fel)6 1 (Xa) pn +(fe2)6 2 =0 a , where k\ =^ 0 7^ &2- /f ever-?/ s^c/i x a is l-purely inseparable over LL, £/ien i///x 25 said £0 be l-purely inseparable. Theorem 5.2.9 Let vj' LL be an L-field extension. Let xa C v, a > 0. T/ien the following assertions hold. (1) xa is separable l-algebraic over /x if and only if x is separable algebraic over ua. (2) xa is l-purely inseparable over fi if and only if x is purely inseparable over fj,a. Proof. (1) That xa is separable /-algebraic over u implies x is separable algebraic over ua follows immediately from Definition 5.2.7 since the ki there are necessarily in ua. Suppose that x is separable algebraic over fia. Then x is a root of some separable polynomial over ua. Since the coefficients of the polynomial are from /x0, it follows immediately that xa is separable /-algebraic over ji.

173

(2) That xa is /-purely inseparable over \x implies x is purely inseparable over \ia follows immediately from Definition 5.2.8 since the k\ there are nec­ essarily in /i a . Suppose that x is purely inseparable over \ia. Then there exist n G N U {0} and fci, k2 G fia such that k2xpn + k\ = 0. The desired result is immediate from Definition 5.2.8. ■ For an L-field extension z///z, xa C v is separable /-algebraic and /-purely inseparable over [i if and only if xa C [i. For an L-field extension v/fj, (L regular), xa C i/ is separable algebraic and purely inseparable over \x if and only if xa is neutral over fi. Theorem 5.2.10 Let v/p be an L-field extension. Then the following asser­ tions hold. (1) vj\x is separable I-algebraic if and only if vaj'fia is separable algebraic VaGL. (2) vj[i is I-purely inseparable if and only if vaj fia is purely inseparable V ae L. Proof. Suppose that vj'fi is separable /-algebraic (/-purely inseparable). Let x G va, a > 0. Then xa C v and so xa separable /-algebraic (/-purely inseparable) over [i. Hence x is separable algebraic (purely inseparable) over \ia by Theorem 5.2.9. If a = 0, then va = F = fia. The converse also follows easily. ■ Theorem 5.2.11 Let v/ ji be an L-field extension and £ G L(v/fi). Then the following assertions hold. (1) vj \± is separable I-algebraic if and only if v/£ and £/'/i are separable I-algebraic. (2) v/fi is I-purely inseparable if and only if v/£ and £/fi are I-purely in­ separable. Proof. Suppose that v/ ji is separable /-algebraic (/-purely inseparable). Let a G L. Then va/^a is separable algebraic (purely inseparable) by Theo­ rem 5.2.10. Thus va/£a and £a/Ma are separable algebraic (purely inseparable) by standard results from crisp field theory. Hence v/£ and £//i are separa­ ble /-algebraic (/-purely inseparable) by Theorem 5.2.10. Conversely, suppose that v/£ and £/fi are separable /-algebraic (/-purely inseparable). Let a G L. Then va/^a and £ 0 /^a are separable algebraic (purely inseparable) by Theo­ rem 5.2.10. Thus va/jia is separable algebraic (purely inseparable. Hence vj\i is separable /-algebraic (/-purely inseparable) by Theorem 5.2.10. ■ Theorem 5.2.12 Suppose that L is a chain. Let vj'[i be an L-field extension, where the image of fi is finite. Let xa C v, a > 0. If xa is l-algebraic (separable

174

l-algebraic, l-purely inseparable) over /x, then /d(xa)/fi is l-algebraic (separable I-algebraic, l-purely inseparable). Proof. Let be L. Suppose that b < a. Then by Theorem 5.1.12, (id(xa))b = f^b((xa)b) = fJ a. Then (fi(xa))b = Hb and (id(xa))t>/Lib is trivially algebraic (separable algebraic, purely inseparable). Thus the desired result follows from Theorems 5.1.26 and 5.2.10. ■ Let K be a, subfield of F and let x G F. Then it is known that x is separable algebraic over K if and only if x G K(xp). This fact motivates the next result. Theorem 5.2.13 Let v//d be an L-field extension. (1) If vj/d is separable l-algebraic, then V £ G L{yjii) and V x G F either £(x) = v{x) or£(x) =£(xp). (2) Suppose that L is regular. If V £ G L(yjii) and V x £ F either £(x) = i/(x) or £(x) = £(x p ), £/&en vjii is separable algebraic. Proof. (1) Let x G F and a = i^(x). Since vj LL is separable /-algebraic, x is separable algebraic over fia. Let £ G L(v/fi). Then x is separable algebraic over fa. Thus there exists n G N,fc^G fa, fc» not all 0, z = 0 , 1 , . . . , n, such that a; = ]Cr=ok;(z p ) 2 - Hence £(*)

> > >

A { ^ ( ^ ) | i = 0,l,...,n} A{f(fc i )A{((xP) i )|i = 0 , l , . . . , n } aA^P).

Thus either £(x) > a, in which case £(x) = i/(x), or £(x) > £(x p ), in which case f(x) = £(x p ). (2) Let x G i/*. Let i/(x) = a. Set £ = /x((xa)P). Then f G L ( I / / M ) . Hence either £(x) = z/(x) = a > 0 or £(x) = f (a*) > a > 0. Thus x G (/z((xa)P))* = /x*(((x a ) p )*) = /i*(x p ) by Theorem 5.1.12. Hence x is separable algebraic over /d*. Thus */*//i* is separable algebraic. Therefore, u/fd is separable algebraic by Theorem 5.2.3. ■ In the following example, we show that the concept of /-algebraic closure for crisp field extensions does not carry over to L-field extensions. Example 5.2.14 Let L = [0,1]. Let F = P(t), where P is a perfect field of characteristic p > 0 and t is transcendental over P. Let v = 1F. Define the fuzzy subset fj, of F as follows: /d(z) = 1 if z G P and LI(Z) = i/(i + 1) if

175

y E FP\FPi+1 for i = 0 , 1 , . . . . Since n^F** = P, \x is defined on all of R Thus ji is a fuzzy subfield of F. Define the fuzzy subsets & of F as follows: &(z) = lifzeP, £i(z) = i/(i + l)ifze F\F?i+\ and &(*) = fi(z) if %+1 z E F? \P for i = 1,2,.... Then & E L(u/fj) and & C & + 1 /or i = 1,2,.... Clearly, v = U g ^ i - £e£ x a C &. If a = 1, then x E (&) a = P = jj,a and so x is trivially algebraic over \ia. Suppose that 0 < a < 1. Then jia = P 7 ^ /or some j . Hence x is algebraic (purely inseparable) over /i a . Thus xa is l-algebraic (lpurely inseparable) over \i by Theorem 5.1.26 (Theorem 5.2.9). Hence &/// is l-algebraic (l-purely inseparable) for i = 1,2, Now vj\x is not l-algebraic since for x E F\P, X\ C zv, but x is not algebraic over \i\ = P. That is, v is a union of an ascending sequence of fuzzy intermediate fields of u/fi each l-algebraic (l-purely inseparable) over /x, but v is not l-algebraic over JJL. Thus fi cannot have an l-algebraic (l-purely inseparable) closure in v.

5.3

Composites, Linear Disjointness, and Sep­ arability

Unless otherwise specified, F is assumed to have arbitrary characteristic in this section. Definition 5.3.1 Let v/p be a L-field extension and let £, tr E L(I///JL). Define the L-subset £7r of FbyVxE F, n

^(x)

= \f{A{C(yi)A7r(zi)

Ii = l,...,n} Ix =

^2viZi,neN}. 2=1

Then £TT is called the composite

of £ and ix.

We have that £, IT C £7r since l{i} C £ and l{i} C IT. The definition of the composite GH of two subfields G and H of F can be found in [67, p. 63] or [261, p. 179]. We recall from crisp field theory that the composite GH in F is an integral domain, but not necessarily a subfield of F. For example, if K is a subfield of F and x and y are elements of F which are algebraically independent over K, then the composite K(x)K(y) is not a subfield of F. Theorem 5.3.2 Let u/fi be an L-field extension. Let £, 7r E L{yj\i). is an L-subring of F.

Then £7r

176

Proof.

Let x, y G F. Then

(fr)(x) {£*)(*) A (Sn)(y) (£TT)(2/) = = =

(V{A?=1 {^K) A A T-K(zi)} T ( ^ ) } |I x = £ E??==! iu,^ i^Z ii) ^n € N}) = 1U(^») N A(V{A™=1=1{«(!ij)A5r(«i)} {£(^) A T T ^ ) } |I V y -= T,T=i £ 7 = u1J^vJ',mm €€ N}) A(V{A^ » {W A TT( 7r(^)} V {{AA?^= 1="1L{i U K ) A £ ((^^)) A 7r(zi) n(Zi) A VJ)} I\

This latter expression is less than or equal to both 9q

V{A£=1 |1zx + y2/ == ^ f c 5 f c , ?g6GNN}} =1{£(t {£(**) fc ) AAn(s k)} V{AL *(**)} fc=i and

atffa) V{A{ =1 {^(r h ) V{AJL

zj/ AT TT A T ^^ )) }} 1I xy

// == Y^rh9hJeJ eN} N} h=i

w z

since x = Y%=i i * 2/ = Y^jLi j j implies x+y = Yl7=i wizi + YZjLi uivJ WiU ZiV an< s m c e and xy = Yl7=iYlT=i( j)( j) ^ i{wiuj) > €(wi) ^ ^(uj) a n d Tr(zi^) > 7r(zi) A T T ( ^ ) . Thus (£TT)(X + y) > (<£TT)(X) A (f?r)(y) < (fr)(xy). Now

(fr)(-

(£TT)(-Z) -x)

= = = = =

and

u v

V { A ?=1={C(u ) AA7r(2: 7 r ( 2 ,i)} ) } | --■* x ==E X ; ?zl=WiZi,n i^^i,TiG N N}} V{A? 1 U K;.) £= V { A ?=1= {«u ) A7r(^)} A T T ( ^ ) } I ax: -= E£IU(l U C-Wi)zi,n -^^n G 6 N} V{A? 1 ^ KH) n {-Wi)zi,n G N} V{A?=1=1{Z(-Wi) A7r(Zi)} | x-T x = £?=i(-<)*," V{A? {£(--Wi) ATr(zi)} (f7r)(a:). (fr)(*)-

Clearly, (£7r)(0) = 1 = (£7r)(l). ■ (f7r)(l). Hence £7r is a L-subring of F. B Theorem 5.3.3 Suppose that L is regular. Let v/ji be an L-field extension and let £, n G L{yj\i). Then the following properties hold. (l)£*Cnr* = (£n7r)*.

(2) (&)*=£***. (3)V ae L\{0}, ia-Ka C (^7r)0; and z/ |£(F)| < oo, |7r(F)| < oo and L is a chain, then $,a^a — (C7r)a- B Theorem 5.3.4 Suppose that L is regular. Let v/fi be an L-field extension and let £, n G L{yj\i). Ifir/fi is algebraic, then £7r is an L-subfield of F. Proof. Since ir/fi is algebraic, TT*/^* is algebraic by Theorem 5.1.15. Hence the composite £*7r* is a subfield of v* and is algebraic over £*. Now £*7r* = (£TT)* by Theorem 5.3.3. Also, £*TT* = ^*(TT*) = (f (TT))* by Theorem 5.1.12. Hence (f (?r))*/£* is algebraic. Thus f (7r)/f is algebraic. Hence £[7r] = f (7r) by Theorem 5.1.16. Now £[7r] C £7r since £[7r] is the smallest L-subring of F

177

which contains f and 7r and £TT is an L-subring which contains £ and ir. That f M 2 ^TT follows by Theorem 5.1.8. Hence ^n = £[n] = £(7r). Thus £TT is an L-subfield of F. ■ The definition of linear disjointness for field extensions can be found in [67, p. 161] or [261, p. 109]. Definition 5.3.5 Suppose that L is regular. Let v/^x be an L-field extension and let £, tr G L{yj\i). Then £ and 7r are said to be linearly disjoint over ji if £* and 7r* are linearly disjoint over /i* and £ n 7r = /i. By Theorem 5.3.2, £77 is an L-subring of F. By the notation v = £ ® 77, we mean that 1/ = £77 and that £, 77 are hnearly disjoint (over £ fl 77). By the notation v* = £* 0 77*, we mean the tensor product of £* and 77* over £* n 77*. Examples are easily constructed of L-field extensions u/fi with £, 7r G L(yj\x) such that £* and 7r* are hnearly disjoint over //*, but £ D 7r D /i. When this is the case, (£ fl 7r)//x is neutral and £ and 7r are hnearly disjoint over £ D 7r by Theorem 5.3.3(1). Theorem 5.3.6 Suppose that L is regular. Let v/£, £//i, and I/J//J, be L-field extensions such that £?/> is an L-subfield of F. If v and ip are linearly disjoint over fi (= i/fl^), then (1) v and £T/> are linearly disjoint over v fl (£^>); (2) £ and ip are linearly disjoint over \x. Conversely, if conditions (1) and (2) hold with u fl (£t/>) = £, then v and ip are linearly disjoint over /i. Proof. Suppose that v and ip are hnearly disjoint over zz. (1) v* and ip* are hnearly disjoint over fi* = v* fl ip* = {y fl ip)*. Thus 1/* and (£7/;)* = €**!>* a r e linearly disjoint over 1/* n (£V>)* = (^ n (£V0)* hY [67> Lemma, p. 162]. Hence v and £^/> are hnearly disjoint over v fl (£V0(2) Now /z C £0-0 C vDip = /z. Thus £ 0 ^ = //. Now v* and -0* are linearly disjoint over /J,* and so £* and ?/>* are hnearly disjoint over /z*. Hence £ and T/> are linearly disjoint over /z. For the converse, we have that 1/* and (£i/>)* = £*V>* a r e hnearly disjoint over £*. Also, £* and V7* are hnearly disjoint over /z*. Hence by [67, Lemma, p. 162], 1/* and ^* are hnearly disjoint over /z*. Now i/Hip = vC\ (£V>) Hip = %nip = fi. Thus i/ and ^ are linearly disjoint over /z. ■ There are some differences between [67, Lemma, p. 162] and Theorem 5.3.6 which should be noted. Example 5.3.7 Let L — [0,1]. Then zv/£, £//z, and ip/fj, are fuzzy field exten­ sions such that v and ip are linearly disjoint over zz, ^fl(£'0) 7^ £> an^ £^ — ^V7 even though £ ^ v : Let K = P{i) and F = K{s, tp ), where P is a perfect

178

field of characteristic p > 0, t is transcendental over P, and s is a root of the polynomial yp + ty + t iny over K. Then yp + ty + t is irreducible over K by Eisenstein's criterion. Hence K(s)/K is separable algebraic. Define the fuzzy subsets v, £, /x, and ty of F as follows: v = I 5 , where S = K(s)] ( 1

Z(z) = \ I [ 0

fi(z)=l

( 1 \ ^ 0

ifz£P(s?),

ifzeS\P(sr), otherwise;

ifzeP, ifzeS\P, otherwise;

ip = l j , where J = K(tp ). Then v, £, /i, and I/J are fuzzy sub fields of F and v* = S = £*, /x* = K, and ip* = J. By Theorem 5.3.4? £tp is a fuzzy subfield of F. Now v* and ip* are linearly disjoint over /x* since S/K is separable algebraic and J/K is purely inseparable. Also s = sp(—t-1) — 1. Thus (£ip)(s) > £(5P)A^(-* - 1 ) A(£)(-!) = 1. Hence (£ip)(s) = 1 = u(s). However, £(s) = \. Thus ^fl (£V) 7^ £. Hence v and ^ip are not linearly disjoint over £. Let z G F. Then z = Yl^Zo ^ s * ' where the ki G J. Now £ip = 1 on J since £tp D V> and if) = 1 on J. Thus

(WW

> A{(^)(^)|i = 0,l,...,p-1} > A{(^)(fe) A («0(*')l* = 0,1,... ,p - 1} = 1.

Hence £ip = 1F = vip. Note also that since £(s) = ^, ^ and £i/> are not equal onCTheorem 5.3.8 Suppose that L is regular. Let i//£, £//LI, and t/>//x 6e L-field extensions such that v and \p are linearly disjoint over /x (= v H ip) and £ip is an L-subfield of F, Then (1) implies (2) implies (3), where (1) £//x is compatible and /x is neutrally closed in £?/>; (3)vCi(W)

= £.

Proof. (1)=>(2): Let x G £*V*- Since £* and V* are linearly disjoint over /i*, x = YJi=iVi ® *»> where 7/; G £*, 2* G V* for i = l , . . . , n . Let 1 = £1 ? *2> • • • ,U € t/>* be linearly independent over /x* and be such that z; = E j = i *ii*i» % € /x*, j = 1 , . . . , r; i = 1 , . . . , n. Then s <8> 1 = £ ? = 1 yf ® D j = i ^ j = Z j = i E i L i kijVi ® * j a n d s o 2

179

YA=\

k

HVi = X-

N

°W £(x)

> > =

A{€(knyi) | i = 1 , . . . , n} A{$(fcii)Af(i/0 | i = l , . . . , n } A{^(yi) | i = l , . . . , n } ,

where the last equality holds since ^/ji is compatible and there exists yi £ fi*. Thus £(x) > A{f(3/i) A ^ i ) | i = l , . . . , n } . Hence £(x) > (£i/>)(x) and so £(x) = (f-0)(x). Let x € fi*. Since ^z is neutrally closed in £^>, /z(x) = (£?/>) (^O > €(x) > n(x). Hence £(x) = (£ip)(x). Thus condition (2) holds. (2)^(3): ( i / n ( ^ ) ) * = i / T l f y (by Theorem 5.3.3) = £* (by [67, Lemma, p. 162]). Now i/ D f and fV = £ on £*• Thus i/ n (£V>) = € on £*• Since

Theorem 5.3.9 Suppose that L is regular. Let v/£, £//z, and tp/jj, be L-field extensions such that £ip is an L-field extensions. If v and ip are linearly disjoint over fj,, £//J, is compatible, and \x is neutrally closed in £rp, then v ^ £ implies that vif; ^ £ip. Proof. Now v* and I/J* are hnearly disjoint over fi*'. Thus if u* ^ £*, then (yijj)* = v*i>* ^ £** = (£)* and s o ^ T^ £V>- Suppose that i/* = £*• Let x G i/* be such that z/(x) > £(x). By Theorem 5.3.8, (vip)(x) > v(x) and (^)(x) = £(x). Hence vip y£ £ip.M Definition 5.3.10 Suppose that F has characteristic p > 0. £e£ x ^ -^ F an^ leti e NU{0}. Define the L-subsetxp% of F as follows: V x E F, xp% {%) = \{z) if x = zp for some z € F and x(x) — 0 otherwise. In the remainder of this section, we assume that F has characteristic p > 0. Theorem 5.3.11 If u e L(F), then vpi G L(F) V z € N. Proof. Let x, y € F and i e N. Then vpi (xpi - yp%) = vp\(x - y)p%) = v(x — y)> v(x) A v{y) = vp\xpt) A vp\ypt). Suppose that either x ^ Fp% or pi p p p y f. F . Then v \x - y) > 0 = */ *(:r) A v \y). Similarly, vp% (xy'1) > vp\x) A vp\y) for y ^ 0. ■ Theorem 5.3.12 Let

x

6 L F . Then V i j ' G N U {0}, (x p ') p i = xp^ •

180

Proof. Let x G F Then (xpi)pi (xpi+j) = (xpi Y ((xpi f) = X P V ) = x{x) = yf J (xpZ J). Suppose that x £ Fp% 3. Let k be the largest nonnegative integer such that x = yp for some y G F. Then k < i + j . If fc < j , then (x p l ) p J (x) = 0. Suppose that k > j . Then there exists a nonnegative integer m such that fc = j + m and m < i. Thus ( x P V 0 * 0 = (xpi Y ({y^ Y) ^ p pTn ) = 0. Clearly, x * w ( x ) = 0. Thus ( x ^ V = x p * + '. ■ x \y Theorem 5.3.13 Let v/fi be an L-field extension and let n G N U {0}. If upn C /i, £/ien V xa C z/, (x a ) p U C yu. Conversely, i / V x a C */, (x a ) p C /z, £/ien i X C /x. Proof. Suppose that vpn C yu. Let xa C z/. Then a < i/(x) = vpn {xp ) < p n(x ). Thus (x a ) p = (xp )a C /i. Conversely, suppose that x a C v implies (xa)pn Q V- Suppose that x £ Fpn. Then vpn(x) = 0 < fi(x). Let a = z/(x). Then xa C z/ and so ( x p n ) a = (x a ) p n C /i, i.e., v>pn(xpn) = v(x) = a < fi(xpn). Thus zX C p. ■ Theorem 5.3.14 Let x £ £ F - T/iera iTie following conditions hold. ft) V a € L\{0}, {Xa)pi = (Xpi)a for i = 1 , 2 , . . . ; r^X*"" = ( x p " ) * / o r t = l , 2 , . . . . Proof. By Theorem 5.3.12, it suffices to prove the result for i = 1. (1) a* G (xa) p & x eXa& X(x) > a & Xp(xp) >a^xpe (Xp)a- If p p p V £ (x )a, then x (y) > a > 0 and so y = x for some x e F. (2) x p € x * p ^ x 6 f < ^ x(*) > 0 ^> x p (^ p ) > 0 4* a* G (x p )*- ■ Theorem 5.3.15 Let f, TT G L F . 77ien (f TT)** = f ^ V * /or z = 1,2,.... Proof. By Theorem 5.3.12, it suffices to prove the result for i - 1. Let xeF. Then

(e-*p)(xp) = v{AU(e(yi)**p(zi)) =

P

I xp =

EtiVi^neN}

V{A- =1 (^((^) ) ATT ((^) P )) | xp = £ ? = i t e ) , W , (*i)P € (e)* = ( D P , W e (7Tp)* = 7T*P, i = 1,... ,n;n G N} P

= V{A?=1K(^) A7r(Vi)) | x = J27=i^i^i

€ T,

Vi G 7r*,i = 1 , . . . , n ; n G N }

.= (fr)(x) = (£*)*(**). Now £PTI-P C (^TT)P and so if x £ F*, (£ P TTP)(X) = 0. Thus f»W = (£TT)P. ■ Theorem 5.3.16 Let i///x fee an L-field extension. ITien (/i( x ))p' = M p i (x p i ) /or» = 1,2,....

Let \ € LF and £ C i/.

181

Proof.

It suffices to prove the result for i = 1. Let z G F. Then

(Mx))p(*p) = =

(Kx))(z) A{eWUeiW/i),Qx}

= A{^(Z") |£" e I ( / / / ) , ? 2 xp} > A { ^ ( 2 P ) | VeX(i/P/ M p ),V'2xn -

(MP(XP))(*P).

We now show that the above inequahty is actually an equahty. Let ip G L(yp/fip) be such that ip D xp- Define the L-subset £ of F as follows: £(z) = xp(zp) for aU z G F. Then £ G L(F). Since f (*) = xP(zp) < vp{zp) = v[z) V zeF^Cu. Now £*>(zP) = f (*) = ^(2 p ) > Mp(*p) V x p (z p ). Thus Q / x V x Since f = ^ o n F p , we have our desired conclusion, i.e., (fi(x))p = Mp(xp)- " Definition 5.3.17 Suppose that L is regular. Let v j\i be an L-field extension. Then vj\i is said to be separable if ji and vp are linearly disjoint over pP. Theorem 5.3.18 Suppose that L is regular. Let v/fi be an L-field extension. Then the following assertions hold. (1) v* I'[i* is separable if and only if vp and fi are linearly disjoint over p v H /x and {yp C\ fi)/fip is neutral. (2) vI\i is separable if and only if v* /'/J,* is separable and vv D fi = jip. Proof. (1) i/*/[i* is separable <^> (v*)p and fi* are linearly disjoint over p p (/x*) <^> {v )* and /i* are linearly disjoint over (/J,P)* <=> vp and fi are linearly disjoint over vp D /i- and (yp fl fi)/'/J,P is neutral. (2) The result here follows from (1). ■ Theorem 5.3.19 Suppose that L is regular. Let z//£ and %/fj, be L-field ex­ tensions. (1) If vjJJ, is separable, then £/'/i is separable. (2) If v/^ and £/fi are separable, then v/fj, is separable. Proof. (1) v*/fi* is separable and so £*/^* is separable. Now p p = i / p n / i 3 p £ H [i 2 Mp- Thus £p fl fi = \±p. Thus the desired result follows from Theorem 5.3.18. (2) i/*/£* and £*/M* are separable. Thus v*/fi* is separable. Now i/ p fl/iC p v D £ = £p. Hence vp D \i C £p n fi = fip. Thus up n fi = fip and so again the desired conclusion follows from Theorem 5.3.18. I Let K be a subfield of F. It follows routinely that F/K is separable if and only if I F / I K is separable.

182

Example 5.3.20 Let L — [0,1]. vjii is a fuzzy field extension which is separa­ ble algebraic, but not separable: Let F = P(t), where P is a perfect field of char­ acteristic p > 0 and t is transcendental over P. Define the fuzzy subsets v and JJL of F as follows: v = lp and fi(z) — 1 if z £ Fp and fi(z) — ^ otherwise. Then v and \x are fuzzy subfields of F. Since v* = /J,*, v*/fJ>* is trivially separable algebraic. Thus v/fi is separable algebraic. Now (vp C\ n)(tp) = v(t) Afi(tp) = 1 and /J>p(tp) = fi(t) = ^. Thus vp D fi ^ \ip. Hence vj'ji is not separable. Example 5.3.21 Let L = [0,1]. v/ fi is a fuzzy field extension which is sepa­ rable and separable algebraic, but does not satisfy the separability condition appearing in Theorem 5.2.13 and so is also not separable l-algebraic: Let F = P(t), where P is a perfect field of characteristic p > 0 and t is tran­ scendental over P. Define the fuzzy subsets v and yb of F as follows: v = lp and n{z) = 1 if z G P and fi(z) = \ otherwise. Then v and fi are fuzzy subfields of F. Since v* = //*, v*/fi* is trivially separable algebraic. Thus vj[i is separable algebraic. Suppose that z £ Fp. Then [yp D fJ>)(z) = vp{z) A [x[z) = 0 A ± = 0 = /ip(z). Suppose that z (£ P. Then [yp n fi)(zp) = v(z) A fi(zp) = 1 A \ = \ = fi(z) = fip(zp). Suppose that z G P. Then {yp D/i)0) = 1 = fip(z). Thus vp C\/JL = \xp. Since V*//J,* is separable, v/fi is separable. Define the fuzzy subset £ of F as follows: e(z) ^ '

=

I 1 \ \

ifzGFP, otherwise.

Then £ is a fuzzy intermediate field of vj[i. Now v(t) = 1 ^ £(t) ^ €(tp). Theorem 5.3.22 Suppose that L is regular. Let v/fi be an L-field extension. (1) If u/fi is separable and algebraic, then v j/J, is separable algebraic. (2) Suppose that V £ G L(u/fi) andV z G F, either £(z) = £(zp) or £,{z) = v(z). Then z///x is separable. Proof. (1) Since v/y is separable and algebraic, i/*//z* is separable and algebraic. Thus v* l\i* is separable algebraic. Hence vj\i is separable algebraic. (2) By Theorem 5.2.13, vj\x is separable algebraic. Thus i/*/fi* is separable. Hence it suffices to show that vp n fi = fip. Let z G F. Then {vp D fi)(z) = vp{z) A fi(z) = 0 = fip(z) if z i Fp. Now {vp n y)(zp) = u(z) A /i(zp) = n(z) since either /i(z) = fi(zp) or fi(z) = v(z). Thus {yp n y)(zp) = np\zp). Hence p VPD/J, = ii M

5.4

Finite-Valued L-Field Extensions

In this section, we are concerned with finite properties of L-subfields.

183

Let v G L(F). We recall that v is said to be finite-valued if and only if \v{F)\ < oo; otherwise v is said to be infinite-valued. Theorem 5.4.1 Let {F{\i G N} be a collection of subfields of F and let {a,i\i G N} be a subset of L. (1) Suppose that Fi C F i + i , ax = 1, and a{ > ai+1 V i e N . Define the L-subset v of F as follows: y(z\ = { )

jai \ 0

ifze FAFi-i V* G N, where F0 = 0 if z G F\UieN F{.

Then v G L(F). (2) Suppose that Fi D Fi+1 and a{ < ai+i < 1 V i G N. Define the L-subset v of F as follows:

( en ifzeFi\Fi+x VieN v(z) = I l ifzenieNFi, [ 0 Thenue

ifzeF\Fx.

L(F).

Proof. The desired result follows from Theorem 5.1.2. ■ Throughout the remainder of this section K denotes a subfield of F. Theorem 5.4.2 (1) If F is a finite field, then every L-subfield v of F is finitevalued. (2) Suppose that L is a dense chain. Then [F : K] < oo if and only if V v G L{F) such that v D 1^, v is finite-valued. Proof. Assertion (1) is immediate. Suppose that there exists v G L(F) such that v D IK and v is infinite-valued. Then v{F) contains either a strictly ascending infinite sequence or a strictly descending infinite sequence of elements of L. Hence either F/K has either a strictly descending infinite sequence or a strictly ascending infinite sequence of intermediate fields, respectively, by Theorem 5.1.2. Thus [F : K] = oo. Conversely, suppose that [F : K] = oo. Suppose that F/K is algebraic. Then there is a strictly ascending chain of intermediate fields {Fi \ i G N} of F/K. By Theorem 5.4.1, there exists v G L(F) such that v D IK and v is infinite-valued. Suppose that F/K is not algebraic. Then there exists t G F which is transcendental over K. Let Fi = K(tn%), where n is an integer > 2, z = 1,2, Then the Fi form a strictly descending infinite sequence of intermediate fields of F/K. By Theorem 5.4.1, there exists v G L(F) such that v 2 I K and v is infinite-valued. ■ The converse of Theorem 5.4.2(1) is of course not true. Let Q denote the field of rational numbers. Since Q has no subfield other than itself, every L-subfield v of Q equals 1Q.

184

Theorem 5.4.3 Suppose that L is a dense chain. (1) Suppose that F has characteristic p > 0. Then F is finite if and only if every L-subfield v of F is finite-valued. (2) Suppose that F has characteristic 0. Then [F : Q] < oo if and only if V v G L(F), v is finite-valued. Proof. The proof follows from Theorem 5.4.2 and the fact that V v G L(F), v D l p , where P is the prime subfield of F. ■ Theorem 5.4.4 Suppose that L is a dense chain and that F/K is finitely generated. Then F/K is algebraic if and only ifVi/E L(F) such that v ~^\K, v is finite-valued. Proof. If F/K is algebraic, then [F : K] < oo. The desired result now follows from Theorem 5.4.2. ■ Theorem 5.4.5 If F/K has no proper intermediate fields, then every v G L(F) such that v D IK is one or two-valued. Conversely, if L has more than two elements and every v G L(F) such that v D IK is one or two-valued, then F/K has no proper intermediate fields. ■ Theorem 5.4.6 F/K is simple if and only if there exists z G F such that V v G L(F) such that U'DIK and V x G F , V{Z) < v{x). Proof. Suppose that F/K is simple. Then F = K(z) for some z G F. Let v G L(F), where v D 1K and let x G F. Then x = (ZZ=o kizi)(E?=o hjzj)~^"> where the ki, hj G K. Thus

> >

i/(fco) A . . . A v(knzn) v{?)

A is(h0) A . . . A

v(hmzm)

since v(kiZ%) > v(ki) A v(z%) = v(zl) > v(z) for z = 0 , 1 , . . . ,ra and similarly u(hjZj) > v{z) for j = 0 , 1 , . . . , m. Conversely, suppose that there exists such a z G F. Suppose that F ^ K(z). Then there exists x G F such x £ K(z). Let a G L be such that a < 1. Define the L-subset v of F as follows: v{y) = 1 if y G K(z), v(y) = a otherwise. Then v G L(F), V^IK, and i/(x) < v(z), a contradiction. Thus F = K(z). ■ Theorem 5.4.7 Suppose that [F : K] < oo. Then F/K has a finite number of intermediate fields if and only if there exists z G F such that V v G L(F) such that VO\K and V x G F, v{z) < v(x).

185

Proof. The desired result follows from Theorem 5.4.6 and Artin's theorem on primitive elements [67], i.e., F/K has a finite number of intermediate fields if and only if F/K is simple. ■ Suppose now that F is a finite field. Then F has characteristic a prime, say p. We let K° denote the multiplicative group of K. Let v G L(F). Let v° be the L-subset of F° defined by v°(x) = v(x) V x G F°. Then i/° is an L-subgroup of F°. Suppose that L is a chain. Then (z/0)* = (x) for some x G (i/ 0 )*. Let a G v°(F°) and let na denote the smallest positive integer such that (t/°)a = (nax). Then i/° = ({(xn*)a\a G v°(F°)\{0}}) by Theorem 2.1.13. This result corresponds to the crisp result that F° is cyclic. Let 7T = l p , where P is the prime subfield of F. Then the following result corresponds to the crisp result that F = P(x), where x is any generator of F°. Theorem 5.4.8 Suppose that L is a chain, F is a finite field, and v G L(F). Then v = <{(xna)a\aev(F°)\{0}}), where (y0)* = (x) . Proof. Suppose that z G (^°)*\{0}. Then by Theorem 2.1.13, v{z) = v°(z) = ({(x n -) 0 |a G ^°(F°)\{0}}> (z) = ((xn>)b) (z) for some b G ^(F)\{0} since v is finite-valued. By Theorem 1.2.14, <(*">)„> (z)

= < < <

v{(0r»»)6)*(z)|i€Z} V{(Eto(Pi) v (Pi)((^ n o )-) i )W | P i e Z p , » = 0 , l , . . . , f c ; fceNU{0},a€!/°(F())\{0}} 7 r ( { ( ^ ) a | a € i/>(F°)\{0}})(z) (by the fact that the v(pi) = 1) v{z).

Hence we have the desired result for z / 0. Now u(0) = 7r(0). H Theorem 5.4.9 Suppose that L is regular, F is a finite field, and v G L(F). Then v is a direct sum of m L-subgroups /ii (i = 1 , . . . , m) of the additive group of F such that (/ii)* is cyclic and of order p, i = 1 , . . . , m. Proof. Since v* is finite, v is finite-valued. Since also pis* = (0), v is a direct sum of L-subgroups m C v such that the (//;)* are cychc and of order p by Theorem 2.1.6. Since i/* is a direct sum of m cychc groups of order, [30, Proposition 1(b), A. V. 93], we have that there must be m /ii. ■ Suppose that F is finite. The following theorem shows that not only do the automorphisms of F leave the subfields of F invariant, but also the L-subfields. Let / be an automorphism of F. Since / is one-to-one, f(y){y) = v(x) V y G F, where y = f(x) and v G L(F).

186

Theorem 5.4.10 Suppose that F is a finite field and v G L(F). If f is an automorphism of F, then f(v) = v. Proof. We first show that f{ya) = f(v)a V a G L. Now y G f(u)a & (f(v))(y) >a& (f(v))(f(x)) > a, where y = f(x) <=> v(x) > a <S> x G va & y = f[x) G f(va) since / is one-to-one. Now f(va) = vaV a e L since every subfield of F is left invariant under / [30, Proposition 3(a), A. V. 94]. Since fiya)

5.5

= / M a , f(v)a

= "a V a G L. T h u s f(u)

= V. ■

Separability and Modularity

We assume in this section that F has characteristic p > 0. Let v G L(F) and fi G L(v). Proposition 5.5.1 Suppose thatL is regular. Ifv/fJ> is separable, thenvp is separable V n G N.

//ip

Proof. It suffices to prove VP//J,P is separable. vv and fi are linearly disjoint p over \x . Thus vp n /x = /xp. Hence ^ p2 n /xp = (^p H ^ ) p = (^ p ) p = V? by Theorem 5.3.12. Now i/*//z* is separable and so z/*p//z*p is separable. Thus vvI\xv is separable. ■ Proposition 5.5.2 Suppose that L is regular. Let £, 77 G L{yl\x). that £* and 77* are linearly disjoint over /z*. T/ien W (f ^ 7 / ) / ^ *5 neutral; (2) £ and 77 are linearly disjoint over £ fl 77.

Suppose

Proof. (1) By Theorem 5.3.3, (f n 77)* = £* fl 77* = /x*. (2) The desired result follows from Definition 5.3.5 and the fact that /x* =

(*ni/)*. ■ Example 5.5.3 Le* Z, = [0,1]. Let F = P(x) and K = P(x p ), where P is a perfect field of characteristic p > 0 and x is transcendental over P. Let v = 1F. Define the L-subset fi of F by

I \

otherwise.

Then /J, G L(v). Now (// H vp){xp) = fi(xp) A vp{xp) = fi(xp) A i/(x) = 1

187

and fip(xp) = (j,(x) = ^. Thus ji 0 vv ^ \iv. Hence vj\i is not separable. However, vj[i is neutral and thus separable algebraic. Now (fi n z/p)* = fi* D (vpy = ii*r\(v*)p = F " = ([JL*)P = (fjpy by Theorem 5.3.14. Thus (fiPivP)/^ is neutral. Example 5.5.3 also shows that a separable algebraic L-field extension need not be separable according to the current definition of separable. This illus­ trates another difference between the concepts of separable /-algebraic and separable algebraic since a separable /-algebraic L-field extension is necessarily separable when L is regular, Theorem 5.2.13. A straight forward argument shows that (£77)* = £*rj* when L is regular and that £, 77 C £77. Proposition 5.5.4 Suppose that L is regular. Let£, 77 G L(v). Then v = £(8)77 if and only if v* = £*
V a € L.

Proof. (1) Immediate. (2) x G (£ fl 77)a 44> (£ D 77) (x) > a «=> £(x) A 77(0:) > a <^> £(x) > a and 77(2;) > a <^> x G £ a H 77a. (3) x G (£77) a <£> (£77) (x) > a <^> 3 n G N, 7/i, Zi G F such that £(?/;) > a and 77(2:;) > a for i = 1 , . . . , n and x = YA=I Vizi ^ x e ZaVa- B Proposition 5.5.6 Suppose that L is regular. Let £, 77 G L(i/). Let X = £* H 77*. Suppose that £* n 77* 2 ^ - TTien £ and 77 arie linearly disjoint (over £ D 77) i/ and only if £ a and 77a are linearly disjoint over K V a G L\{0}. Proof. Since £* fl 77* D X, ( a n 7/a D X V a G L\{0}. Suppose that £ and 77 are linearly disjoint. Then £* and 77* are linearly disjoint over K. Since K Q ia Q £* and K Cna Cn*, the desired result follows. Conversely, suppose that £ a and 77a are Hnearly disjoint over K V a G L\{0}. Let 6 G L\{0} and

188

a G Lh) where Lh = {a | 0 < a < b}. Since & C £ a , r]a and £& are linearly disjoint over K. Let a: G 77*. Set c = rj(x). Then c > 0. Hence cAbeLb and so x 6 UaeLbVa- Thus 77* = Ua6Lb^a and so 77* and & are linearly disjoint over K. Since b is arbitrary and £* = U a 6 L b £ a , £* and 77* are linearly disjoint over K. Thus £ and 77 are linearly disjoint. ■ Proposition 5.5.7 Suppose that L is regular. Let £, 77 G L(^). £e£ if = £* fl 77*. Suppose that £* D 77* D K. (1) Suppose that £ and 77 are finite-valued and L is a chain. If v = £ ® 77, tfien i/a = £ a ® K ^7a V a E A W ^ ; i/ z/a = £a 0 K Va V a G L\{°}> then z/ = £ 77. Proof. (1) Since v = £ ® 77, 1/ = £77. Thus i/a = (£77^ = £a77a V a G L\{0} by Proposition 5.5.5. Since £ and 77 are linearly disjoint, £ a and 77a are linearly disjoint over K V a G £\{0} by Proposition 5.5.6. (2) By Proposition 5.5.6, £ and 77 are linearly disjoint. Now v = £77 since Va = ZaVa Q (^)a

C Va V a G L \ { 0 } -

■

Definition 5.5.8 i///i is sazrf to 6e modular linearly disjoint (over vp fl u) V n G N.

if and only if vp

and fi are

Proposition 5.5.9 Suppose that L is regular. vj\i is modular if and only if v*//J<* is modular. Proof. Now vpTl and \i are linearly disjoint over vpn fl \i «=> (vpn)* and /1* are Unearly disjoint over {vpn fl//)*. By Theorem 5.3.3, (upn D/J,)* = (z/ pri )*D^* and by Theorem 5.3.14, (vpn)* = (v*)pTl. Hence vpn and u are linearly disjoint over vpTl D \x <£> {v*)pn and \i* are hnearly disjoint over (v*)pTl D //*. ■ Corollary 5.5.10 Suppose that L is regular. If v*/p* is separable, then v/ji is modular. H Suppose that L is regular. By Theorem 5.2.3, we have that v/p is purely inseparable <=> I/*//J,* is purely inseparable. Also v j\i has a unique maximal purely inseparable L-intermediate field £ by Theorem 5.2.6. It follows imme­ diately that £ is neutrally closed in v. Theorem 5.5.11 Suppose that L is regular. Let £ be the unique maximal purely inseparable L-intermediate field of v/u. Then V//J, is modular if and only if z//£ is separable and £//z is modular.

189 Proof. Suppose that i///x is modular. Then i/"//i* is modular by Proposition 5.5.9 and so f*//i* is modular by [85]. Thus £//x is modular by Proposition 5.5.9. By [85], v*/£* is separable. Since £ is neutrally closed in v, £p is neutrally closed in vp. Thus £ p = v? on (£ p )* and so £p = v? n £ on (£p)* = £*P = i/"Pnf * = (i/ p nO*. Hence f = z^n£. Thus i//f is separable by Theorem 5.3.18. Conversely, suppose that v/£ is separable and £//i is modular. Then v*/C is separable and £*/^* is modular. Thus i/"//x* is modular by [85]. Hence vj\i is modular. ■

5.6

Neutrally Closed L-Subfields

In this section, F is of characteristic p > 0. Let /x G L(v), where v G -L(-F). Recall that /x is neutrally closed in v if and only if v = \i on /x*. Proposition 5.6.1 Suppose that L is regular. Let £, 77 G L{yj\i). Suppose that fi is neutrally closed in v. (1) vj'/x is separable if and only if I/*//J,* is separable. (2) £ and 77 linearly disjoint over /x z/ and on/?/ if £* and 77* are linearly disjoint over /J,* . Proof. (1) Suppose that JJP is not neutrally closed in vv. Then 3 x G F such that xa C z/p, x a 2 Mp> x 6 Q Mp f° r some 6 and a, where 0 < b < a < 1. Since 6, a > 0, 3 z G F such that x = zp. Thus za Q v, za £ (j,, and zi, C /j, contrary to the hypothesis. Hence /xp is neutrally closed in vv. Suppose that v*/fj,* is separable. Then

(fjpy = fjrp = M* n z/*p = \? n K ) * = (/x n vpy. Thus (/x D vv)j[iv is neutral. Hence ^ f l i / p = \iv. The desired result follows from Proposition 5.5.1. (2) Suppose that £* and 77* are linearly disjoint over ^x*. Then (£ n 77)* = £* n 77* = /x* and so (£ n 77) //x is neutral. Since /x is neutrally closed in 1/, /x is neutrally closed in £ fl 77. Thus £ fl 77 = /x. The desired result now follows from Definition 5.3.5. ■ Let /x G L(y). It was shown in Example 5.2.14 that fj, does not necessarily have an /-algebraic closure in u, i.e., there does not exist 77 G L{yj\i) such that 77//X is algebraic and V xa C 1/ with a > 0, x a algebraic over /J. imphes x a C 77. Proposition 5.6.2 Suppose that L is regular. Let /x G I>(^). Suppose that /\{v{b) I 6 G |x*} > V{z/(:r) | a: ^ /x*}. 7/^x is neutrally closed in v, then \x has an I-algebraic closure in v.

190

Proof. Let /z*c denote the algebraic closure of fi* in v*. Define the fuzzy c subset jj, of F by fic(x) = v(x) if x G ^* c and nc(x) = 0 otherwise. Then \xc G L{yj\x). Let xa C /xc with a > 0. Then xGfi c * = /x*c and so 3 n G N, A:0, fci,..., kn G fi* such that knxn + . . . 4- k\x + k0 = 0. Thus ( ^ n ) a ( ^ a ) n + • • • + (fcija^o + (*o)a = 0 o .

By hypothesis and the fact that \x = ^ on //*, A{/x(6)|6 G ^*} > V{i/(a;) | # ^ /i*}. Thus (ki)a C /i for i = 0 , 1 , . . . ,n. Hence x a is /-algebraic over /x. Thus ^ c //x is /-algebraic. Let ya C i/, a > 0, be /-algebraic over /z. Then 2/ is algebraic over fi*. Hence y G /z*c. Thus 1/(2/) = nc{y). Hence ya C /xc. ■ In Proposition 5.6.2, /xc is neutrally closed in v since fic = u on fic*. Proposition 5.6.3 Suppose that L is regular. Let \i G L{y). Suppose that \x is neutrally closed in v. If v/ fi is purely inseparable, then u/fi is I-purely inseparable. Proof. Let xa C v with a > 0. Then 3 e G N U {0} and 3 b G L\{0} such that (x a ) p e + (—a:6)pe = 0C, where c = a A b and (x6)pC C /j,. Since /x is neutrally closed in 1/, (x a ) p C fi. Hence we can choose b > a. Thus xa is /-purely inseparable over \i. ■ If vj\i is /-purely inseparable, then clearly vj\i is purely inseparable. Thus Proposition 5.6.3 gives a condition under which vj\x is /-purely inseparable if and only if vjfi is purely inseparable, namely that fi is neutrally closed in v. In fact, the assumption that /i be neutrally closed in v yields the result that if xa Q v, a > 0, is algebraic over ji with respect to the polynomial p(y) = ypG +k over //*, Definition 5.1.13, then xa is also /-algebraic over fi with respect to p(y) over //*. Example 5.6.4 Let L = [0,1]. Let F = P(x), where P is a perfect field of characteristic p > 0 and x is transcendental over P. Let v = lp. Define the fuzzy subset JJ, of F by /i(y) = 1 if y G P(xp) and fi(y) = \ otherwise. Then fi G L{y). Now ya + (-y)x = 0 i and (y 6 )* + (-yb)p = 0b V y e F andV a, b G [0,1], a > ^. Thus vjfi is both purely inseparable and l-purely inseparable, yet ji is not neutrally closed in v. Proposition 5.6.5 Suppose that L is regular. Let £ G L[yj\x). (1) Suppose that vj^ is neutral. Then £/// is modular if and only if v/fi is modular. (2) Suppose that £/// is neutral. Then v/£ is modular if and only if vjfi is modular.

191

Proof. (1) i/* = £*. Thus £*//x* is modular <^ z/*//x* is modular. Hence the desired result follows from Proposition 5.5.9. (2) f* = /z*. Thus i/7f* is modular o- v*/fi* is modular. ■ Corollary 5.6.6 Suppose that L is regular. (1) v/fi is modular if and only if vjr/x(n) is modular. (2) I///JL is modular if and only ifv^/fj, is modular, where i/ n ) is the neutral closure of v in some n G L(lp/v). For (1), we note that /i* = /x* and for (2), v* = i/ n )*. ■

Proof.

P r o p o s i t i o n 5.6.7 Suppose that L is regular. Let /x G L(v) and let £ G L(F). Suppose that /x£ and v£ are L-sub fields of F. If fi is neutrally closed in v, then *x£ is neutrally closed in v£. Proof. Now fj, = v on /x*. Thus (/x£)(x) = V{^(yi) A . . . A xx(yn) A £(zi) A . . . A Z(zn) | x = YA=I Vizii 2/i € ^*, *i € £*, i = 1 , . . . ,n, ra e N} = {yk)(x) if x G (/x£)* = M*£*- That is, /i£ = i/£ on (/z£)* a n d s o M£ i s neutrally closed in P r o p o s i t i o n 5.6.8 Suppose that L is regular. Let /x, £ G L(i/) fee sixc/i £/m£ jz£ w an L-subfield of F. Then f i ^ ^ = (xx£)
(^n0 (n) . Proof. (/x£)(n) = ^ on (xx£)* and ^ ^ = vv on /x*£*. Now (/x£)* = M*£* and v = vv since V x 6 F, x / 0, i/i/(x) > i/(l) A i/(x) = v{x) and if a: = Y^l=\ Vizii t n e n K^) > ^(2/i) A ^(^i) for z = 1 , . . . ,n and so v{x) > vv(x). Let x G ( ( / i f l O ( n ) ) * . Then v(x) = (xx H f ) ( n ) W < (^ (n) n £ ( n ) )(x) = /xW(x) A £( n )(x) = i/(x) A v(x) = v(x). Hence z>) n £ ( n ) = (/ifl £) ( n ) on ((xx fl 0 ( n ) ) * - T h u s it; suffices to show that ((/x n £) ( n ) )* = (^ (n) H £ ( n ) )*. Now

((/i n0^)* = (xx n0* = /x* n r = /^(n)* n£(n)* - 0* n £
D i s t i n g u i s h e d L-Subfields

Let F have characteristic p > 0. Let v G L(F) and /x G L(i/). Let £ G L(v/fi). We let xx*°° denote /x*p °°, where p n ^ • P " 0 0 ={k ~ | /cGxx*,nGNU{0}}. We sometimes think of £ as being extended to /J*°°F such that £(z) = 0 V x G ^°°F\F. For L-subfields l^*oo and 77 of /x*°°F, lM*oo77 denotes the composite of l^*oo and 77.

192 An intermediate field D o f a field extension F/K of characteristic p > 0 is called d i s t i n g u i s h e d if D/K is separable and F C Kp °° D. D e f i n i t i o n 5 . 7 . 1 vj'/J, is said to have a distinguished L-intermediate T]if3rjE L{yj\i) such that rj/fj, is separable and v C 1^*0077.

field

P r o p o s i t i o n 5.7.2 Suppose that L is regular. Let 77 G L{yj\i). (1) If rf is distinguished in v*/fi*, then 1^*00^/1^*0077 is neutral. (2) If 77 is distinguished in v/fi, then 77 is neutrally closed in v. Proof. (1) ( V o o i / ) * = n*°°v* = ^*°°77* = (V0077)*. (2) Let d G 77*. T h e n v(d) < (l^oorj)(d) < l M *oo(l)A 77(d) = 77(d) < v{d). ■ Let K b e a subfield of v*. T h e n K is said t o b e c o m p a t i b l e with v if V{^(c) I c 6 v*\K} < A{v(c)

Ic G if}.

We note t h a t every level subfield of v is compatible with v. T h e o r e m 5 . 7 . 3 Suppose that L is regular. If v/fx has a distinguished Lintermediate field, then V*//J,* has a distinguished intermediate field. Con­ versely, if v* I[i* has a distinguished intermediate field which is compatible with v, then vj'/1 has a distinguished L-intermediate field. Proof. Suppose t h a t v/fi has a distinguished //-intermediate field 77. T h e n v C lM*oo77 a n d so v* C (1^*0077)* = (lM*oo)*77* = /j,*°°r]*. Conversely, suppose t h a t v*/fi* h a s a distinguished intermediate field S which is compatible with v. Let 77 b e t h e L-subset of F defined by 77(5) = v(s) if 5 G S a n d 77(5) = 0 if s G F\S. T h e n 77 is a L-subfield of F, 77* = S, a n d 1/* C /x*°°77*. If x G 77*, then v(x) = rj(x) < (l^-oo77)(a;). Suppose t h a t x G v*\rj*. T h e n (lM*oo77)(x)

= =

V{lAl*oo (fex) A . . . A 1^*00 (fcn) A r)(di) A . . . A 7](dn) \ h G/x*°°,d; G77*,2 = l , . . . , n , ^ E l i M i ^ E N } V{7 7 (d 1 ) A . . . A77(d n ) I ki G A X * 0 0 , * G 77*,

i = l,...,n, x = E i L i M i , ^ 6 N } = >

V{i/(di) A . . . A 1 / ^ ) IfciG M * 0 0 , * G 77*, i = l , . . . , n , x = 5^JL 1 fcid i ,n G N } v{x)>

where t h e inequality holds since 77* is compatible with v. ■ P r o p o s i t i o n 5 . 7 . 4 Suppose that L is regular. Let £ be the unique maximal purely inseparable L-intermediate field of v/fi. Then £ is neutrally closed in v if and only if l^*oo n v = £.

193 Proof.

Suppose t h a t f is neutrally closed in v. Clearly, lM*oo (Iv D £ since

M*oo

L e t

3 ^

^

G

p

j f ^ g (J^^

ni/)*

=

M*oonzy* =

^

t h e n

^moQ

ni/

)(x)

=

0 = £(x). Suppose t h a t x G ( l ^ o o f W ) * . T h e n (l^*ooni/)(x) = l M *oo(x)Az/(x) = 1 A £(x)} = £(x), where £(x) = i/(x) since £ is neutrally closed in ZA For t h e converse, let x e f . T h e n £(x) = lM*oo(x) A i/(x) = 1 A v{x) = v(x). ■

5.8

Splitting

We assume F is of characteristic p > 0. Let z/ G L ( i r ) a n d \i G L ( ^ ) . D e f i n i t i o n 5 . 8 . 1 vj\i is said to split if 3 £, 77 G L{yj\i) such that £//i zs purely inseparable, rj/fi is separable, and v — £ (g)^ 77, i.e., z/ = £ (g) 7? a n d ^nry = ^. T h e following example illustrates complications which arise in fuzzifying structure results from t h e crisp case. U//JL does n o t split even though 3 £ G L(I//II) such t h a t £ p C ^ a n d *//£ h a s a separating transcendence basis, i.e., an L-subset x °f F s u c n t h a t X £ v, X is algebraically independent over £ a n d z//£(x) is separable algebraic. Suppose t h a t L is regular. T h e n *//£ has a separating transcendence basis if and only if i/*/£* has a separating transcendence basis. I n fact, x is a separating transcendence basis for z//£ if a n d only if x* is a separating transcendence basis for i/*/£*. E x a m p l e 5 . 8 . 2 Le£ L = [0,1]. Let K = P(xp) and F = K(z, x ) , where P is a perfect field of characteristic p > 0 and x, z are algebraically independent over P. Let H = K{zpx). Now F = K(x) 0 x K(z). Let F = J ®K L, where J = K(x) and L is an intermediate field of F/K. Then H <£ J. Suppose that H C L. Then zpx G L. Now z G JL and so zp G K(LP) C L. Thus x G L which is impossible since L/K is necessarily separable. Hence H g L. That is, H is not contained in any tensor factor of F over K. Define the L-subset v of F by u(y) = 1 if y G H and v(y) = \ otherwise. Then v is an L-subfield of F and v* — F. Let fi = IK- Let £, 77 G L(u/fi) be such that £/fi is purely inseparable and rj/fi is separable. Then £ Pi 77 = fj, since ji is neutrally closed in v. Now £* = J and ^* = K. We show that v ^ £ (g) 77. Now n

( £ 0 77)(2px) = V{A^ 1 {£(x i )A77(7/ i )} I zpx = ^ x ^ n G N } . Suppose that (£ rj)(zpx) = 1. Then 3 n G N, x;, y; G F suc/i tfia* £(x;) > ^ < viVi) for * = 1, • • - ,rc a n d z p x = 5^™=1 Xiyi. Since £ a n d 77 C v, v(xi) = 1 = v(yi) fori = 1 , . . . , n . Thus Xi G # n £ * and?/; G # 0 7 ? * fori = l,...,n.

194 Now HOC = K. Hence zpx G rf and so H C rj* contrary to the fact that H is not contained in a tensor factor of F. Thus (£ ® rj) (zpx) ^ 1. Hence £ ® r\ C v. Now let £ be the L-subset of F defined by

f l £(») = < \

ifyeK, ifyeJ\K,

y 0

Then £ G L(yj\x). Since v* = £* ® K

otherwise. K(Z),

( 0 €"(»)=< 5 { 1

V*/£*

is separable. Let y 6 F. Then

i / y G F\ff, ifyZK\Kr>, i/y€.FfP

and f 0

if ye (F\J) U ( i W * " ) ) =

{ 1

ifyeKP\HP

F\K;

^(y) A ^p(y) = { \ ifye(Jr\ (F^m)) u ((A*) n F") = K\K?; = KP.

Thus £ fl i/p = £ p . Hence i//£ is separable. Now £/fi is purely inseparable of exponent 1. However, as shown in the previous paragraph, there does not exist rj € L(u/fi) such that v — £ 77. iVbw 3 77 G L(y/n) such that u* = £* (g>j<- 77*, namely the L-subset n of F defined by V(y)=\

[ 1 \ y 0

ifyeK, ifyeK(z)\K, otherwise.

Hence v* = (£77)* and so *//(£ ® rf) is neutral. It follows from Theorem 5.1.10 that v — £(zi, (zvx)\). Thus z//£(z±) is neutral since (—zpx)i C ^{zi). Hence V/£(ZL) is separable algebraic. Thus z i is a separating transcendence basis of

vli- 2

We now show that v % \^^r\ guished for i/* Ifj,*. Suppose that .

1 = =

for any rj G L(i//fi) such that rf is distin-

(V~r/)(^) V { V o o ( x ! ) A . . . A V ° ° 0 r n ) A 77(3/1) A . . . A ry(yn) | x{ G /x*°°, yierj*,i = l,...,n,zpx = Y^=i *iVi,n € N}.

Then zpx = ^ = 1 z ^ /or some x i? yi such that 77(7^) > \ for i = 1 , . . . ,n. #ence */(?/;) = 1 and 50 yi G H for i = 1 , . . . ,n. Afo element of H can be in a separating transcendence basis of v*/J since H C J(v*p) and so yi G Hi7!*1*) for i = 1, •.. ,n since 77* is distinguished for i/*//x*. Since i/*//x* /ias

195 inseparability exponent 1 andrf Thus yieHn

is distinguished for v* jfi*, ^*(r/*p) = ^*(z/*p).

/x*(ry*p) = K(zpx) n if(z p ) = i^(^ p2 ).

Hence zpx E fi*°°(zp ), a contradiction. Thus (l^oor])(zpx) ^ 1. Hence v % 1^*0077. 77ms z//)Lx does not /lave a distinguished L-intermediate field. Theorem 5.8.3 Suppose that L is regular. Let £ E L(v/ p). (1) If i//£ has a separating transcendence basis and £//z is purely insepa­ rable, then 3 77 E L(I///JL) such that n/ji has a separating transcendence basis, £77 = £ (8) 77 and z//(£ 0 77) is

neutral.

(2) If v/% is separable and £//z is purely inseparable of bounded exponent, then 3 77 E L{yj\i) such that £77 = £ 0 77 and i//(£ 0 77) is neutral. Proof. If either condition (1) or (2) hold, then 3 an intermediate field S of is*I'/J,* such that S//J,* has a separating transcendence basis when (1) holds and such that S/K is separable if (2) holds and such that v* = £* 0^* S [64]. Define the L-subset 77 of F by n(x) = v(x) if x E S and n{x) = 0 otherwise. Then 77 E L{yl\i) and 77* = 5. Now 77//X has a separating transcendence basis by comments following Definition 5.8.1 if condition (1) holds. Now v* = £*0A/* 77* = (£ 0 77)* and /x* = £* D 77* = (£ D 77)*. Thus £77 = £ 0 77 and i//(f 0 77) is neutral. ■ In Theorem 5.8.3(2), 77/^ is not necessarily separable: Let £//x be neutral, but not separable as in Example 5.5.3. Let v = £. Then i//£ is separable and £//x is purely inseparable of bounded exponent. Now 77 = 1/, but 77//Z is not separable. Suppose that L is regular. Let £ be the unique maximal purely inseparable L-intermediate field of vj\i. If i//£ is separable and 3 77 E L{yj \i) which is distinguished for v/fj,, then v/^n is neutral: v C 1^*0077 and so ^* C (1^*0077)* — /x*°°77*. Thus 77* is distinguished for i/*//z*. Since i/*/£* is separable and £*/V is purely inseparable, 1/* = £* 0M* 77* = (£77)* by [156, Theorem 1, p. 606]. Theorem 5.8.4 Suppose that L is regular. Let£ be the unique maximal purely inseparable L-intermediate field of vj'fi. If vj\i splits, then &//£ is separable and 3 77 E L(V//J,) such that 77 is distinguished for vj\x. Conversely, if £* is compatible with z/, JJ, — 1^*, z//£ is separable, and 3 77 E L(V//J,) such that 77 is distinguished for v/fi, then v/fj, splits. Proof. Suppose that vj\i splits. Then v — £ 0 ^ 77, where 77 E L{yjyi) and 77/// is separable. Clearly, 77 is distinguished for v/[i. Now v*//J>* splits and so i/*/£* is separable. Suppose that £ D vp D £ p . Then 3 x E £* such that (£ n vp){xp) > &(xP). Let a = (£ n up)(xp). Then a > £ P ( ^ ) = £(x). Thus x a C 1/, x a ^ £, and xp C £. However, this contradicts the maximality

196 of £. Thus £ fl vv = £p and so i//f is separable. Conversely, suppose that i//£ is separable and 3 77 G L{y/[x) such that 77 is distinguished for i///x. Since fi = 1^*, /i* C 7/a V a G L. By Proposition 5.6.1 and [156, Theorem 1, p. 606], 1/* = T <Sy 77*. Also /z*°°77* = ^*°°
=

V{Voo(xi) A . . . A Voo(x n ) A 77(2/!) A...A77(y n ) | ^ G /i*°° 2/i G 77*, i = 1 , . . . , n, x = Y!i=\ xiVii neN} V{A{ry(i/i)|xi G //*, a.

= =

Thus 3 a ^ G 77(F) such that a* < a 2 +i < a, z = 1,2,..., {a; | 2 = 1,2,...} converges to a, and 2 G H^L^/i*00 ®M. 77aJ = /i*°° ®M* 77a. Hence Z G (/i*°° ®,x* 77a) H (^* ®M. 77*) = C ®/x* 77a. Since also £ is neutrally closed in v by the maximality of £ in z/, we have by the compatibility of £* with v that (£77)(z)

= =

V{f (*i) A . . . A £(x n ) A 77(7/!) A . . . A 77(7/n) I #; €C,Vi €Va,i = 1, • •. ,TI,2 = X)r=i XiVi,n G N} V{77(yi) A . . . A?7(yn) | x2- G£*,2/z €Va,i = l , . . . , n ,

=

a.

Hence */ C 1^*0077 = £77. Thus 1/ = £77. Clearly, £ D 77 = JJL since /x = 1M*. Hence v/H sphts. ■ Corollary 5.8.5 Suppose that L is regular. Let£ be the unique maximal purely inseparable L-intermediate field of v / \x. Suppose that £* is compatible with v and that fj, = 1M*. If v / \± splits, then v = £ ®M 77 for every distinguished L-intermediate field of v/fi. Proof. Since v/\i sphts, i//f is separable by Theorem 5.8.4. Let 77 be any distinguished L-intermediate field of v/fi. (One exists by Theorem 5.8.4.) The proof of Theorem 5.8.4 shows that v = £®M 77. ■ Conditions for every intermediate field E of F/K to have a distinguished subfield which is contained in one of F/K can be found in [38]. Lemma 5.8.6 Suppose that V intermediate fields E of F/K, E/K has a dis­ tinguished subfield which is contained in one of F/K. Suppose that v D IK- If va/K splits V a € v(F), then V/IK splits.

197 Proof. By [156, Lemma, p. 607], V distinguished intermediate fields rf** of va/K, va = [ya D J) ®K V^ since va H J is the unique maximal purely inseparable intermediate field of vajK. Thus some v^ is in a distinguished subfield 7] of F/K. Hence rfa) = i/a n 77. Thus i/a = (i/a n J ) % ( ^ a n 77). Define the L-subsets /i and £ of F by /x(z) = i/(z) if z E J, /i(z) = 0 otherwise and £(z) = i/(z) if 2: E 77, £(z) = 0 otherwise. Clearly, // and £ are L-subfields of F such that fi n £ = lK. Since ^* = U a > 0 ( ^ a H J ) and £* = U a > 0 (^ a H 77), /z*/-^ is purely inseparable and C/K is separable. Now (f n ^ p ) a = £a H (^ a ) p = (£a)p = (P)aV a E L. Hence ( f ] i / p = £ p . Thus fi/lK is purely inseparable and £/IK is separable by Theorems 5.3.18 and 5.3.3. Let x G F. Set ^(a;) = a. Then n

a > (fj£)(x) = V{Ar =1 {Mii) A£(di)} I x = ^ j i ^ , n E N} > a, 2=1

where the latter inequality holds since x has a representation with ji G i^a fl J and di E i/a H 77 for z = 1, . . . , n. Thus i/(x) = (/zf)(x). Hence v = /zf. Thus v = 11 0 1 K £ by Definition 5.3.5. ■ Lemma 5.8.7 Suppose that not every maximal separable intermediate field of F/K is distinguished. Then 3 an L-subfield v of F such that VZ5\K, V* — F, and V/1K does not split. Proof. If F/K does not split, then the result is trivially true. Hence suppose that F/K splits, say F = J % D, where J/K is purely inseparable and D/K is separable. Let S be a maximal separable intermediate field of F/K which is not distinguished. Define the L-subset v of F by v(x) = 1 if x E S and v(x) = ^ otherwise. Then z/ is an L-subfield of F such that v D 1# and j / * = F We show that v ^ //£ for any L-subfields /x, £ of F such that l ^ C / i , { C i / and fi* = J and £* = L). Then V/\K cannot split since D is an arbitrary distinguished subfield of F/K, [156, Lemma, p. 607]. Suppose that v = /z£ for some such ^ and f. Let x e S, x £ D. Then 1

= =

(jiOCr) V{Af=1{//(2/i) A £(z;) I x = X)[Li 2/i**, 2/i € //*, z; E f V = l , . . . , n ; n E N}.

Thus 3 a representation Yl7=i yiZi °^ x

wl

^-

KVi) > g < *(*) for z = 1 , . . . , n. Since ^ 2 A*> K?/*) = 1 ^ d so y* E S for i = 1 , . . . , n. Hence i/i E /i* H 5 = if, z = 1 , . . . , n. Thus x E D, a contradiction. ■

198

Let F/K be a field extension. If 3 a nonnegative integer e such that K(Fpe)/K is separable, then the smallest such e is called the exponent of inseparability of F/K and is denoted by inex(F/K). Theorem 5.8.8 Suppose that F/K is inseparable. Then the following condi­ tions are equivalent. (1) V intermediate fields E of F/K, E/K splits, and E/K has a distin­ guished subfield which is contained in one of F/K', (2) V L-subfields v of F such that vTl^K, V/1K splits; (3) Either F/K is purely inseparable or F/K is algebraic, splits, and the intermediate fields of J/K appear in a chain; (4) F/K is algebraic and E/K splits for every intermediate field E of F/K. Proof. (1)=>(2): Immediate from Lemma 5.8.6. (2) =>(3): Let E be any intermediate field of F/K which is finitely generated and inseparable over K. Now every L-subfield v' of E such that v1 D lx has a trivial extension to a L-subfield v of F, i.e., v = v' on E and v{x) = 0 V x G F\E. Thus every L-subfield v' of E such that v' D 1K splits over IK- By Lemma 5.8.7, every maximal separable intermediate field of E/K is distinguished. Thus if E/K is not algebraic, then mex(E/K) = 1 and every set of d relatively p-independent elements of E/K is a separating transcendence basis of a distinguished subfield of E/K, where d is the transcendence degree of E/K, [39, Corollary 10, p. 194]. However, since E/K splits and mex(E/K) = 1, Kp H E contains elements which are relatively p-independent in E/K, a contradiction. Thus E/K is algebraic. Hence F/K is algebraic. Thus if F/K is not purely inseparable, then the intermediate fields of J/K appear in a chain by [37, Theorem 2.2, p. 50]. (3)=>(4): Since F/J trivially has a finite separating transcendence basis, the result follows by [37, Corollary 2.4, p.51]. (4)=>(1): Immediate. ■

5.9

Purely Inseparable L-Field Extensions

Theorem 5.5.11 began the description of modular L-field extensions. To com­ plete the description, the structure of modular purely inseparable L-field exten­ sions must be determined. Throughout this section, F is a field of characteristic p > 0 and v, \x G L{F) with / i C i / . Definition 5.9.1 Let v/\i be purely inseparable. Let A be an index set and Vot G L{yj\x) V a G A. Define the L-subset Y\aeA va of F by V x e F, ( 1 1 v<*)(x) = V{A{i/ a (x Q ) | a G A} | x = Y^ x<*), ot£A

aeA

199

where by YlaeA x<* we mean a^ but a finite number of the xa = 0. We define QfrcteAVct to be Y[aeA v<* */ and on^V if V 0 € A, Y\aeA vp and vp are linearly disjoint over ji (their intersection), where Ap = A\{(3}. We recall from Section 5.1 that YlaeA va is an L-subfield of F. Theorem 5.9.2 Suppose that L is regular. Let i//fi be purely inseparable. Let va G L{yj[i) V aeA. (1) Ifu = ®aeAva, then v* = ®aeAi/a*. (2) Suppose that (i is neutrally closed in v. Ifv — r i a e ^ VOL an<^ u* = ^ • . a g i l ^ a ) * , then

V

=^)flfa£A^oc

Proof. (1) As in Theorem 5.3.3, i/* = F L e ^ K ) * - Now V /? G A, YlaeA, "<* and up are hnearly disjoint over \i imphes Yla€A(3(va)* and (up)* are linearly disjoint over /x*. 3Ii< (2) Now YloceA^^y ^ (upY a r e linearly disjoint over /x* V/3 G A. Thus VQL anc v a r e nneai riaG^/3 * & "ly disjoint over their intersection V j3 G A. Now V /? G A, (Y\aeA Va^vp)* = M* a n d since /x is neutrally closed in z/, Ilae.4 ^ a C\vp = /J,. ■ A sufficient, but not necessary condition for a purely inseparable field ex­ tension to be modular is that it be a tensor product of field extensions whose intermediate fields are chained, [40, Proposition 1.3, p. 540]. For this reason and also because of Example 5.9.13 below, the description of modular purely inseparable L-field extensions vj\x need some restrictions on vjix. Theorem 5.9.3 Suppose that L is regular. Suppose that vj\i is purely in­ separable and fi is neutrally closed in v. Let / i a G L(u/fi) V a G A. If v* = ^ . a e ^ l ^ a ) * } where the intermediate fields o/(/x a )*/xi* are chained, then vj\x is modular and there exists £ G L(V//JL) such that v/£ is neutral, £ = ^ . a g ^ ^ j where ua G L(i//fj), and ua = v on (i/*)* Va G A. Proof. For all a G A, define the L-subset va of F by ua(z) = v{z) if z G (/xa)* and ua(z) = 0 otherwise. Then */* G L(i///x). Let £ = UaeA^Since v* = < g > ^ ^ e A ^ ) * and A4 ls neutrally closed in v, £ = (g^aG-A*^ by Theorem 5.9.2(2). Now £* = UaeA^Y = u*- T h u s vfe i s neutral. I/*/JX* is modular by [40, Proposition 1.3, p. 540] and so vj\x is modular. ■ We now give a relationship between relative p-bases of vj\i and generating sets of v/fi when vj\i is purely inseparable of bounded exponent e, i.e., vv C /x. First, we note that the existence of a relative p-basis of v/fj, is equivalent to the existence of a minimal generating set of sorts. vj\i may not have a relative p-basis, but may have a minimal generating set of another sort. We now make these ideas precise.

200

Definition 5.9.4 Let x be an L-subset of F such that x^zvThen s (1) x i called a minimal generating set of V/JJ, i / V x G x * , (Mx ~~ x))(x) = 0 and v = fi(x)] (2) x is called a weak minimal generating set of vj \x if V x G x*> Kx) 3 Kx ~ x) and v = /z(x); (3) x is called relatively p-independent in v/fi i/V a; G x*, (fi(vp)(x — v x))(x) = 0; and v = [i{y ){x)\ (4) X is called a relative p-basis ofv/fi if x is relatively p-independent in vj'n and v = fi(vp)(x)Example 5.9.5 Let L = [0,1]. Let F = P(x), where P is a perfect field of characteristic p > 0 and x is transcendental over P. Define the fuzzy subset v of F by v(z) = 1 if z G Fp and v(z) = \ if z G F\FP. Then v is a fuzzy subfield of F. Now v* = v*p(x) and so x is a (relative) p-basis of v* (over v*p) and a minimal generating set of v*/v*p. Now vp{z) = 1 if z G Fp , vp(z) = \ ifz G Fp\Fp\ and vp(z) = 0 if z G F\FP. V z G i / V * p , v{z) = \ and so p p (xi) ^ v {z\). Thus v ^ vp(z\). Hence v/vp has neither a relative p-basis nor a minimal generating set. Now xi £ vp(xp). However, v = vp(x\, (xp)i). p Thus vjv has a weak minimal generating set x, namely, the fuzzy subset x of F defined by x(xP) = 1> x(x) = ^> and x{z) — 0 otherwise. With regard to Theorem 5.9.2, we note that v has the sup property and v/vp is compatible, but vjvp is not strongly compatible. Proposition 5.9.6 Suppose that v/fi has bounded exponent e. Let x be an L-subset of F such that x Q v. Then x i>s a ujeak minimal generating set of v/fi if and only if x is a weak minimal generating set of v/fi(vp). Proof. v = fi(x — x) if a n d only if v = fi(vp)(x — x) since vpG C fi. Hence p if v = /i(z/ )(x), then /x(x) 3 M x ~ x) implies fi(vp)(x) D n(vp)(x ~ x)', and if v = Mx), then fJ,(vp)(X) 3 K"P)(x ~ x) implies fj,(x) D Mx ~ x). ■ Proposition 5.9.7 Suppose that vj\i has bounded exponent e. Let x be an L-subset of F such that x^vThen x ^ a relative p-basis of vj/j, if and only if X is a minimal generating set of vj \x. Proof. Suppose that x is a relative p-basis of vj'ji. Then v = fi(vp)(x) a n d so pe v = v>{y )(x) = fi(x)- Clearly, (v>{vp)(x-x))(x) = 0 implies (fj,(x-x))(x) = 0 V x G x*- Conversely, suppose that x is a minimal generating set of v/fi. Then clearly v = fi(vp)(x). Suppose that 3 x G x* such that x G /z*(x*\{#})- Now /x*(x*\{#}) = /x(x ~" XY- Thus (^x(x — x))(x) > 0, a contradiction. Hence x £ ^*(x*\{ x }) a n d so x* is a minimal generating set for v*/fi*. Thus x* is a

201

relative p-basis of I/*/M*> [217]. Hence x is maximally relatively p-independent in vj[i. Since v = /z(j/ p )(x), X is a relative p-basis of vj\i. ■ We close this section by giving an existence result of weak minimal gener­ ating sets for purely inseparable L-field extensions. Let S be a set of L-singletons such that if xa, Xb G <S, then a = b > 0. Define the L-subset x(5) of F as follows: x(S)(x) = a if xa G S and x(«S)(z) = 0 otherwise. Define /*(£) = A*(x(«S)). Then /x(5\{x a }) - MxO?) ~ *a) V x G F. Hence we call 5 a minimal generating set of v/ fi if ^ = /z(«S) and fi(S\{xa})(x) = 0 V xa G 5 . We call 5 a weak minimal generating set of vj\i if i/ = /i(5) and ^(5\{x a }) C /x(5). Thus 5 is a (weak) minimal generating set of i///x if and only if xs is a (weak) minimal generating set of u/fi. We recall that fi is said to neutrally closed in v if i/ = /x on /z*. Lemma 5.9.8 Suppose that L is a chain. Suppose that vj\i is purely insepa­ rable, ii is neutrally closed in v, and A{fi(x) \ x G /z*} > V{v(x) \ x G i/"\/i*}. Suppose that is* = //*(r), where r is purely inseparable over fi* of exponent e > 0. T/ien zy = fi({(rpTlu)u \ u el}, where I = {u G v{F) \ vu D M*}\{0} and nu GN is such that vu = /x*(rp u) V u G I. Proof. If x G //*, then i/(x) = fi(x). Let x G i/*\/i*. Then i/(x) = v for some v e I. Now i/v = /i*(r p n v ), 1 < n v < e. Thus x G fi*(rpnv) and x ^ /i*(r pTlu ) = vu V u e I such that w > v. Thus z/(x) > /i({(r p n t l ) u | n G /})(x) > (fJ<((rp v)v))(x) = v = v(x), where the first inequahty holds by Theorem 5.1.10 and the hypothesis concerning A and V. ■ Theorem 5.9.9 Suppose that L is a chain. Suppose that v/n is purely insepa­ rable, fx is neutrally closed in v, and A{/i(x) | x G //*} > V{v{x) | x G i/*\/z*}. Suppose that v* = /i*(r), where r is purely inseparable over /i* of exponent e > 0 . Then{(rp u)u \ u G / } 25 a weak minimal generating set of v/ji. Proof. By Lemma 5.9.8, {(rpnu)u | u G / } is a generating set for vj[i. Let v G / . Now / is finite since [v* : /z*] < oo. Thus either v has an immediate predecessor w (v > w) in / or v is the smallest element in / and so vv — v*. Suppose that w exists. Then rpTlv G vv = fi*(rpTlv) and rpTlv £ vu = fi*(rpTlu) \/ u>v. Thus (M{(r p n u )u I u ^ A W ) ) ^ ) = w < v = K ^ O - Suppose that v is the smallest element in I. Then rpn £ vuM u G / such that u > v. Thus (K{(rpnu)u | u G A M H X O = 0 < v = i/(r). Hence { ( r p n u ) n | it G / } is a minimal generating set of i///x. ■ Lemma 5.9.10 Suppose that L is regular. Suppose that vj\x is compatible and (j, is neutrally closed in v. Let C G L{u/fi) be such that C//i is algebraic. Let 7

202

be an L-subset of F such that 7 C 1 / and 7 is a weak minimal generating set 0/^(7)///. If £ and ^(7) are linearly disjoint over /J,, then 7 is a weak minimal generating set o/C(7)/C Proof. Let y e 7*. Since vj^i is compatible and \i is neutrally closed in v, M7 ~~ y)/^ ls compatible and /i is neutrally closed in £^(7 — y). By making the following identifications in Theorem 5.3.8, ^(7 - y) for f, C for 77, and 1/ for ^77, we have that fi(j - y) = C/x(7 - y) = C(7 - 2/) o n ^ 7 - 2/)*, where the latter equahty holds since £//x is algebraic and so CM7 ~~ 2/) 1S a n ^-subfield of F. Now /i(7 - 2/) C ^(7) and £(7 - 2/) n ^(7) = ^(7 - 2/) by Theorem 5.3.8. Hence C(7 — y) ¥" C(7) e l s e C(7 — 2/) H /z(7) ^ /x(7 — 2/)- Thus 7 is a weak minimal generating set of C(7)/C " Theorem 5.9.11 Suppose that L is a chain. Suppose that u//i is purely insep­ arable, i//fi is compatible, and \i is neutrally closed in v. Let va € L(I///J,), a £ A, A a nonempty index set. Suppose that (^ a )* = p*(ra) and v = (8>M,aGi^a, where ra is purely inseparable of exponent ea > 0 over /J,* V a G A Let S* = {(r£"a )ua I Ua € J a } , where Ia = {^a € i/ a (F) I K ) u Q 3 /^*\{0}}, OLE A. Let S = UoceA^ct- Then the following assertions hold. (1) S is a weak minimal generating set of /j,(S)/fj,. (2) S is a weak minimal generating set of v/{i if and only if v — [i{S). Proof. (1) Let x<* = x(Sa), a € A. By Theorem 5.9.9, Sa and so x<* is a weak minimal generating set of va//i V a e A. Let (3 e A. Let 7^ = x(S\Sp). By hypothesis n{ip) and ti{xp) are linearly disjoint over fi. By Lemma 5.9.10, Xp is a weak minimal generating set of IJ>(lp)(xp)/V>(lp)- Since /? is arbitrary, it follows that 7/3 U xp = x($) is a weak minimal generating set of /z(x(«S))///. Thus 5 is a weak minimal generating set of /J,(S)/H. (2) The desired result here is immediate from (1). ■ Theorem 5.9.12 Suppose that L is regular. Suppose that F = ^>iyLlKK{xi). Suppose that there exists an intermediate field H of F/K such that H/K is simple and H is not contained in any simple tensor factor of F/K. Then there exists £ e L(F) such that £ D 1K, £* = F, and £ is not a tensor product of L-subfields of F whose supports are simple over K.

203

Proof. Define the L-subset £ of F by £(x) = 1 if x G H and £(x) = ^ otherwise. Then £ is an L-subfield of F, IK Q K, and £* = F Suppose there exists & G L(£/1K), i = 1 , . . . ,m, such that £* = ^ x f e ) * and (&)* = #(3fc) for some ^ G F, z = 1 , . . . , m. We show that £ ^ Il£Li & s o that £ 7^ ^i2=i^Let 2 G F be such that if = K(z). Then ( f t . . . &*)(*) = V{A{€i(zijk) | i = 1 , . . . ,m; A; = 1 , . . . ,n} | z = ] £ £ = 1 *ijfc • • -^njfc, n = rij E N, j G J } . Now £(z) = 1. Suppose that (ft .. . £ m )(z) = 1. Since the ft are finite-valued, there exists j G J such that 2 = ]Cfc=i zijfc • • • zmjk and ^i(zijk) = 1, i = 1 , . . . , m; fc = 1 , . . . , ra. Since & C ft 2^fc G i7 fl ft*. Suppose z has exponent r over if. Then zijk has exponent r over K for some Zijk. Thus K(zijk^j — H and so H C ft*, a contradiction. ■ Example 5.9.13 Le£ Z, = [0,1]. Le£ F = AXx*"3, yp~l), where K = P(x, y), P is a perfect field of characteristic p > 0, and x, y are algebraically in—i —i dependent indeterminates over P. Let H = K(xp + yp ). Suppose there exist intermediate fields K\, K2 of F/K such that F = K\ ®K K2 and K\ /K and K2/K are simple, and H C K\. Then K\jK has exponent 3 and K2/K has exponent 1. Now H/K has exponent 2 and K\/K is simple. Hence H = K(KP) C K(Fp) which is impossible since xp~2 + yp~X £ K(FP). Thus by Theorem 5.9.12 the L-subfield £ of F defined by £(z) = 1 if z G H and £(2) = \ otherwise, is such that £ / 1 K is n°t a tensor product of L-subfields of F whose supports are simple field extensions. Hence V ix, v G F such that F = K{u) ®K K(v),£ D £u ®i K £ u , where £ n , £ v G L(£/1K) are such that (£ u )* = K(u) and (£")* = K{y). It is easily seen that {f C \K. We note that £/l/c is modular since £*/K is modular.

5.10

EXERCISES

Let F be a field of prime characteristic p. Let /x, v be L-subfields of F such that v 3 /i, where L = [0,1]. 1. Suppose that vj\x is separable. Prove that the following statements are equivalent. (a) Every fuzzy maximal relatively independent subset x of vl\i that v/fi(x) is separable algebraic.

ls

suc

h

(b) Every relative p-basis M of i/*//z* is such that i/*//x*(M) is separable algebraic. 2. Suppose that vj\i is separable. Prove that the following statements are equivalent. (a) u/fi has a finite separating transcendence basis.

204

(b) Every maximal relatively p-independent subset x °f vj\x is such that X is a separating transcendence basis of vj[i. (c) Every maximal relatively p-independent subset \ °f vl'H is s u c n that X is a transcendence basis of v•/' ji. 3. Let F = K(x) be a pure transcendental field extension of K. Let /i be a fuzzy intermediate field of F/K. If F = K(y) for some y G F, then prove that /i(x) = /J,(y). 4. Let F = K(x) be a pure transcendental field extension of K. Let /i be a fuzzy intermediate field of F/K. Let a G (J<(F) be such that \xa D K. Then prove that there exists z G F such that \ia — K(z) and fi(z) = a. 5. (Fuzzy Luroth's Theorem [15]) Let F = K(x) be a pure transcen­ dental field extension of K. Let JJ, be a fuzzy intermediate field of F/K. Prove that the following assertions hold: (a) either fi(F) = { a i , . . . , an} for some n G N or ^ ( F ) = { a i , . . . , a n , . . . } ; (b) for each a; G fJ>(F), /iai = ■K'(^) for some Zi e F and /i(^) = a;; (c) { ( ^ ) a i | 2 = 1,2,...} is a weak minimal generating set of /i over K if M(-F) = { a i , . . . , a n , . . . } and { ( ^ ) a i | i = 1 , 2 , . . . , n } is a weak minimal generating set of /x over if if /x(F) = { a i , . . . , an} for some n G N. 6. (15) Let F/.K" be a field extension. Prove that F/K is simple if and only if every intermediate field of F/K is fuzzy simple, i.e., if it has a minimal generating set S over K such that if xa,yb e S with a > 6, then

205

Chapter 6

STRUCTURE OF L-SUBRINGS AND L-IDEALS In this chapter, we continue our presentation of fuzzy commutative ring theory. Our primary goal is to present the requisite material for the understanding of fuzzy algebraic geometry. The results here are primarily those of [1, 2, 47, 48, 84, 90, 104, 105, 133, 137, 141, 143, 144, 145].

6.1

Comparison of Radicals

Throughout this chapter R is a commutative ring and n is an L-ideal of R. Definition 6.1.1 (1) \i is called C-prime z/V £, ?/ G i?, either n(xy) = fi{x) orfi(xy) = n(y). (2) Let fj, G LI(R). Then fi is called C-primary ifVx,yER either jji{xy) = n(x) or fi(xy) < ^(y171) for some m G N. Proposition 6.1.2 fi is C-prime (C-primary) if and only if fia is prime (pri­ mary) V ae /J,(R) U {b G L | b < fi(0)}. Proof. Let fia be a prime ideal of R V a G /i(R) U {b G L \ b < /x(0)}. Then \x G LI(R). Let x,y G R and let n(xy) = a. Then xy G [ia> Hence x G /i a or y G /ia- If x G // a , then a = fi(xy) > /i{x) > a. Hence fi(xy) = fi(x). Similarly, if y G fia, then n(xy) = fi(y). Conversely, let a G fi(R) U {b G L \ b < ^(0)}. Then \ia is an ideal of R. Let x,y G R and let xy G /i a - Then n(xy) > a. Now

206

fi(xy) = [i(x) or fi(xy) = n(y). Suppose fi(xy) = fi(x). Then fi(x) > a. Hence x G /J,a-

The proof for the case of £-primary L-ideals is similar. ■ P r o p o s i t i o n 6.1.3 Let \x G LI{R) be C-prime. Then fi(R) is a totally ordered set with the least element /x(l) and the greatest element /x(0). Proof. Let x,y G R. Then ^(xy) — fi(x) or n(xy) = /i(y). Suppose fi(xy) = fj>(x). Then fi(x) = n(xy) > /Z(T/). The second part of the assertion follows easily. ■ Let QM = {77)77 is a £-prime L-ideals of R such that 77 D ji). Definition 6.1.4 Define the L-subset IZdl(fi) of R by Tldl(fi) = DrjeQ^rj. Clearly, IZdl(fi) is an L-ideal of R. L e m m a 6.1.5 (1) Udl{p){0) = ^(0). (2) K(fi) C ndl(fi). Proof. (1) By definition, /x(0) < Tldl(fi)(0). Define the L-subset 77 of R by r)(x) = /z(0) V xeR. Then 77 G Q^. Hence /i(0) = ftdi(/x)(0). (2) Let 77 G Q^. We first show that 7](x) = 77(2;*) V i G N. Now 77(2;*) > 77(0:). Let 77(a:t) = a. Then xl £ r]a. Since 77a is a prime ideal of R, x e na. Hence rj(x) > a. Thus 77(2;) = ^(x1). Now ^(x x ) < rj(xl) = 77(2:) V i G N. Hence ft(/x) ClZdl(fi).m For all a < /z(0), let Va = {I I / is a prime ideal of R such that / 2 V>a\Then y ^ = fi/ep o J. Recall that Pp — {£ I £ *s a prime L-ideal of i?, //* C £* and [i C £}. For all a < ^(0), let Va = {Va I r7 G ? „ } and let

<2a = K U e <2M}. L e m m a 6.1.6 Suppose that L is a chain, /x(0) = 1, and fi has the sup property. Then V aeL,Va = Va-

207

Proof. Let naeVa. Then 77 G V^ and so 77 D /i. Thus rja D fia. Now either na= Roi na = 77*. Hence 77a G P a . Thus P a C p a . Let / G Va. Define the L-subset 77 of R by 77(0:) = 1 if x G / and n{x) = b if x £ / , where 6 = V{/x(a;) I a £ I } . Since ^ has the sup property, b < a < 1. Hence 77 is an L-ideal of i? and 77a = / . Since L is a chain, b is a prime element of L. Thus 77 G 7 ^ . Hence I = r)ae Va. Thus 7>a C P a . ■ Lemma 6.1.7 Suppose that L is a chain and fi has the sup property. Then V

a<M0), Q a = 7 V Proof. Let rja e Qa. Then 77 G QM and so 77 D /x. Thus na D /j,a. Now 77a is a prime ideal of # . Hence na e Va. Thus Q a C P a . Let I eVa. Define the L-subset 77 of R by r)(x) = /x(0) if x G / and 77(2;) = b if x £ I, where 6 = V{^(x) I x £ I}. Since ^ has the sup property, b < a < fi(0). Thus rja = I and 77 is an £-prime L-ideal of R. Thus 77 G QM. Hence I = rja e Qa. Thus 7>a C Q a . ■ Lemma 6.1.8 Suppose that L is a chain and \i has the sup property. Then V a < /i(0), K(fi)a = y/Ji^ = 1Zdl(fi)a. Proof. Suppose that x G 7£(^) a . Then 7Z(fi)(x) = b > a for some b and so x G y ^ C yZ/To". Suppose that # G y ^ - Then x2 G /i a for some i G N. Thus ^(# z ) = c > a for some c G fJ>(R). Hence x% G ^ c and so x G ^/I^. Thus 7l(fj,)(x) > c> a. Hence x G 7£(/i) a . Thus 7£(/i)a = v'Ma- That lZdl(/i)a = y/Jt^ follows from Lemma 6.1.7. ■ Remark 6.1.9 Let L = [0,1]. Let (i be a fuzzy subset ofL defined by fi(x) — 0 if x 6 Z \ (2), \x{x) = 1 - 7 if x G ( 2 i _ 1 ) \ (2*) for i = 2 , 3 , . . . . 77ien /i is a ./kzzT/ zciea/ ofL and 7Z(fi)(2) = 1. Hence 2 G 7^(^)i 6w£ 2 ^ ^/JST = {0}. Hence 7£(M)I ^ \/^T- * « ; 1 = V{/x(2*) I i G N} and 1 £ {^(2*) | i G N}. Thus /x does no^ /lave £/ie S7jp property. Hence Lemma 6.1.8 need no^ 6e £7~ue if \x does not have the sup property. Theorem 6.1.10 Suppose that L is a chain and \i has the sup property. Then n(fi) = VxU(fjL). ///i(0) = 1, then IZ(fi) = ndl(fi) = y/ji. Proof.

n{i\le

By Lemmas 6.1.6 and 6.1.8, Va < /i(0), 7£(/x)a = lZdl(/u,)a = yJT^ =

Va} = n{77a 177a e Va) (when ^(0) = 1) = n{77a \vePfl}

I 77 G V^])a = (y/P)a. Also result follows. ■

K(IJL)(0)

= (n{n

= /x(0) = 7taZ(/z)(0). Thus the desired

Definition 6.1.11 Let £ 6e an L-ideal of R. Then £ is TZdl-primary only if\f ^ , ( € LI(R), i / o ( C ^ zrap/zes v C £ or C C ftd/(£).

if and

208

Theorem 6.1.12 Let L be a dense chain and let £ G LI{R). ing conditions are equivalent. (1) £ is primary; (2) £ is IZ-primary; (3) £ is IZdl-primary.

Then the follow­

Proof. We have that ft(£) C Kdl(£) C V£ by Lemma 6.1.5. Suppose that £ is primary. Let v, £ be L-ideals of i? such that v o £ C £ and i / ^ ( . Then C C V?- By Theorem 3.9.10, C C ft(£). But then C C ftd/(£). ■ Corollary 6.1.13 Let L be a dense chain. Suppose that £ is an L-ideal of R such that £ satisfies any one (and hence all) of the properties of Theorem 6.1.12. Then K(£) = 7fcfl(0 = V? • Proof.

6.2

By Theorem 3.9.10, ft(£) = V£- ■

7^-Primary L- Represent at ions

In this section, we assume that R has an identity. Lemma 6.2.1 Let x G R. Then l^nj = l^n-i^o 1 ^ for all n — 1,2, Proof. Now (l(a;»-i) ol ( x ) )(y) = V{l(a.n-i)(u) M(x)(y) \ u,v G R,y = uv}. Thus (l^n-i^o 1(X))(R) = {0,1} and so it suffices to show that l(x^(y) = 1 if and only if (l^n-i)O l^)(y) = 1. However, this follows easily since y G (xn) n_1 if and only if 3 u G ( x ) and v G (x) such that y = uv, e.g., u = rxn~x and n v = x, where y = rx ', r G i ? . l Definition 6.2.2 7/ £ zs an IZ-primary L-ideal of R, then 7£(£) is called the associated prime L-ideal of £, and £ is called an IZ-primary L-ideal belonging to the prime L-ideal 7£(£); or IZ-primary for 7£(£). Theorem 6.2.3 Let £ and 77 6e nonconstant L-ideals of R. Then £ is Hprimary and rj = 7£(£) 2/ ana7 on/y i/

W ( C , c 7e(0; (%) If [i and v G LI(R) such that jiov C^ and ji % £, i/ien ^ C 77.

209

Proof. Suppose that £ is 71-primary and 77 = 7£(£). Then condition (2) holds. Since £ C ft(£), condition (1) holds. Conversely, suppose that conditions (1) and (2) hold. If p and v are Lideals of i j such that / / o ^ C ^ and /z g £, then z/ C 77 by condition (2) and so ^ C 7£(£) by condition (1). Thus £ is an 7l-primary L-ideal of ii. Hence 7£(£) is a prime L-ideal of R by Theorem 3.9.7. Since £ and 7£(£) are two-valued and ^(00*0 = f 0*0 f o r a11 x £ O^OO)** i n o r d e r t o s n o w t h a t V = ft(0 it suffices to show that 77* = ft(£)*. Clearly, £* C 77* C ft(£)* = V ^ by Corollary 3.8.5. Let x G >/£* a n d let n be the smallest positive integer such that xn G £*. If n = l, then x G 77*. Assume that n > 1. Now (re) C y ^ , {xn~l) g £*, and (z n ) C £*. Thus l( x n-i)0 1<X> = \{xn) c £, l D u t ^ 2 £ a n d thus C ^ V- Hence condition (2) of Theorem 6.2.3 holds. ■ Proposition 6.2.5 Let /z G LI(R). If /z zs an intersection of IZ-primary Lideals, then /z* is an intersection of primary ideals and fi(0) = 1. Proof. Suppose that /z = Pl£a, where each £ a is 7£-primary. Then /z(0) = 1, since £«(0) = 1 for all a and /z* = D(£ a )* as in Lemma 3.1.12. Each (£ a )* is a primary ideal by Theorem 3.9.4. ■ Proposition 6.2.6 Suppose I is an ideal of R such that I is an intersection of primary ideals. Then for all a G L, a is a prime element, a ^ 1, lja is an intersection of IZ-primary ideals, where lia(x) = 1 if x G I and lia(x) = a otherwise.

210

Proof. Suppose that I = C\iJi, where Ji is a primary ideal of R. Clearly, l/ tt = nilJia. Now (lj i o )* = Ji is primary, lj i o (0) = 1 and ljia(R) = { l , a } . Thus 1 j i o is 7^-primary by Theorem 3.9.4. ■ Theorem 6.2.7 Let fi G LI(R). Let u be such that u(0) = 1 and u(R) = {l,c}. Suppose c = ai A a2 A . . . A a n , ty/iene aj G L, 1 < j < n. Le£ ^* = Ai fl ^2 H . . . PI Am, where Ai is an ideal of i? V 1 (R)\ < \£l\ + 1. Proof. Since & is two-valued and ^(0) = 1 for all i G fi, we have that A{£;(x)|2 G Cl}\x G R} can have no more that |fi| -f 1 elements. ■ Definition 6.2.11 Let /J, G LI(R). A representation of /i as a finite intersec­ tion, A4 = £i f l . . . fl £ n , of IZ-primary L-ideals of R is called an IZ-primary L-representation (or L-decomposition) of \i. It is called irredundant or reduced if no & contains n " = 1 ^ and the & have distinct 7£(£;), i — 1 , 2 , . . . , n. Theorem 6.2.12 Suppose that L is a chain. Then every L-ideal fi of R such that /x(0) = 1 and \i is finite-valued has an IZ-primary L-representation if and only if every ideal of R has a primary representation.

211

Proof. Suppose that every ideal of R has a primary representation. Let fi(R) = { a i , . . . , a r , a r +i = 1} with ai < 02 < . . . < a r < 1. Then /zar+i = //* C fjLar C . . . C fiai = R. Let Xi = l(,x a . +1 ) 0 . for i = 1,2,..., r. Then /x = Ai f l . . . HAr. Since ^ a i has a primary representation, A^ has an 7£-primary L-representation, by Proposition 6.2.6, say, A; = H^L.^ij, i = l , . . . , r . Thus \i = n [ = 1 n ^ i j £ij is an ^-primary L-representation. Conversely, let I be an ideal of R. Then 1/ is an L-ideal of R such that (1/)* = I. As in the proof of Proposition 6.2.5, I has a primary representation. ■ Theorem 6.2.13 Suppose that L is a chain. Suppose that R is Noetherian. Then every L-ideal \i of R such that /x(0) = 1 and \x is finite-valued has an IZ-primary L-representation. ■ Theorem 6.2.14 Let L be a dense chain. Then every L-ideal ji of R such that /J,(0) = 1 has an IZ-primary L-representation if and only if R is Artinian. Proof. If R is Artinian, then every L-ideal of R is finite-valued by Corollary 3.10.4. The desired conclusion holds from Corollary 6.2.13 since every Artinian ring with identity is Noetherian. Conversely, suppose that every L-ideal fi of R such that ^(0) = 1 has an 7^-primary L-representation. Then every L-ideal // of R such that ytx(0) = 1 is finite-valued by Proposition 6.2.10. Thus every L-ideal v of R is finite-valued since the L-ideal fj, of R is finite-valued, where fj,(x) = 1 if x G v+ and n(x) = v(x) if x £ v*. Hence R is Artinian by Corollary 3.10.4. ■ Theorem 6.2.15 Let /J, G LI(R). If \i has an IZ-primary L-representation, fi = £1 f|... n£ n , then fi has a reduced IZ-primary L-representation. Proof. Suppose that &!,...,&,. G { f i , . . . , £ n } are such that T^&J = • • • = ft(6J = V- Let £ = nrjssl iir Then £ is ^-primary and 1Z(Q = rj by Theorem 6.2.4(1). Hence \i = fi D . . . fl£^, where the ^ have distinct L-radicals. If ^ D f l j l ^ for some i, then ^ is deleted. ■ We now show that the 7£-primary L-ideals appearing in a reduced 1Zprimary L-representation of an L-ideal are not necessarily unique. Example 6.2.16 Let R denote the polynomial ring F[x,y] over a field F and let L = [0,1]. Define the L-subsets //, £ifc, £2 of R by

u(f) = l ]

1 I

1

I 5

iffei^'Xy)

otherwise

otherwise

212

k = 2 , 3 , . . . and nf)==

[ 1 \

1

iffe(x) otherwise.

Then \x = £ifcPl£2 is a reduced IZ-primary L-representation of \i for k = 2,3, Theorem 6.2.17 Let fi G LI(R). Suppose that /J, has an IZ-primary L-representation, fi = £i f l . . . D £n- Then //* = (£i)* f l . . . D (£ n )* is a primary repres­ entation of fj,* and if it is reduced, then the representation \i = £i f l . . . fl£n is reduced. Proof. By Proposition 6.2.5, /i* = (fi)*n.. .fl(f n )* is a primary representa­ tion. Suppose that ^ D rij&€j for some i. Then (&)* D (n?^£j)* = rij^(£j)*, a contradiction. Suppose that 7£(&) = T^(^) for some 2 ^ j . Then 7£(&)* = 7£(£j)*. But 7£(&)* = >/|i7 f° r e a c n *• Thus ^/l^ = \ftJl, a contradiction. ■ The following example shows that the converse of Theorem 6.2.17 does not hold. Example 6.2.18 Let I and J be prime ideals of R such that I C J C R and let L = [0,1]. Let fi be the L-ideal of R defined by ( 1 fi(x) = < \

{ 0

ifxel ifxeJ\I

ifxeR\J.

Define the prime L-ideals r\ and n' of R by f A

,, v

/ l

ifxel

[ l if x e J

"(*) = ( 0

ifxiJ

Then [i = 77 C\rf and n* = 77* Dn^. The representation [x = rj fl 7/ is reduced. However, 77* C rj^. Hence the representation [i* = 77* fl 77^ is no£ reduced. Definition 6.2.19 ^4 prime L-ideal 77 o/i? zs ca/Zed a prime L-ideal divisor offiifr/Dfi and 77* D ^*. A prime L-ideal divisor 77 of \i is called minimal or isolated if there does not exist a prime L-ideal divisor v[ of \x such that rjf C 77 and r[ ^ 77. Theorem 6.2.20 Let fi e LL(R). Suppose that \x has a reduced Tl-primary Lrepresentation, \x = £1 n . . . fl fn. Then ji has a finite set {771,..., 77™}, m
213

Proof. Let 77 be any prime L-ideal divisor of/z. Then 77 D 1Z(fi) = 7£(£i)n... n7£(£ n ). Let 77^ = 7£(£i), i = 1 , . . . , n. Then 77 D 771 o . . . o 77n and so 77 D 77; for some 2. Thus 77 contains some 77; which is minimal among 771,..., 77^. Hence if we select those 77^ in {771,..., r/ n } which are minimal and reindea;, then we have n(fi) = 77in...n77 m . ■ Corollary 6.2.21 Let fj, e LI(R). Suppose that fi has an H-primary L-represe­ ntation, fi = £1 fl . . . fl £ n and fi* has the reduced primary L-representation, \i* — (£1)* H . . . fl (£ n )*. If (T7J)* = 7£(£j)* is a minimal prime L-divisor o//z*, then r/j = 7£(£j) is a minimal prime L-divisor of JJL. Proof. If 77 is any prime L-divisor of fi, then as in the proof of Theorem 6.2.20, 77 D 77^ for some i. Thus if 77 C T?J and 77 7^ 77j, then 77^ D 77^ and 77j ^ 77i. Hence by the minimahty of (T?J)*, (77^)* = (77;)*. But this contradicts the assumption that the above representation of /z* is reduced. Hence 77^ is minimal. ■ The following example shows that the converse of Corollary 6.2.21 does not hold. Example 6.2.22 Let R denote the polynomial ring F[xy y] over a field F and let L = [0,1]. Define the L-subsets /i, £1, and £2 of R by iffe(x2,xy) if f e(x2,xy,y2)\(x2,xy) otherwise

( 1 /*(/)=< h y ^

1 5

&(/)

otherwise

1

{\

iffe(x) otherwise.

Then fi is an L-ideal of R and £1 and £2 o,re IZ-primary L-ideals of R. It follows that fi has the IZ-primary L-representation, /x = £1 fl £2. By Theorem 6.2.17, this representation is reduced since fi* = (£1)* H (£2)* ^s a reduced primary Lrepresentation of /i*. Afow ^/£i7 = y/(x2,xy,y2) = (x,y) and v/f^T = yf$) — (x). Thus (x) is the only minimal prime L-divisor o//x* = (xr,xy) . Let 77; = ft(£i), i = 1,2. ITien 77^/) = l t / / e ( i J y ) , Tfr(/) = ± otherwise; m{f) = 1 if f € (x), 772(/) = 3 otherwise. Since rji % rj2 and 772 2 ^l? ^1 a n ^ ^2 are 60^/1 minimal prime L-divisors of \x. That is, 771 is a minimal prime L-divisor of ft, but (771)* is not a minimal prime L-divisor of /i*. Theorem 6.2.23 Let /i £ LI(R). Let L be a chain. Suppose that /1 has a reduced IZ-primary L-representation, /i = £ifl.. .fl£ n . Then fi(R) = U™=1£i(R).

214

Proof. Suppose that there exists k such that for all x G i?\(£fc)*, /x(x) =fi €k(x)- Then for all x G i?\(£&)*> there exists j x such that £jx(x) < &(#) since fi(x) = A™=1£i(a;). Thus for all x G R\(€k)*, Kx) = M & ( x ) | i = 1,2, . . . , n ; 2 ^ fc}. Since ffc(a;) = 1 for all x G (&)*> AjLx&fc) = A{^(x) | 2 = 1,2, . . . , n ; 2 7^ A:}. Thus /x = D^fc^i, a contradiction that the given representation of /x is reduced. Hence for i = 1,2,..., n, there exists X{ G R\(£i)* such that /x(a;i) = £(xi) < 1. The desired result now follows from Proposition 6.2.10. ■ We close this section with results concerning uniqueness properties of an 7^-primary L-representation of an L-ideal of R. Theorem 6.2.24 Let xx G LI{R). Suppose that /x has a reduced IZ-primary representation, xx = fi fl . . . fl fn, n > 1. The prime L-ideals r]i = ?£(&), 2 = 1, 2 , . . . ,n, are uniquely determined by the criterion that a prime L-ideal 7] of R is one of the rji if and only if there exists an L-ideal £ of R such that £ 2 M an^ f1 '• C is IZ-primary for 77. Proof. We first show that each rji satisfies the given criterion. Since /x C rij^i^j, there exists an L-ideal £ of R such that £ g /x and £ C Dj^i^j. (C = H ^ i O or l ( c ) , c G n ^ * W ) Now C6 C (n j ¥ i&)& C n? = 1 & = /x. Thus & C /z : £. Since C 2 & (else C C /x) and C(^ : C) S xx C £ , /x : C C *7»- A l s o *7i = ft(6) Q ft(xx: C). Hence (1) of Theorem 6.2.3 holds, i.e., /x : C C 77^ C ft(/x: £). Now suppose 7^; C /x: £ and 7 % xx: £, where 7 and ^ are L-ideals of R. Then 7C 2 A* and hence 7C 2 &; for if 7C C &, then since £ £ H ^ f j and 7C C £ we would have 7C C Hj^j and so 7C C Dj^j fl & = xx. Thus 7 ^ C / i C ^ and so ip C rji. Hence (2) of Theorem 6.2.3 holds, i.e., 7 ^ C /x : £ and 7 £ M : C imphes I/J Qrji. Conversely, let 77 be any prime L-ideal of R and let £ be an L-ideal of i? such that C 2 /x and xx : C is ^-primary for 77. Now /x : £ = ( n g ^ ) : C = H?=1(& : C), where the latter equality holds by Theorem 3.1.42. Thus

77 = n(fi: c) = ft(n?=1(&: 0) = njL^te : c), where the first equality holds by hypothesis and the last equality holds by Theorem 3.8.2 and the fact that & C & : £ and &(0) = 1 and so (& : C)(0) = 1. Now consider those & such that C 2 &• Since £(& : C) <= &> & • C Q Vi- Also 6 C fi : C- Thus 77; = ft(&) C ft(& : £). Hence (1) of Theorem 6.2.3 holds. Suppose ^yip Q & : C and 7 g & : £, where 7 and ^ are L-ideals of i?. Then 7C g &. Thus 77/^C C & and so ip C Vi. Hence (2) of Theorem 6.2.3 holds. Thus & : C is 72,-primary for 77;. Hence 7£(& : £) = 77*. Suppose C C ^-. Then £j : C = li?- Thus 77 is an intersection of some of the 77;, contains their product and hence contains one of the 77;, and is contained by the same 77;. Thus 77 = 77; for some 2. ■

215

Theorem 6.2.25 Let /i G LI(R). Suppose that \x has a reduced IZ-primary L-representation. Then the number of IZ-primary L-ideals in any such represe­ ntation is unique as are the associated prime L-ideals. Proof. Let \x = £x n . . . H£ n = £{ n . . . n £ ^ be any two reduced ^-primary Lrepresentationsof/z.Let7/i = 7e(^),z = 1 , . . . ,n, and 77^. =n(£'j)J = l,...,ra. Now any n'j satisfies the condition of Theorem 6.2.24 since it is an associated prime L-ideal. But then rj. must be one of the rji by Theorem 6.2.24. Hence we have the desired result since 771,..., rjn are distinct as are the 77^,..., 77^. ■ Lemma 6.2.26 Let fj, £ LI(R). Suppose that fi has an IZ-primary L-representa­ tion, /i = £1 f!... fl £ n , and let rji = TZfa) for i = 1,2,..., n. Define the L-ideal £< ofR byV xeR, £;(*) = M 7 ( z ) I 7 e ZJ(iJ), VC 6 LI(R) such that /x : C g W, C C 7} /or z = 1,2,... ,n. T/ien £^ C £; for i = 1,2,... ,n. Proof. Suppose /i : ( g ^ - Then C ^ £^. Thus there exists an L-ideal 7 of R such that £7 C /j, C & and 7 g 77^, e.g., 7 = /i : £. Thus C £ £* since & is 7^-primary. Hence £^ C £^. ■ Definition 6.2.27 Le£ ^x 6e an L-ideal of R having a reduced IZ-primary Lrepresentation. The IZ-primary L-ideals occurring in any such representation of [i are called IZ-primary L-components and their uniquely determined set of IZ-radicals are called the associated primes of \i. The IZ-primary com­ ponents of /i whose IZ-radicals are isolated primes of /x are called isolated IZ-primary L-components of /i. More generally, if 771,... ,77^ are associated primes of \x such that no other associated prime of \x is contained in any of them, then the intersection of the TZ-primary L-components, £1 H . . . fl £&, is called an isolated component of //. Theorem 6.2.28 Let fi G LI(R). Suppose that /x has a reduced IZ-primary representation, /z = £1 fl . . . fl £ n . Let rji = 7£(£i), and let £^ be defined as in Lemma 6.2.26, i = 1,2,... ,n. Then any isolated component of /i, say, £^ D . . . fl £ifc, is the intersection of those £J for which r]j is a maximal element of {Vii,---,Vik}Proof. Now £< is an L-ideal of R such that £z' C £;. Let i/j be the intersection of those £i (i = 1 , . . . , n) whose L-radicals 77; are contained in rjj, where r]j is maximal in {77^,..., rjik}. Clearly, £^ D . . . fl £ifc is the intersection of these Vj. (Recall that if 2 £ {21,..., i*}, then 77; g rjj.) It will thus suffice to show that each Vj = £^. Let v — []£;, where the product is taken over those i such that

216

Vi % Vj- Then v C n&, where the intersection is taken over those i such that 7]i 2 Vj ■ Thus v % rjj else £; C 77^ for some i and so 77; C 77^, a contradiction. Thus we have i/i/j C (n£;)^- C (n&) HVJ = fi and so v C /j, : i/j and v £ rjj. Thus fi: i/j £ rjj and so z/j C fj. On the other hand, if 77; C 77^, then ^ C ^ (if fJ>'- C £ Vj, then certainly ji: (> £ rji iox rji C rjj), and hence by Lemma 6.2.26, £i S & a n < i s o £j Q &• Thus ^ C i/j, the intersection of those & for which ^i S %*• Therefore, ^ = ^-. ■ Theorem 6.2.29 Let ji be an L-ideal of R having a reduced IZ-primary Lrepresentation. Then the isolated components of JI, and in particular, the iso­ lated IZ-primary L-components of ^, are uniquely determined. Proof. Immediate from Theorem 6.2.28 and the uniqueness of the Lassociated primes of \x. ■ If rji and rjj are prime L-ideals of R, then rji D rjj implies that (rji)* 2 (Vj)*Hence if ji = £1 f l . . . D £ n is a reduced ^-primary L-representation and /x* = (£1)* H . . . fl (£ n )* is a reduced primary L-representation, then (&J* Pi . . . fl (&*)* *s a n isolated component of /x* implies that &x D . . . fl ^fc is an isolated component of ji. The following example shows that the converse is not true. Example 6.2.30 Let jx, £1, £2, Wi, and 772 be as in Example 6.2.22 Then (x) is the only isolated component of ji. However, since rji $Z 772 and 772 2 Vi » £1 and £2 are both isolated IZ-primary L-components. Note also that £1 and £2 are unique. However, (£1)* is not unique since JI* = (x2,xy,yk} n (x) /or any fc = 2 , 3 , . . . .

6.3

^-Primary Representations

In this section, we assume that R has an identity, L is a chain, and ji G LI(R). Now ^ has the inf property if and only if < well-orders fi(R). If ji has the inf property, then we let fi(R) =

{ji(0)}U{ai\ieI},

where / is a well-ordered index set such that V i, j G / , i < j if and only if a* < aj and /x(0) 7^ a; V i G I. Proposition 6.3.1 Suppose that ji has the inf property and \ji(R)\ \ > 2. Then 3 L-ideals ^ of R, i € / , such that ji = OieI^, \v[il(R)\ = 2, and fi(R) = U i€ //iM(/J).

217

Proof. For aU i G I define the L-subset ft] of R by V x G R, /i [il (x) = M(O) if x G (J,ai+1 and a^l\x) = ai otherwise. Then u^ is an L-ideal of R V i G 7. Let x G R. Suppose that x G fiai\lJ>ai+1 for some i G 7. Then /x(x) = a; = A{aj I j G I,j >i} = /i(0) A (Aja^- | j G 7, j > i}) = (njeIfi^)(x). Suppose that x G /i*. Then /xW(x) = /x(0) V i e / . Hence /x(x) = /i(0) = (n»GjjuM)(:r). Clearly, \^(R)\ = 2 V i G 7 and xx(i?) = UieI^(R). ■ Corollary 6.3.2 Suppose that fi has the inf property and | fi(R) \ > 2. If /x is C-prime (C-primary), then 3 C-prime (C-primary) L-ideals u^ of R, i G 7, such that u = n»€//xW, | xxM(i?)| = 2 , and /x(#) = U i G //i^(i?). Proof. Let /xW be defined as in Proposition 6.3.1. Since (i is £-prime (£primary), /^(o) and xxai are prime (primary) V i G 7 by Proposition 6.1.2. Hence V i G 7, (/i^)M(o) = /x ai+1 and (/i^) a . = i? are prime (primary). Thus fjfi is £-prime (£-primary) V i G 7 by Proposition 6.1.2. ■ Corollary 6.3.3 Suppose that /J, has the inf property, | /x(7£)| > 2, and zx(0) = 1. 7//i is C-prime (C-primary), then 3 prime (TZ-primary) L-ideals u^ of R, i G 7, sucA that /i = R^j/x^, |/x[i](^)l = 2 , and ^(iJ) = \JieIu^(R). Proof. Let [M be defined as in Proposition 6.3.1. Since /x(0) = 1, / J ^ ( 0 ) = 1. Hence the desired result follows from Theorems 3.5.5 and 3.7.2. ■ Definition 6.3.4 A representation of yi as a finite intersection, /x = £i D . . . D £n> of C-primary L-ideals £; of 7£ (2 = 1 , . . . , n) 25 called an C-primary Lrepresentation of /x. It is called irredundant or reduced if no £; contains C\{£j | j = 1 , . . . , n, j 7^ i) and TZdl^i) are distinct, i = 1 , . . . , n. Theorem 6.3.5 Suppose that /x 25 finite-valued and /i(0) = 1. Then /x /m5 an C-primary L-representation if and only if /i has anTZ-primary L-representation. Proof. Let /x = Hf=1^ be a £-primary L-representation. By Corollary 6.3.3, each & = n ? ! ^ , where each ^ is 7^-primary. Thus /x = n^ = 1 n ^ i j &j is an ^-primary 7^-representation of xx. The converse follows since an 7£-primary 7^-representation of /x is an £-primary 7/-representation. ■ Theorem 6.3.6 Suppose that /x(0) = 1 and /x = n™=1£i, where & 25 an L-ideal of R such that |fi(i?)| = 2 /or i = 1 , . . . , n. Then /x = n™=1£* 25 an irredundant C-primary L-representation of [i if and only if it is an irredundant TZ-primary L-representation of u.

218

Proof. Since &(0) = 1 and & is finite-valued, 7kfl(&) = ft(&) by Theorem 6.1.10,z = l , . . . , n. ■ Example 6.3.7 Le£ i? denote the ring of integers and let L = [0,1]. Define the L-subset rj of R by

( 1 V(x)={

\ { I

t/*€(0) ifxe(2)\(0) ifxeR\(2).

Then rj is a fuzzy C-prime ideal of R. Define the L-subsets /J, and v of R by , . (lifxe **> = ( j , .

(lifxe

"<*> = {*

(2) lfxeR\(2)

(0)

*/xS*\<0>.

Then fi and v are C-prime L-ideals of R. Now n = rj and rj = /JLD V are both irredundant C-primary L-representations ofr). Hence the number of C-primary ideals in an irredundant C-primary L-representation of an L-ideal of R need not be unique. Proposition 6.3.8 If every L-ideal /i of R has an C-primary then every ideal of R has a primary representation.

L-representation,

Proof. Let ji = flJL^i be an ^-primary L-representation of fx. Then Va < )tx(0), fia = ( H ^ ^ a = njL1(^»)o and so every level ideal of fi has a primary representation. Since every ideal of R is a level ideal of some L-ideal of R, the desired result follows. ■ It is clear that the family of L-ideals of R which have £-primary L-representations is larger than the family of L-ideals of R which have ^-primary Lrepresentations. Suppose that L is a dense chain. Then by Theorem 6.1.12, £ is a primary L-ideal of R if and only if £ is an 7^-primary (7£dZ-primary) ideal of R. By Corollary 6.1.13, if £ is a primary L-ideal, then ft(f) = ftcK(f) = ^ One can give a definition of a primary L-representation of an L-ideal /i involving y/JL similar to that of an 7£-primary representation of /J,. Hence if an L-ideal of R has a primary L-representation, then it has a 7^-primary (7&fZ-primary) representation and the representations are identical.

6.4

L-Prime Spectrum of a Ring

Let X = {rj | 7] is a prime L-ideal of R}.

219 For x

an

i-subset of i?, let ^(X) = {v e X | X C 77},

and X(x) = X\V(x), the complement of V(x). Clearly, V(x) = V((x)). If f* is an L-ideal of i?, then V(/x) = V( v //I). Theorem 6.4.1 Let X = {X((i) \ fi is an L-ideal of R}. Then the pair (X, T) is a topological space. Proof. Consider the L-ideals \i and v of R defined by ji(x) = 0 and v{x) = 1 V x E i?. Then V(/x) = X and V{v) = 0. Thus 0, X E T. Now let /i and i/ be any L-ideals of i?. Then 77 E V(/i) U V(i/) <4> 77 E X and either /i C 77 o r z / C 7 7 = > 7 7 E 3 C and / i f l z / C r / ^ r / E V(/z) fl V(v). We now show that the above imphcation is actually an equivalence. Suppose that 77 E X and /j, D v C 77. Then /iz/ C 77 since jiv U i € n ^ C 77 <=> 77 E V(Uien/x»). Thus riignVOxi) = F ( U i G n ^ ) € T. Hence UienX(fii) = X(Ui€ft/i;) E T. ■ The topological space (X, T) is called the L-prime spectrum of R and is denoted by L-Spec(R), or X for convenience. Let V(x) = {77 E X I 77(x) = 1},

x eR. We let Spec(i?) denote the set of all prime ideals of R. Theorem 6.4.2 Let a, be L\{0} and x, y E R. (1) X{xa) n X(yb) = X((xy)c), where c = a A 6; (2) Suppose that L is regular. 3 a E L\{0} such that X(xa) = $ if and only if x is nilpotent if and only if X(xa) = 0 V a E L\{0}; (3) IfBaE L\{0} such that X(xa) = X, then x is a unit and a is not prime (and so a = 1 if L is a chain). (4) The set {X(x a ) | x E R, a E L} is a basis for X; (5) Let a E L\{1} and Xa = {77 E X | 77(7?) = {a, 1}}. Then X a is compact. (6) Suppose that L is regular. Then Spec(R) is a dense subspace of X. Proof.

(1) 77 E X(xa)

fl X(yb)

& xa % 77 and yb £ 77 => (XT/)C g 77 <^>

77 E X((xy) c ). Now suppose that (xy)c £ 77. Suppose that xa C 77. Then 77(2;) > a > c. Now 77(2:2/) > 77(2;) > c. Thus (xy)c C 77, which is a contradiction. Hence xa £ r). Similarly, yb £ V- Hence the above imphcation becomes an equivalence.

220

(2) Suppose that X(x a ) = 0 for some a G L\{0}. Then V(xa) = X. Since L is regular, 0 is a prime element of L. Hence xa C.\P for every prime ideal P of R. Thus x is in every prime ideal of R. Hence x is nilpotent. Suppose that x is nilpotent. Then x is in every prime ideal of R. Thus r)(x) = 1 V rj G X else x fi 77*. Hence rca C 77 V 77 G X and V a G £\{0}. (3) Suppose that X(x a ) = X for some a G £\{0}. Then xa % rj V 77 G X. Hence x not a member of any prime ideal of R. Thus x is not in any maximal ideal of R. Hence x is a unit. Suppose that a is L-prime. Let I be any prime ideal of R. Define the L-subset 77 of R by 7](y) = 1 if y G I and 77(7/) = a otherwise. Then 77 is a prime L-ideal of R and xa C 77. Hence X(a;a) ^ X, a contradiction. Thus a is not L-prime. (4) Let X(/x) G T. Let 77 G X(/x). Then either JJL{X) > rj{x) or ji(x) and 77(0:) are not comparable for some x G R. Let /i(x) = a. Then xa % 77 and so 77 G X(x a ). Now V{ji) C F ( x a ) and so X(x a ) C X(/i). (5) If a is not prime, then X a = 0 and the result holds. Suppose that a is prime. Clearly, {X(xb) H X a \ x £ R and either a < b < 1 or a and b are noncomparable} is a basis for X a . Let. {X((xi)t,) HX a | x G i? z G A, and either a < 6 < 1 or a and 6 are noncomparable} be an open cover of X a and let K be the set of all such 6's. Let c = V{b\b G K}. Let 77 G X a . Then 77 G (U{X((xi)6) I i G A, 6 G IT}) H X a . Thus 3 z G A, 6 G if such that ( a ; ^ g 77. Hence either 6 > 77(0;;) or 6 and rj(xi) are noncomparable. Thus either c > 77^) or c and 77(xi) are noncomparable. Hence 77 G X((xi) c ). Thus

{X((xi)c)nxa I xeR,ie

A}

is an open cover for X a . Hence

x a = u({x((^) 6 )nx a |ZGA}) - u({x((^)c) |2GA})nx a = (i\n{7((xi) c )|ieA})ni f l = (x\y(uiGA((x;)c)))nxa = x a \(y(u GA ((^)c))nx a ). Thus V(Ui€A((a:i)c)) D X a = 0. Let P be any proper prime ideal of R. Let fi e LR be such that /i(x) = 1 if x G P and ji{x) = a if x £ P. Then /x G X a . Hence /x ^ V(Ut€A((^*)c)- Thus 3 j G A such that (a^) c g /x. Hence either c > H(XJ) or c and /i(xj) are noncomparable. Thus Xj £ p . Hence {x{ \ i G A} is not contained in any proper prime ideal of R. Thus ({xi | z G A}) = i?. Let 1 = 7*1 xi + . . . + r n x n for some n , . ..,rneR. Suppose 3/i G F(U^ = 1 (£;) c )nX a . Then U" =1 (xf) c C /i. Thus / J ( # ; ) > c V i = 1 , . . . , n. Suppose there exists i, 1 < i < n such that ji{xi) ^ 1. Then a = fi(xi) >c>bVbeK, a

221

contradiction. Hence fifa) = 1 V i = 1 , . . . , n. Thus x» 6 /x* V i = 1 , . . . , n. Hence 1 6 / / * , which is a contradiction since /x* ^ i?. Hence V(U? i ) c )n3C v(ut =:l{1 (xXi)c) nxaa = 0. Thus {X((xi)c) fl X a | x 6 i?, 1 < i < n] is a finite subcover for X a . Hence X a is compact. (6) Let 2) = {1/ | / 6 5pec(i?)}. A typical closed subset of X is n x € s ^ ( ^ i ) , where 5 C R. A typical closed subset of 2) is 2) n n x € sV(a;i). That Spec(#) is a subspace of X now follows by the identification of Spec(i?) with 2). Let 77 6 X, 77 ^ 2). Then 3x 6 .R such that n(x) = a, where 0 < a < 1 and a is prime. Let X(xi) be such that 77 6 X(xi). Now 77* is a proper prime ideal of R. Since 77(2;) ^ 1 , 1 ^ 6 X(xi). Thus each open subset of X containing 77 contains an element of 2). ■ Theorem 6.4.3 (1) For all /x 6 X, closure of /x, written {/x}, equals V(/d). (2) Let F be a subset of X and let F denote the closure of F. Then F C V ( 1 M ) , w/iere M = H^eF^. Proof. (1) Since V(fj) is a closed set and /x 6 V([i), {/1} C F(/x). Let cr ^ {/x}. Then there exists an open set X(0) such that a 6 X(#) and /i ^ X(#). This imphes that 5 £ a and K / i . Thus a £ V(/z). Hence V(AO C {/X}. (2) Clearly, lM(x) = l& £(x) = 1 V £ 6 F. Therefore, if 77 6 F, then 1M Q 77 and consequently 77 6 V(1M)- Thus V ( 1 M ) being a closed set containing F, contains F. ■ The following example shows that equality need not hold in Theorem 6.4.3(2). E x a m p l e 6.4.4 Let R = Z and let L = {l,a,6,0}, where a and b are noncomparable. For allp 6 R, p a prime let 77^, 77^ 6 LR be such that rj%(x) = 1 if x e (p), 77£(x) = aifxg(p) and n^(x) = 1 2/ x 6 (p), 77^(0:) = b if x <£ (p). Then X = {r]p X {77^ |I p e6 R,p i?,p a prime} U {r^ {77^ \| p G 6 R, i£, pa p a prime}. Let F = {775, 77^}. T/ien

fea)*n(77S)* (■ft), n fof). = (2)n(3) = (6) = M. Now

V(1M) --= {vl,*&, 7?2, Vbs} and

a \peR,p a prime} U {IJ§}. ^(3a) = {Vp V(3a) == {v P I P e -R,p a prime}u{^}.

222

Hence X(3 a ) = {rj^ | p G R,p ^ 3,p a prime}. Thus £(3 a ) is a basic open set and_r\\ G X(3 a ). Since X(3a) H (F\{r&}) = / ~ 1 ( ^ / ) ( x ) — fKt'Kx) V x e R=> rf(f{x)) = £'(/(*)) V x G R =» rf =J'. Hence / i s injective. If V(x\) is a basic closed set in L-spec(R), then f~1(V(xi)) is a basic closed set in L-spec(R/) since f-HV(xi))

= =

WeL-spec(R')\f(r,')(x) WeL-spec(R')\r,'(f(x))

= l} = l}

= vv(*h)Hence / is continuous. Let rj\ £' G L-spec(Rf). Since / " H 7 / ) is constant on Ker / , it is clear that f~l{n') G V(lKerf)- If V £ VO-Kerf)^ then 77 is a prime L-ideal of i?, constant on Ker / . Since / is an epimorphism, it follows from Theorem 3.5.11 that f(rj) is a prime L-ideal of R''. This defines a function g:V{\Kerf)^L-spec{R'), where #(77) = f(rj). Clearly, # = f"1. To prove the continuity of g, let be a closed set in L-spec{R'). Then

T\vv{x))

V'{f(x)i)

= /(V"(/(*)i) = {/too I »/(/(*)) = 1} 1 1

= {/- (V)l/- (»?')W = i} =

V(x) n V(lKer/)

which is a closed subset of L-spec(R). Finally, suppose that / is an isomor­ phism. Then Ker / = {0} and ^(lKer / ) = V(fh) = L-spec(R). M Theorem 6.4.6 Let f be an homomorphism of R onto the ring R'. If each prime L-ideal of R is constant on Kerf, then L-spec(R) is homeomorphic to L-spec(R').

223

Proof. The functions / and g of Theorem 6.4.5 are well defined, continuous, and inverses of each other. Since all L-prime ideals are constant on Ker / , Ker / C 77* V 77 G X. Hence V(lKerf) = V(0i) = L-spec(R). ■ Corollary 6.4.7 If I is an ideal of R such that I C fl^^T?*, then L-spec(R) and L-spec(R/1) are homeomorphic. Proof. Let / be the natural homomorphism of R onto R/I. 77 G X is constant on Ker/. ■ Corollary 6.4.8 L-spec(R) and L-spec(R/ H^x v)

are

Then each

homeomorphic.

Proof. L-spec(R) and L-spec(R/y/R) are homeomorphic by Corollary 6.4.7. Now R/y/R and R/ H^x V a r e isomorphic since (n^xv)* — ^escV* = v^R- B L e m m a 6.4.9 Let L be such that for alia G L\{1}, there exists b G L, a
\ b

otherwise.

Then 77 G X. Hence 77 G X(xt) for some x G {xi}i^A and t e K. This implies that #t ^ ^ However, 77(x) > b > a > t = xt(x). Thus xt C 77, a contradiction. Hence a = 1. ■ Corollary 6.4.10 Let L be a chain. Let K C L\{0}, {xi}ieA Q R and X = \j{X((xi)t) \ i e A, t e K} for some index set A. Then V{t\teK} = l. ■ Lemma 6.4.11 Let K C L\{0}, {X*}» € A C i?, a G L\{0} and X G i?. Suppose x is not nilpotent and X(xa) C U{3C((xi)t) | z G A, £ G i f } . J4ZSO suppose V{t | £ G i f } = b is a prime element. Suppose a and b are comparable. Then b> a. Proof. Suppose a > b. Since x is not nilpotent, there exists a prime ideal P of R such that x £ P. Define 77: R —> L by

„(„) /v y = / ! *

^ b

if

^eP

otherwise.

Then 77 G X. Now x a (x) = a > b = rj(x). Hence xa^.n and so 77 G X(xa). Thus V G £(?/*) for some y G {zi};GA and t £ K. This implies that T/J g 77. However, v(y) >b>t = yt(y) and so yt C 77, a contradiction. Hence 6 > a. ■

224

Corollary 6.4.12 Let L be a chain. Let K C L\{0}, {xiheA Q R, a e L\{0} and x G R. Suppose x is not nilpotent and X(xa) C U{£((xi)t) \ i G A, t G i f } . Then V{t\teK}>a.M The proof of the following lemma is straightforward. L e m m a 6.4.13 Let a,b G A W - Suppose a
\ieA,teK}

= \j{X((xi)b)

| i G A}.

Proof. Since t < 6 V £ G if, it follows that \j{X((xi)t) | z G A,£ G i f } C U{3t((xi)b) | z G A}. Let /x G U{jC((xi)6) | z G A}. Then fi G X((^) 6 ) for some i G A. Write y = X{. Then i/& g ^. This implies that either 6 > ji{y) or 6 and /x(y) are noncomparable. First suppose that b > fi(y). If t < /j,(y) V t G if, then 6 < /x(y) a contra­ diction. Hence there exists t G K such that either t > /x(?/) or t and ^(y) are noncomparable. This implies that yt % fi and s o / i G 3£((y)t) C L){X((xi)t) \ ieA,t€K}. Now suppose that b and /x(?/) are noncomparable. Suppose V t G if, t and /i(y) are comparable. Let A = {t e K \ t < fi(y)} and B = { t G K \ t > fi(y)}. If B = 0, then V t e K, t < fj,(y) and so 6 = V{t | £ G i f } < /x(y), a contradiction. Hence there exists t G K such that £ > fi(y). Thus yt ^ ^ and so /x G 3£((2/)t) C \j{X((xi)t) | z G A, t G i f } . If there exists £ G if, such that t and /x(y) are noncomparable, then clearly, yt £ A^- Hence /x G X((y)t) C U{X((xi)t) | i G A,t G i f } . Consequently, U{X((^) 6 ) | z G A} C \j{X((xi)t) \ i G A,t G i f } . Hence U{X((x i ) t ) | z G A,* G i f } = U{£((^) 6 ) | z G A}. ■ L e m m a 6.4.15 Let Xi e R, 1 < i < n and b G L. Then n

Proof. We prove the result for n = 2. Let /x G X((xi + x2)b)- Then (xi + #2)6 2 ^- Hence either 6 > fi(xi + # 2 ) or b and /x(:ri + a?2) are noncomparable. First suppose that b > JJL{X\ + z 2 ). Suppose b < /x(:ci) and b < )Lx(x2). Then 6 < ^(a^) A //(a;2) < fi>(xi + x 2 ), a contradiction. Hence either (6 > ^x(xi) or b and /x(a;i) are noncomparable) or (b > /x(a;2) or b and ^(#2) a 16 noncomparable). Thus either (x{)h % /x or (x2)& g /i. Hence /x G £((xi)&) or A* € X((x 2 ) 6 ), i.e., /x G £((an) 6 ) U 3t((x 2 ) 6 ).

225

Now suppose that b and /J,(XI -f x2) a r e noncomparable. If either (b and n(x\) are noncomparable) or (6 and ^(#2) are noncomparable), then as before li G 3£((a;i)&) U3£((#2)&). Suppose b and ^(xi) are comparable and b and //(a^) are comparable. If b < fi(xi) and b < ^(£2), then b < /z(xi) A fi(x2) < [i(xi + X2), a contradiction. Hence either 6 > ^(^1) or 6 > /x(x2). Thus either [i G X((x!) b ) or fi G X((x2)b), i.e., // 6 £((#i)&) U X((x 2 )b). Consequently, ^((^1+^2)5) Q ^((^i)6)UX((x2)b). The general result follows by induction. ■ Lemma 6.4.16 Let a G L\{0} and x G R. Then X(xa) = X(x^) V n G N. ■ Theorem 6.4.17 The topological space (3£,T) is compact. Proof. If L\{1} has no prime element, then X = 0 and the result is trivially true. Suppose £\{1} has prime elements and let a be a prime element of L\{1}. Let {3£((xi)t) I i G A,£ € K C L\{0}} be a cover of X by its basic open sets. Let b = y{t I t G K). Then by Proposition 6.4.14, {X({xi)h) \ i G A} also covers X. Thus X = U{X((xi)b)\i e A}

=

xl(UiGA(xi)b))

=

£(Ui G A(x;) 6 ).

Hence V(Ui£A(xi)b) = 0- Let P be a proper prime ideal of R. Define /i : R —» L by (x\ = f 1 if x G P ^ ' ^ a otherwise. Then /x G -£. Thus /i G -£(UieA(#i)6) and so Ui€A(xi)6 £ M- This implies that there exists i G A such that (xi)fc g! /i. Hence either b > fi(xi) or b and /i(xi) are noncomparable. In either case, Xi £ P. Thus {x^ \ i G A} is not contained in any proper prime ideal. Hence ({xi \ i G A}) = R. Let 1 = Yl7=i Tixi f° r some ri G R, 1 < i < n. Suppose V^^x^i) ^ 0 and let a G V(U^=1(xi)b). Then U™=1(a;i)6 C a. This implies that a(x^) > b Vz = 1,2, . . . , n . Suppose there exists z, 1 < i < n, such that a(x{) ^ 1. Now n

a(l) = aiJ^UXi)

> A?=1a(riXi) > A? =1 a(xi) > A?=1& - b.

i=i

Now a € X and so a G U{jt((xi)b)|z G A}. Thus Eli G A such that a G jC((xj)b). This implies that (xj)b % a. Thus either b > OL{XJ) or b and Q;(XJ) are noncomparable. However, ct(xj) > a ( l ) > 6, a contradiction. Hence a(xi) = 1 V 2 and so 1 G a*. Hence i? = a*, a contradiction. Thus V(\J7lz=:1{xi)b) = 0. Consequently, {£((xi)&) | i = 1,2,... ,n} is a subcover of X. Hence X is compact. ■

226 Proposition 6.4.18 Let 21 C X. Define 3(21) = fVeotMThen 3(21) is an L-ideal of R. 3(21) is called the ideal of 21 in R. ■ Proposition 6.4.19 For all 21 C X, V(3(2t)) = 21, rtfte c/osure o/21 m X. Proof. Let fi G 21. Then 3(21) = rv € a/u C ^. Hence /x G ^(3(21)). Thus 21 C F(3(2t)). Hence 21 C F(3(21)) = ^(3(21)), since F(3(21)) is closed. Now let V(fi) be a closed subset of X such that 21 C F(/z), where /J, is an L-ideal of # . Then /x C rj for all 77 G 21. Hence /z C rv e a/x = 3(21). Thus F(3(21)) C V(/i). Hence F(3(21)) = 21. ■ Definition 6.4.20 A topological space 2U is called irreducible if for any de­ composition W = A\ U A2, where Ai and A2 are closed subsets o/2U, we have either W = A\ or 2U = A2. Theorem 6.4.21 Let 21 be a closed subset of X. Then 21 is irreducible if and only z/3(21) is an L-prime ideal. Proof. Suppose 21 is irreducible. Prom the definition of 3(21) it is clear that 3(21) is nonconstant. Let xai yb be such that xaoyb C 3(21), where x,y G R and a, 6 G L. Then xa o yb e C\^e^fi and so xa o yb C fi for all >LX G 21. Thus for all ^ G 21, either xa C fj, or yb C fi. Hence 21 = (21 D V(:r 0 )) U (21 D VX2/&)). Since 21 is irreducible and 21D V(xa) and 21D V(yb) are closed, it follows that 21 = 21 n V{xa) or 21 = 21 n V(yb) and so 21 C V(xa) or 21 C V(yb). Hence xa C 3(21) or 2/6 C 3(21). Conversely, suppose that 3(21) is an L-prime ideal of R. Suppose 21 = 21i U 212, where 21x and 212 are closed subsets of X. Thus 3(21) C 3(21x) and 3(21) C 3(212). Also 3(21) = 3(21i U212) = 3(21i) fl 3(212). This implies that 3(21i) o3(212) C 3(21i) H3(212) Q 3(21). Since 3(21) is L-prime, either 3(21i) C 3(21) or 3(212) C 3(21). By Proposition 6.4.19, it follows that either 21 = 2ti or 2t = 2t 2 . ■ Definition 6.4.22 Let 21 be a closed subset of X and /x G 21. Then /J, is called a generic point of% z/21 = {^}, the closure of ji in X. It is known that if ty_Q X, where X is a topological space, then 2) is irreducible if and only if 2) is irreducible. Proposition 6.4.23 Let 21 be a closed subset of X and let fj, G 21 be a generic point of 21. Then 21 is irreducible.

227

Proof. Since \i is a generic point of 21, 21 = {//}. Since {[i\ is irreducible, {/x} is irreducible. Hence 21 is irreducible. ■ Theorem 6.4.24 Let 21 be a nonempty closed irreducible subset ofX. Then 21 has unique generic point. Proof. Since 21 is closed irreducible, 3(21) is L-prime and so 3(21) e X. Write 93 = {3(2i)|. Then 3(53) = 3(21). By Proposition 6.4.19, ^(3(33)) = ®. Hence ^ W ( a ) ) = {3(21)}. Thus {3(21)} = ^(3(21)) = 21 = 21, since 21 is closed. Hence 3(21) is a generic point of 21. Next we show that 21 has a unique generic point. Let fi be any generic point of 21. Then 21 = {/x} = V^(3({/i})) by Proposition 6.4.19. Since F ( 3 ( M ) ) = V{n), 21 = V(/J). Hence 3(21) = 3 ( V ( A O ) . However, 3(Vr(/i)) = /i and so /JL = 3(21). Hence 21 has a unique generic point. ■ Example 6.4.25 Let L — [0,1]. Let R = R[x,y, z] denote the polynomial ring in the algebraically independent indeterminates x, y, z over R, the field of real numbers. Consider the system of fuzzy equations (xs)2 ~ Vt = 0 i 4

(x5)2zu

= Ox.

Let fi = Ux2 — y)i, (x2z)i \ , the L-ideal of R generated by the fuzzy singletons, (x2 — y)i and (x2z)i. Define the L-subsets r\i of R, i = 1,2,3,4,5, as follows: m(u) = l 1 ^ue(x2'^) v ' 1 0 otherwise ^

(u\=f1 ' | 0 \

i ue

f (x2 -Viz) otherwise

1

V4(u) = < i ty>H = S l

otherwise

if u e (z) otherwise ifue (0) otherwise,

Then rji is a primary L-ideal of R for i = 1,2,3,4, 5 and

ii = 771 n 772 n 773 n 7?5 n ry5. We Ziave

V*7ifa)= I 0

ifue (x,y) otherwise

228

_ f 1

ifue

(x)

Vm{y) V ^ M ~= {1 I7 otherwise otherwise fori i = = 2,4, 2,4,5. = r\ii for 5. Also and xy/rjl pi m=v

3 ( f (AO) ==

Vmri \ / % r ~\Vmri v ^ 4 r i-v/^5-

T/ie solution of the system of fuzzy intersection equations is {(x,x2,0) | x G R} U {(0,0,2) | z G R} together with t = ~, s A u = | . iVow y^T and .^772 display the crisp part of the solution while y ^ 3 , y ^ and ^7/5 display the fuzzy part. Also V(M) = uLi^(v^)3(V(/i)) == nf ^VM === z(VM) n?=1=i3(ny^)) 3(V(V5fc)) = = 1 h— rLiy/vl, :l\A/i, and y/rjl is a generic point for 3(V(y/rji)i i = 1 , . . . , 5.

6.5

Quasi-local L-Subrings

Since most of the important results in the remainder of the chapter require L to be a chain, we assume in the rest of the chapter that L is a chain. We will remind the reader of this at the beginning of the remaining sections. We recall that a commutative ring with identity is called quasi-local if it has a unique maximal ideal. Let R be a subring of a commutative ring S with identity 1 such that 1 G R. In this section, we examine the extension of L-subrings and L-ideals of R to S. If %R is a multiplicative system in R [261, p.46] and R^i is the quotient ring of R with respect to 9DT, then we give necessary and sufficient conditions for an L-subring (L-ideal) /i of i? to be extended to one fj,e of Rm (of S) such that the images of \i and \ie coincide. This allows us to give necessary and sufficient conditions for an L-subring fi of R to be extended to a quasilocal L-subring of Rp, where P is a prime ideal of R, and also necessary and sufficient conditions for ji to be extended to an L-subfield of 5, where 5 is the quotient field of an integral domain R. We also show that there is a one-to-one correspondence between the prime L-ideals ji of R such that /x* = {x G R | n(x) = /x(0)} is disjoint from VJl and all prime L-ideals of R^\. In this section, we introduce the concept of a quasi-local L-subring of a quasi-local ring. Let il be a nonempty subset of L. There is no loss of generality in using il as an index set in the following results. Proposition 6.5.1 Let {Ia \ a G 11} be a collection of ideals of R such that R = U/ 0 . Suppose that for all 6, a G il, b > a if and only if h C Ia> Define the L-subset 11 of R by for all x G R, fi(x) = y{a V{a I| x G fj.(x) 6 7/«}. a }. Then fi is an L-ideal of R.

229 Proof. Let x,y G R. Suppose t h a t ii(x - y) = ra. T h e n x - y G Ia for all a <m a n d x - y £ Ib for all b > ra. T h u s either (1) x, y e Ia for all a < m and either x ^ Ife for all b > m or y £ Ih for all b > ra or (2) there exists n G L such t h a t for all a G it, n < a < ra, x <£ Ia and y £ Ia. If case (1) holds, t h e n either JJL(X) = ra or fi(y) — ra. If case (2) holds, then fi(x) < n a n d fi(y) < n. In either case / x ( x - y ) = m > fi(x)Afi(y). Now suppose t h a t /i(xy) = m . T h e n xy G Ia for all a < m a n d xy £ 4 for all 6 > ra. Hence x ^ Ih a n d t/ ^ 7 6 for all b > ra. T h u s fi(xy) =m> fi(x) V /x(y). Hence /x is a n L-ideal of # . ■ E x a m p l e 6.5.2 Let L = { l , a , 6 , c , 0 } fee sucA Jfta* l > a > c > 0 , l > 6 > c > 0 and a and b are noncomparable. Let i l = { l , a , 6 , 0 } and let R = Z. Let A = {0}, J a = ( 2 ) , J 6 = (3) and I0 = Z. Let fi e LR be such that li(x) = V{t | x G It, t G i l } . Now fi(2) = a, /x(3) = 6, and ^(5) = 0. Thus fj,(2 + 3) = fi(5) = 0 < c = a Ab = ji{2) A /x(3). Thus \x is not an L-subring of R. Hence Proposition 6.5.1 need not be true if L is not a chain. P r o p o s i t i o n 6 . 5 . 3 Let {Ra \ a G it} be a collection of subrings of R such that R = URa. Suppose that for all 6, a G it, b > a if and only if Rb C Ra- Define the L-subset [i of R by for all x € R, n(x) = V{a | a G RajThen fi is an L-subring

of R.

I

P r o p o s i t i o n 6 . 5 . 4 Let fi be an L-subring of R. Then for all y G R, y a unit, and for all x G R, /^(xy-1) > /i(x) A fji(y) if and only if fx(y) = /x(y _ 1 ) for every unit y in R. In either case /x(l) > /x(y), where y is a unit. ■ Suppose t h a t R is a quasi-local ring. Let M denote t h e unique maximal ideal of R. Let fi be an L-subring of R. We say t h a t /i is a q u a s i - l o c a l Ls u b r i n g of R if for all x G R and for all y G R such t h a t y is a unit, / x ( x y - 1 ) > H(x) A fj,(y), or equivalently fi(y) = ^(y"1)Since one of t h e defining properties of an L-subfield fi of a field F is t h a t for all x, y G F , ?/ ^ 0, ^{xy'1) > fi(x) A fJ>(y), it seems n a t u r a l t o require t h a t this property b e used in defining a quasi-local L-subring of a quasi-local ring. T h e following results also motivate t h e definition of a quasi-local L-subring. P r o p o s i t i o n 6 . 5 . 5 Let R be a quasi-local ring. If fi is quasi-local L-subring of R, then for all a such that 0 < a < / i ( l ) , / i a is a quasi-local ring, and Mr\fia is the unique maximal ideal of \ia. Proof. T h a t \xa is a subring of R follows from known arguments. Let y G {ia, y £ M. T h e n y~x G R. Since \x is quasi-local, ji{y) = / x ( y _ 1 ) . Hence y'1 G / i a . ■

230

Proposition 6.5.6 Let R be a quasi-local ring. Let fj, be an L-subset of R. If for all a G /i(R), Ma is a quasi-local ring with unique maximal ideal M D /za, then fi is quasi-local L-subring of R. Proof. That /x is an L-subring of R follows from known arguments. Let y G R be such that y is a unit. Now y € fia for some a G fi(R). Since y <£ M, y $L M D /i 0 . Since M D \xa is the unique maximal ideal of ^ a , y~l G fia- Hence y G /xa if and only if t/ - 1 G /i a - Thus fi(y) = /i(y _ 1 ). ■ Proposition 6.5.7 Let R be a quasi-local ring. (1) If fi is a quasi-local L-subring of i?, then fi* is quasi-local and / i * D M is the unique maximal ideal of /J,* . (2) If Rf is a subring of Ry then Rr is quasi-local with unique maximal ideal M C\R' if and only if 1R> is a quasi-local L-subring of R. Proof. (1) Let y G /x*, y £ M D fi*. Then y"1 exists in R. Now /i(2/ _1 ) = /j,(y) > 0 . Thus*/" 1 G^*(2) The desired result follows from Propositions 6.5.5 and 6.5.6. B Proposition 6.5.8 Let R be a quasi-local ring. Let {Ra \ a G 11} be a collec­ tion of quasi-local subrings of R with unique maximal ideals Ma of Ra, a G it, such that R = URa and for all 6, a G it, b > a if and only if Rt C Ra- Assume that {Ra | a G it} is closed under intersection. Define the L-subset /x of R by for all x G i?, /j,(x) = V{a | x G Ra}. Then fi is quasi-local if and only if Ma = M C\ Ra for all a G it. Proof. By Proposition 6.5.3, ji is an L-subring of R. Suppose that ji is quasi-local. For all a G it, M H Ra C Ma. Suppose y G M a , y £ M C\ fia for some a. Then y £ M. Hence y'1 exists in R. Since y G M a , y"1 £ Ra. Thus Kv'1) < M(2/) s m c e {^a | a G it} is closed under intersection, a contradiction. Conversely, suppose Ma = M fl Ra for all a G it. Let ?/ G R be a unit. Then y £ M and so y £ Ma for all a G it. Suppose ?/ G i? a . Since ?/ ^ M a , 2/ _1 G i? a Hence y e Ra ii and only if t/" 1 G i? a - Thus /x(y) = ^O/" 1 )- ■

6.6

Extension of L-Subsets

The results of this section lay the foundation for the results concerning exten­ sions of L-subrings and L-ideals in the next two sections. We recall that L is a chain.

231

Proposition 6.6.1 Let X be a nonempty subset of a set S and let fi be an L-subset of X. If v is an extension of /i to an L-subset of 5, then \ia fl v\> = LH> for all 6, a such that 0 < a a if and only if v^dVa, and (3) for all 6, a G fi(X), b > a, fia fl vh — [ih, then \x has an extension to an L-subset fie of S such that (fie)a 2 va for aU a G fi(X). Proof.

Define the L-subset /ie of 5 by for all y 6 5, fie(y) = V{a | y G i/ a }.

If V £ va, then fie(y) > a and so y G (fie)a- Thus va C (/ie)a- We now show that jjie is an extension of \i. Let x G X. Suppose /i(x) = a. Then x G fia- If x E fib for some 6 > a, then /z(x) > o > a, a contradiction. Hence x ^ //& for all 6 > a. Since also a: G /xa C i/a and x ^ 1/5 for all b > a by condition (3), fie(x) = a. ■ Example 6.6.3 Let L = {l,a,&, 0}, where a and b are not comparable. Let S — Z and X = 2Z. Define the L-subset /1 of X by fi(x) = 1 z/x = 0, /z(x) = a i / i € 4Z\{0}, and ^z(x) = b if x G 2Z\4Z. T/ien ^(X) = {l,a,6}. Let v be a fuzzy subset of S such that v\ = ji\ = {0}, va = 2Z, and v\> = Z. Then (1) \Jva = Z = S, (%) Vs, t G /i(X), s > t <£> vs C i/t, and (3) Vs, £ G /z(X), s>t, /itnvs= fis. Now /xe(6) = a V 6 ^ 6 = /x(6). Tftu* Proposition 6.6.2 does not AoZd z/ L is not a chain. Theorem 6.6.4 Let X be a nonempty subset of a set S and let \i be an Lsubset of X such that fi has the sup property. If*B = {ya \ a G fi(X)} is a collection of subsets of S which satisfies (1), (2), and (3) of Proposition 6.6.2, then fi has a unique extension to an L-subset fie of S such that (fie)a = va for all a G fi(R) and fie(X) = fi(X). Proof. Define fie as in Proposition 6.6.2. Then for all y G S, fie(y) — y{a\y G Va\. Thus x G (fie)a if a n d only if fie(x) > a if and only if x G va. Since \i has the sup property, fie(X) C fi(X). Thus fie(X) = fi(X). Let 77 be an extension of \i such that n a = va for all a G fi(X) and r](X) = fi(X). Then rfa = (fie)a for all a G fie(X). Hence it follows that 77 = fie. ■

232

6.7

Extension of L-Subrings and L-Ideals

Let R be a subring of S. If / is an ideal of R, we let Ie denote the ideal of S generated by / . We recall that L is a chain. T h e o r e m 6.7.1 Let R be a subring of S and let fi be an L-ideal of R such that ji has the sup property. If UaGA,(jR)(/ia)e = S and for all b, a G fi{R), b > a, jjba n (fJ>b)e — M&> then fi has a unique extension to an L-ideal fie of S such that (ne)a = (fia)e for all a G fi(R) and fie(R) = n(R)Proof. Since b > a if and only if /i& C fia for all 6, a G fi{R), the condition Ha H (Hb)e — H>b imphes that b > a if and only if (/^>)e C (/i a ) e - Hence the (/i a ) e satisfy the conditions of Proposition 6.5.1. If we let va = (fia)e in Theo­ rem 6.6.4, then the desired result follows from Proposition 6.5.1 and Theorem 6.6.4. ■ Let R have an identity. Let 9Jt be a multiplicative system in R. Let N = {x E R\ mx = 0 for some m G Wl}. Then TV is an ideal of R. Unless otherwise specified, we assume N = (0), i.e., 371 is regular. Let R<m denote the quotient ring of R with respect to Wl. Since JV = (0), we can assume that R C Rm. If fi is an L-subring of R, we assume that /x(l) = /x(0). T h e o r e m 6.7.2 Let fi be an L-subring of R such that fi has the sup property. Then fi can be extended to an L-subring of fie of Rgji such that for all x, y G R, y a unit, fie(xy~l) > fie(x) A fie{y) if and only if for all b, a G fi{R), b > a, Ha^(lJ>b)wib = V>b, where Tib — Tlilfib for all b G fi(R). If either condition holds, fie can be chosen so that {fie)a = (fJ>a)wia for all a G fi(R) and fie(R) = fi(R). Proof. Suppose that fie exists. Suppose that xy~l G (fian(fib)<mb), xy~x £ fib, for some 6, a G fi(R) with b > a and where x G fib, y G Tib. Thus b > fiixy'1) = ^(xy'1) > /ie (x) A fie (y) > b since fib C (fie)b, a contradiction. Thus fia n {fib)mb = V>b- Conversely, suppose fia n (fib)mb = f*b for all b > a. Since R = Uaefl{R)fia, Tt = U Wla and Rm = U(fj,a)fma. Let va = (fj,a)wia in Theorem 6.6.4. Then by Proposition 6.5.3 and Theorem 6.6.4, fi has an extension to an L-subring fie of Rm such that {fie)a = (fia)ma for all a G fi(R) and fie{R) = fi(R). Let y G Rm be a unit. Then y G (fia)ma, V £ (Vb)<mb for some a and for all b > a since fie(y) = W{a\y G (fia)ma}- Hence y = cd'1 for some c e fia and d G Tla. Since y is a unit, c G 5DT and so c G 9Ka. Hence dc'1 G (fia)<ma. Now d c - 1 £ (/x&)<m6 for all b > a else cd" 1 = ( d c " 1 ) - 1 G (M6)a«5.ThusAie(y) = ^ e ( 2 / - 1 ) . B Let P denote a prime ideal of R and let i?p denote the quotient ring of R with respect to the multiplicative system cP, the compliment of P in R. Then Rp is a quasi-local ring.

233

Corollary 6.7.3 Let \i be an L-subring of R such that \i has the sup property. Then fi can be extended to a quasi-local L-subring /xe of Rp if and only if for all b) a E fi(R), b> a, jj,an {Hb)pb = M6, where Pb = P fl jib for all s E fi>(R). Proof. The desired result follows from Theorem 6.7.2, if we note that cPa = cPdfia for all a E /J,(R) and if we identify cP with Wl and cPa with Tla in Theorem 6.7.2. ■ Corollary 6.7.4 Let R be an integral domain and let Q denote the quotient field ofQ. Let fi be an L-subring of R such that \x has the sup property. Let Qa denote the smallest subfield of Q which contains \ia for all a E /i(-R). Then \x can be extended to an L-subfield of Q if and only if for all b, a E j^(-R), b > a, V>a^Qb = fibProof.

6.8

Let P = (0) in Corollary 6.7.3. ■

Extension of Prime L-Ideals

We assume that L is a chain in this section. Let R and S be rings and let / be a homomorphism of R into S. If I is an ideal in R, then the ideal f(I)e (or simply Ie) is defined to be the ideal of S generated by / ( / ) and is called the extended ideal, or extension, of / . If J is an ideal of 5, the ideal J c = / _ 1 ( J ) is called the contracted ideal, or the contraction, of J. For the rest of this section we assume that R and S are commutative rings with identities and / ( l ) = 1. Let v be an L-ideal of S. The L-ideal vc = / _ 1 ( ^ ) is called the contracted L-ideal, or the L-contraction, of v. For all x E R, !/"(*) = i/(/(a:)).

Proposition 6.8.1 Let r\ be an L-ideal of S. Then (r)c)* = (r)*)c. Proof.

xef-\ri.)

x G (vc)* O r)c{x) =

=

T?C(0)

<S> r](f(x)) = r}(f(0)) <^> f(x)

£t),^

(ihy.m

Let ffll be a multiplicative system in R. We do not assume the ideal iV = {x E R | xm = 0 for some m E 931} equals (0) . Proposition 6.8.2 Let n be an L-ideal of

RSJJI-

Then

0T). = m*r= Mce = v*-

234

Proof. That (r/ce)* = ((^ c )*) e follows from the definition of the extension of an L-ideal and the fact that (77°)* is a level ideal. (See Theorem 6.7.1.) Now c c e ce ce (T?C)* = (ry*) by Proposition 6.8.1. Thus ((v )*) = (r?*) . In Rm, (r?*) = 77* [261, Theorem 15(c), p.223]. ■ Proposition 6.8.3 Let \x be an L-ideal of R. Then in i?gjt,

and ifWl is prime to ^*, then (fJ>*)ec = fi*. Proof. By Proposition 6.8.2, (/xec)* = ((^ e )*) c . Prom the definition of the extension of an L-ideal, we have (fie)* = (/^*)e and so ((/xe)*)c = (^*) ec - By [261, Theorem 15(b), p.223], /x*ec = //*. ■ Theorem 6.8.4 The mapping \i \—► [ie is a one-to-one mapping of the set of all prime L-ideals fi of R such that /J,* is disjoint from OT onto the set of all prime L-ideals of RyuProof. By [261, Corollary 1, p. 224], ft(//*)6 is a prime ideal of R^i since //* is a prime ideal of R, where h is the natural homomorphism of R onto R/N C Rm. Now h(iJL+) = ft(/z)* by Lemma 3.2.5. Thus ft(^*)e = (ft(/i)*)e. Now (M^)*) e = (MAO C )* by t n e definition of h(/j,)e and by the fact that (/i(/i)e)* is a level ideal of R^. Hence (ft(/x)e)* is a prime ideal of R%i. Let x G Ker ft = N. Then xm = 0 E / i * for some m G 9JI. Hence x E fi* since fJOl fl p* = . Thus Ker ft C /x*. Hence ft(/z*) C h(R) since /z* C R. Thus ft(^)* C ft(-R) by Lemma 3.2.5. Hence h(fj)(R) = fi(R). Since h(fi)e(R) = ft(/x)(il), h(^)e(R) = fi(R). Thus h(fi)e (= /xe) is a prime L-ideal of R^. Let 1/ be any prime L-ideal of R<m> Then v* is a prime ideal of R?m and 1/* = ^*ce = (z/ce)* by Proposition 6.8.2. Now v*c = ft-V*) is a prime ideal of R by [261, p. 220]. Since (z/*)c = ( 1 / % by Proposition 6.8.1 and (vce)* = v+ by Proposition 6.8.2, in order to show that uc is a prime L-ideal of R and vce — v, it suffices to show that vce(R) — u{R) = vc(R). Let x G R. Then ft(*/0(ft(z))

= = = =

V{^ c (r)|ft(r) = ft(:r)} V{*/(ft(r))|ft(r) = ft(x)} v(h(x) i/ c (x).

Thus z/ c (#) = h(vc){R). Now h(vc){R) = ft((*/c)e)(i*) by the definition of the extension of an L-ideal. Now h((vc)e){R) = uce(R) by notation and so i/ c (ii) = uce(R). u(R) = {1, a}, 0 < a < 1. Since vc{x) = v{h(x)) for all x G # , i/c(i?) C i/(fl). Now z/c(x) = v{h{x)) = 1 for all x G i? imphes ft(i?) C ^ which

235

in turn implies v* = Rm since 1 G h(R), a contradiction. Thus vc(R) = v(R). Hence [i i—► fie is an onto map. In order to show that \x \—► fie is one-to-one it suffices to show that /i = jiec. By Theorem 3.5.9, it suffices to show that u(R) = fiec(R) since /z* is disjoint from 9K imphes that Wl is prime to u+. Now for x G R, Vec(x) = fie(h(x))

= h(tf(h(x)) = =

h(ji)(h(x)) \/{fi(r) I h(r) = h(x)}.

Thus /zec(0) = u(0) since Ker h C ^ . Now V{u(r)\h(r) = h(x)} = 1 or a, where 0 < a < 1. Hence {1} C fiec(R) C /x(iJ). Since (^ ec )* = ^z*, we must have fiec(R) = /i(R). Thus fiec = /x. ■

6.9

L-Topological Spaces

We assume that L is a chain in this section. Let R have an identity and let M be a maximal ideal of R. Structural properties of R can at times be determined by the existence of a field F C R with the same identity as R such that R = F + M. We call such a field a coefficient field of R with respect to M. For example, if R is a complete equi-characteristic local ring with unique maximal ideal M, then R has a coefficient field, [198], the structure theorem of complete local rings. Other results on coefficient fields can be found in [155, 181, 197]. The purpose of this section is to determine to what extent these concepts can be carried over to L-subrings of a commutative ring with identity. The notion of a quasi-local L-subring has been introduced in Section 6.5. Due to the importance of the M-adic topology in local ring theory, we examine L-topological concepts in this section and the next. In Section 6.11, we fuzzify the idea of a coefficient field. Let 5 be a nonempty set. Define the L-subset (j) of S by (x) = 0 Vx G S. If 7Z is a collection of L-subsets of S, we define \Jv^^y = 0 if 1Z = 0. In the following definition, we consider an L-topology on an L-subset. Definition 6.9.1 Let \ be an L-subset of a set S. Then T is an L-topology on x if T is a collection of L-subsets of S such that V /i G T, \x C x and (1) ^ T ; (2)WxaCx,BfiET such that xa C fi] (3) ifTlCT, then U^en^ e T; (4) if \i, v G T, then \x n v G T. If T is an L-topology on x, then (x, T) is called an L-topological space and the elements ofT are called open L-subsets with respect to x-

236

Definition 6.9.2 Let x be an L-subset of a set S. Let (x, T) be an L-topological space. Then (x, T) is called pseudo-Hausdorff if V xa, yi C x? % ^ V, a zfiQ ^ b, 3 p, u ET such that xa C /x, yb C z/, and pCw = (p. Definition 6.9.3 Le£ <S and T 6e collections of L-subsets of a set S. Then S is said to generate T when peTif37ZCS such that p = {Jue^u. If S generates T, then S is said to be a basis for T. Definition 6.9.4 Let x be an L-subset of a set S and let S(x) = {#a|#a Q x}Let p be a function from S(x) x S(x) into the rea^ numbers. Then p is called a pseudo-metric on x if (1) V xa, Vb C x, p(xa, Vb) > 0; (2) V xa, yh C x, p(xa, Vb) =0 if and only if x = y or a = 0 or b = 0; (3) V Xa, yb £ X, P(xa, yb) = p{Vb, Xa)]

(4) V xa, yb, zc £ X, pfca, * c ) < p(xa, 2/6) + Kite, *c) w/ien b > a A c. Theorem 6.9.5 Let x be an L-subset of a set S. Let S and T be collections of L-subsets of S such that S is a basis for T. Then (x, T) is an L-topological space if and only if (1) V Xa C X, Xa C U„esV,

(2) V /x, v E 5, V x a C /x D ^, 3 7Z C 5 suc/i £/ia£ x a C U^g^C C / i D ^ . Proof. Suppose that (x, T) is an L-topological space. Let xa Q x- By (2) of Definition 6.9.1, 3 p E T such that xa Q p. Since S is a basis for T, /x = Uvznv for some TZC S. Thus :ra C L^eft^- Hence (1) holds. Let /x, ^ E 5 and xaC pC\v. Then /x fl z/ E T by (4) of Definition 6.9.1. Since S is a basis for T, /x D i/ = U^-fcC for some 1ZCS. Thus x a C U^e^C Hence (2) holds. Conversely, suppose that (1) and (2) hold. Then (1) of Definition 6.9.1 holds since S is a basis for T and since is the empty union of elements from S. Let Tl C T. Then V p e 71, 3 J^ C S such that xx = U ^ i / . Hence {J^znP = U ^ ^ U„ € j- M v £ T since 5 is a basis for T. Thus (3) of Definition 6.9.1 holds. Let xa C x- By (1), x a C U„e,sZA Now \Jvesv E T since 5 is a basis for T. Hence (2) of Definition 6.9.1 holds. Let /x, v eT. Then p = U^exC and v = U7<Et77 for some I, J CS. Thus pHv = (UC) n (U7) = U U (C n 7). By (2), V xa C C H 7, ^a C Ue€,c£ C £ n 7 for some /C = /Cc,7,Xa C 5 . Thus C n 7 = U{U^€icC I * a C C n 7 } . Hence /xni/ = U U U { U ^ I x 0 e C n 7 } e T . Thus (4) of Definition 6.9.1 holds. ■ If p is an L-subset of i?, we define the L-subset 0M of i? by #M(x) = 0 if x ^ 0 and 0/z(x) = /x(0). We sometimes write 6 = 9^ when there is no chance for confusion. Theorem 6.9.6 Let p be an L-subring of R and let {pi | i E 1} be a descend­ ing chain of L-ideals of R such that pi C p and p(0) = Pi(0) V i E I. Let

237

5 = {xa + fXi | xa C //, i G / } and Zet T &e generated by S. Then (/i, T) zs an L-topological space. If Hid & = 9^ and fli € //i* = ( f l ^ / ^ ) * , tfien (n, T) zs pseudo-Hausdorff. Proof. We have that xa C x a -f ^ since (x a + ^i)(x) = a A /ij(x - x) = a A ^i(O) = a = x a (x) V i e / . Thus (1) of Theorem 6.9.5 holds. Suppose that 2C C {xa + Mi) H (?/6 + /ij). Now either /i^ C /J,J or /ij C ^i, say ^j C ^ . Then VwGi?, (^c + Mj)W

= <

cA^j(wz) (yb + /jLj)(z) A Mj(™ - z)

= =

(6 A Mj(> - 2/)) A ^ ( u > - 2) 6 A /ij-(2: - 2/) A /Zj(w - z)

<

=

bAfj,j(w-y)

(since [ij(w — z + z — y) > fij(w — z) A fij(z — y)) (2/& + Mj)M-

Thus zc + fij C yb + 11j. Also (zc + fij){w)

= < = = < =

cA[ij(wz) (xa + fJ>i)(z) Afii(w — z) (a A [ii(z - x)) A fii(w - z) a A fi{(z — x) A /J,i(w — z) a A iii{w — x) (xa +iii)(w).

Hence zc + Mj C xa + Mi • Thus z c C z c 4- Mi £ ( x a + Mi) H (2/6 + Mi)-

Hence (2) of Theorem 6.9.5 holds. Suppose now that D^ifii = 0 and Diein* — (Hie/Mi)*- Consider x a , f/^C/i with x ^ y. Then V z G / , x a C x a + ^i ^ d 2/6^2/6 + Mi- Now V z e R, ((xa + Mi) H (2/6 + Vi))(z)

= = = < = <

(*a + Mi) W A (2/6 + Mi) W (a A fii(z - x)) A (6 A ^(z - y)) a A Hi(z - x) A b A \ii(z - y) a A b A m{x — z + z — y)} a Ab A fii(x— y) fj,i(x-y).

Now fii(x — y) = 0 for some i e I else x - y G ri;6/M* = (^ieiVi)* = 0* = {0}. Thus for this z, (x a + m) H (2/6 + Mi) =•■

238 Theorem 6.9.7 Let p be a psuedo-metric on x- V xa C x, V d > 0, /e£ ^d(^a) = { 2 / 6 ^ x 1 p(xa,yb) a}. Let

B = {ud{xa) \ xaCx,d>

0}.

Let B generate C. Then (S(x), C) is a topological pseudo-Hausdorff space. Proof. Let xa E «S(x). Then xa let yh E ^d(x 0 )ni/ c (z c ). Then p(xa, yb))A(e-p(ybj zc)). Then i/m(2/6) Let wu E Vm(yb). Then u>b>a.

E Vd(xa) V d > 0. Let vd{xa), ve(zc) E B and Vb) < d and p(yb, zc) < e. Let m = (d-p(xa, C 7 ^ ( ^ ) 0 ^ ( 0 by the following argument: Now

/>(Wu, *a) < p ( ^ u , 2/6) + p(2/6, X a ) < 771 + p(y 6 , S a ) < d

since b>uAa. Thus iyu E vd{xa). Hence Vm{yb) C Ud(xa). Similarly, ism(yb) Q ve(zc). Thus (<S(x), C) is a topological space. Let x a , 2/6 E <S(x)> x ¥" V, fl ^ 0 ^ 6. Let d = p(a:0,2/5). Then d > 0. Suppose that zc E i/d/2(xa)^^d/2(Vb)Then p(x a , z c ) < d/2 and p(2/6, ^c) < d/2. Thus p(xa,yb) < p{xa, zc) + p(2/6,*c) < d/2 + d/2 = d since c > aAfc, a contradiction. Hence ^72(^0)^1^/2(2/6) = 0- Thus the desired result holds. B Theorem 6.9.8 Let p be an L-subring of R and let {pi \ i E N} be a descend­ ing chain of L-ideals of R such that pi C p, p(Q) = pi(Q) V i E N. Define p from S(p) x S(p) into the real numbers by V x a , 2/6 E S(p), p(xa, 2/6) = 2~* */ xa-yb a Ac. If b = 0, then a = c = 0. Hence we assume in the following that a A b A c > 0. Suppose that p(x a , 2/6) = 2 _ i and p{yh, zc) = 2~j, where j > i, say, and so p{ D pj. Then pi(x -y) >aAb and pj(y -z)>bAc and so pi(y — z) > b A c. Thus Pi(x-z)

= >

p^x-y + y-z) Pi{x - y) A pi{y - z)

> =

(a A 6) A (6 A c) a Ac

239 and so xa — zc C pit Thus p(xa, zc) < 2 \ Hence p(xa,zc)

< p(xa,yb)

+ p(yb,zc).

Suppose that p(xa, yh) = 0 and p(yh, zc) = 2~j. Then x = y (since a > 0 < b) and yb~ zc C pj. Thus Pj (x — z) = pj (y — z) > b A c > (a A c) A c = a A c. Hence xa — zc C /^-. Thus p(xa, zc) < 2 _ J . Hence p(xa,zc)

< p(xa,yb)

+ p(yb,zc).

Suppose that p{xa, yb) = 0 = p(?/6, zc). Then x = ?/ = z. Thus /o(x0, z c ) = 0. ■ Lemma 6.9.9 Let 71 be a collection of L-subsets of S such that V p G 7£, /x C x- £e* S(ft) = {5(/i) \ pell}. (1) If pi,..., /i n G 7£, thenS(pi) n . . . fl 5(/x„) = S(/ii H . . . n/x n ). (%) IfU^enlJ<(R) is finite, then S(U^en^) = Us^esin^ip). Proof.

(1) x a G S(/ii) fl . . . H S(pn)

<=> /ii(ar) > a, i = 1 , . . . , n <$ a <

P\{x) A ... A /xn(^) o (MI n . . . n Mn)0*0 > « <=> xa G 5(/ii n . . . n /in). (2) xa G ^(U^e^M) ^ (^fienp){^) > a ^ 3 p E 1Z such that /x(x) > a (since U/x€^/x(i?) is finite) <£> xa C p for some // G ^ ^ xfl G <S(M) for some peU^ xae Us^)es(n)S(p). ■ Theorem 6.9.10 Suppose that L is finite. (1) If {x,T) is an L-topological space, then ( <S(x)> S{T) U {0}) is a topological space. (2) If (S(x), «5(T)) 25 a topological space, then ( x , T U {(/>}) 25 an Ltopological space. Proof. (1) Let xa G S(x)- Then x a C x a ^ d so 3 p G T such that xa C /x. Hence x a G 5(/x). Now let S(ft) C S(T), where U C T. Then U^^M G T. Thus U s(/x ) 6 5(^)5(/i) = SiU^np) G 5 ( T ) by (2) of Lemma 6.9.9. Let S(*xi),..., 5(/i n ) G 5 ( T ) . Then pu ..., pn G T and so px n . . . fl/in G T. Thus

S(/*i) n . . . n 5(/in) = 5(/ii n . . . n pn) e S( T) by (1) of Lemma 6.9.9. (2) Let xa C x- Then xa G <S(x) and so 3 S(p) G <S(T) such that xa G 5(/x). Hence xa C jz. Now let 1Z C T. Then U5(M)€<s(^)5(/x) G 5 ( T ) . Thus S(\Jfl£iifi<) G tS(T) by (2) of Lemma 6.9.9. Hence \J^-R,p G T. Let pi,...,pne T. Then S(JUI), . . . ,5(/x n ) G 5 ( T ) . Thus

5(/ii n . . . n pn) = 5(/ii) n . . . n 5(/in) G S(T) by (1) of Lemma 6.9.9. Hence p\ f l . . . fl pn G T. ■

240

6.10

Complete L-Subrings

Once again we assume that L is a chain. Definition 6.10.1 Let p be an L-subring of R and let {(xn)an \ n EN} be a sequence of L-singletons such that (xn)an C p and an > 0 V n G N. Let p be a pseudo-metric onS(p). Then {{xn)an\n G N} is called a Cauchy sequence if V d > 0, 3 k> 0 such thatV n, m>k, p((rr n )a n , (ar m )a m ) < d. If3xae S(p), a > 0, such that V d > 0 , 3 / c > 0 such that V n > f c , p((xn)an, xa) < d, then {(xn)an \ n £ N} is said to converge to xa. If every such Cauchy sequence converges in S(p), then p (or S(p)) is said to be complete with respect to p. Let p be an L-subring of R and let {pi \ i G N} be a collection of L-ideals of R such that p D pi D A*;+i a n d ^(0) = A^(0)> i = 1,2, Suppose that HizNPi = 9^ and Di^u* = (^ieNf^i)*- Let p be the pseudo-metric defined in Theorem 6.9.8. Let p* be the metric on R defined by V x, y G R, p*(x, y) = 2~l if x — y E p*, x — y £ /x*+1 for i = 0 , 1 , . . . , where p0 = p and ^ZQ = -R> P*( x >

y ) = 0 i f x - y e nieNM* (= M ) . In the remainder of this section, the previous paragraph is assumed. We use the equivalences p(xa, yb) < 2~2 O xa — yb Q Hi O Hi{x — y) > a Ab throughout. P r o p o s i t i o n 6.10.2 Suppose that an > 0 V n G N. {{xn)an\n G N} is a Cauchy sequence in S(p) (w.r.t. p) if and only if {xn \ n G N} is a Cauchy sequence in R (w.r.t. /9*) and V z G N , 3 k > 0 such that V n, ra > A;, either Pi(xn - Xm) > an A a m or pi(xn — xm) = 0. Proof. Suppose that {(xn)an \ n G N} is Cauchy. Let i G N. Then 3 k > 0 such that V n, ra > /c, (xn)an - (xTn)am C ^ , i.e., Pi(xn - xm) > an A a m > 0. Thus x n — x m G /x* and so {xn \ n G N} is Cauchy. Conversely, let i e N . Then 3 ki > 0 such that V n, ra > fci, x n - x m G //*. Also 3 Ar2 > &i such that V n, ra > /c2, A^i(xn — Xm) > a n A a m since ^ ( x n — xm) > 0. Then V n, ra > k2, \Xn)an

— \Xm)am

£ Mi- ™

Suppose that pi = // on /x* V z G N. Then if ( x n ) 0 n , (x m )a m C ^, we have that (xn)an -{xm) a r n C /x and so either ( z n ) 0 n - ( x m ) a m C p{ or ^»(x n - x m ) = 0. That is, the condition in Proposition 6.10.2 holds. We note this is the case in the crisp case since pi(x) = 1 = p{x) V x G p*. P r o p o s i t i o n 6.10.3 Suppose that an > 0 V n G N. T/ie sequence {(xn)an \ n G N} converges to xa in S(p), a > 0, z/ and on/y z/ {x n | n G N} converges to x in R and V i G N, 3 A: > 0 suc/i £/ia£ V n>k, either pi(xn - x) > anA a or pi(xn — x) = 0.

241

Proof. Suppose that {{xn)an | n G N} converges to xa. Let i G N. Then 3 k > 0 such that V n > f c , (zn)a n - xa C ^ , i.e., /x;(xn - x) > a n A a > 0. Thus xn — x e p* and so {x n | n G N} converges to x. Conversely, let z £ N. Then 3 ki > 0 such that V n > /cl5 Also 3 k2 > ki such that V n > fe, M;(xn — x) > a n A a since /i;(x n — x) > 0. Then V n>k2, (xn)an ~ xa £ Pi- Thus {(x n ) a r i | n G N} converges to xa. ■ Corollary 6.10.4 Suppose thatan > 0 V n G N. If {{xn)an to xa and 7/5, a > 0 < 6, £/ien x = y. Proof.

\ n G N} converges

{x n I n £ N} converges to x and ?/. ■

Corollary 6.10.5 Suppose thatan > 0 V n G N. J / { ( x n ) a n | n G N} converges to xa, a > 0, £/ien {(x n ) a n | n G N} converges to Xb^ b such that 0 < 6 < a. Proof.

/^(x n — x) > an A a > a n A 6. ■

Corollary 6.10.6 (%) / / / i is complete (w.r.t. p), ^/zen p* is complete (w.r.t. ?)■ (2) If ^ = p on p\ V i G N and xx* is complete (w.r.t. p*), then /i is complete (w.r.t. p). Proof. Suppose that m = p on /x* V i G N. Suppose that (x n ) a r x , x a C p. Then (x n ) a r i — x a C p. Hence either ( x n ) a n — xa C p± or /LXi(xn — x) = 0 . Thus the desired result follows from Proposition 6.10.3 B We now illustrate some differences in the fuzzy and crisp cases. We sometimes write (i?, M) when R is a complete local ring with unique maximal ideal M. Example 6.10.7 Let L — [0,1]. Let (R, M) be any complete local ring such that Ml D M i + 1 V i G N. Let p = 1R. Let x* G Ml\Mi+1 V i G N. Then {x* | i G N} is a Cauchy sequence which converges to 0. (1) Let a > 0. Let {an \ an G (0, 1], n G N} be a sequence which converges to 0. Define the L-subset pi of R by pi(0) = 1, pi(x) = 1 - j if x G M J ' \ M J + 1 , j > z, and /ii(x) = 0 z/x G R\Ml V z G N. 77ien /i; zs an L-ideal of R and

{0} c (n/i,)* c n/x* = nM i = {0}. Let i > 0. T/ien 3 /c s?xc/i £/W V n > k, Pi(xn - 0) > 1

> an A a.

242

Thus {(xn)an I n G N} converges to Oa for any a G (0,1]. (2) Define the L-subset ^ of R by ^ ( 0 ) = 1, m(x) = \ if x G Af*\{0}, ^ ( x ) = 0 z/x G R\Ml V 2 G N. T/ien ^ zs an L-ideal of R and as above

(n^y = nrf = {0}. T/ie sequence of L-singletons {{xn)\ \ n G N} converges to Ox since /j,i(xi—0) > 1 A | V i G N. However, the sequence {(xn)i\ n G N} is not Cauchy since V i, n, ra G N, n ^ ra, (xn)i - (x m )i g /x». ■ Proposition 6.10.8 Suppose that an > 0 V n G N. Suppose that fii = fi on /i* V i G N. Suppose that {(xn)an | n G N} is sizc/i £/ia£ {xn \ n G N} converges to x. Suppose that A{fi(x) | x G //*} > 0. T/ien 3 a > 0 s^c/i that {(xn)an \ n G N} converges to xa. Proof. Let a = A{^(z) | x G //*}. Then a > 0. Let i G N. Then 3 A: > 0 such that V n > k, xn — x G //* C /z*. Thus Hi{xn — x) = /i(xn — x) > a > an A a. The desired result now follows from Proposition 6.10.3 ■ Corollary 6.10.9 Let \i be an L-ideal of R. Then (/x(i))* = (/x*)i V i G N. Proof.

^ i + 1 ) = /^)/x and ( ^ ) * = / / > * . ■

Proposition 6.10.10 Let M be a maximal ideal of R. Let \i be an L-subring of R such that /x* = R. Iffi is complete with respect to ( ^ H I M ) ^ , i = 1? 2 , . . . , then R is complete with respect to M 2 , i — 1 , 2 , . . . . Proof.

By Corollary 6.10.6, R is complete with respect to

((/xniM)(i)r = ((MniM)7 = M' V * € N. ■ 6.11

L-Coefficient Fields

Suppose that R has an identity, M is a maximal ideal of i?, and that L is a chain. We call an L-subring /x of i? an L-subfield in R if /i* is a field containing the identity of R and V x e R such that a: is a unit, /J,(X) = /x(x - 1 ). Proposition 6.11.1 Let f be a homomorphism of R into a ring R!'. Let \ £ be L-subsets of R. Then

(i) fix)* = fix*); (2) f(x) = /(C) implies fix*) =

f{C).

^n^

243

Proof. (1) y e / ( x ) * O f(X)(y) > 0 & W{X(x) \ f(x) = y} > 0 <=> 3 x e X* such that f(x) =y &y e /(%*). (2) f(x) = /(C) implies / ( * ) * = /(C)* which by (1) yields the desired result. ■ Let 5 be a subset of R and let /i be an L-subset of R. We define the Lsubset n\s of R by (n\s)(x) = /z(x) if x e S and (n\s)(x) = 0 if x $ S. If S is a subring (ideal) of i?, then /z| s is an L-subring (ideal) of R. Let /z be an L-subring of R such that fi* = R. If v is an L-subfield in R and 7 is an L-ideal of R such that /z = v + 7 and i / f l ) ^ ^ , then z/ is called an L-coefficient field of fj, with respect to 7. If z/ is an L-coefficient field of fj, w.r.t. 7, then /1* = 1/* + 7*, z/* is a field, 7* is an ideal of R, and fi = /x|^« + /i| 7 *. Suppose that F is a field in R containing the identity of R. If ^ = / i | F + H M , then /z| F H A*|M C 0 and i j = F + M : (/z|F D zz|M)* = ( H F ) * n (/z| M )* = F H M = {0} = 0*and R = fi* = (IJL\F + /x| M )* - (/z| F )* + (/z| M )* = F + M. If /z| F is an L-subfield in zz, then zz|F is an L-coefficient field of fi with respect to

/J,\M-

We see that a necessary condition for fi to have an L-coefficient field is for /z* to have one. We now examine to what extent a coefficient field for zz* can be used to yield an L-coefficient field for /z. Theorem 6.11.2 Suppose that F is a coefficient field of R w.r.t. M. Let /z be an L-subring of R such that /z* = R. If (fj,\F)(R) n (/J,\M)(R) Q {0,1}, then

Proof. Let x e R. Then ( / Z | F + / Z | M ) ( # ) = V{(/i|ir)(c)A(/i|M)(™) I x = c+m, c e F, m e M) = 11(c) A fi(m) since (/Z|F)* = F, (//|M )* = -&f, and x has a unique expression as a sum of an element c from F and an element m from M. Now /z(x) = /JL(C + m ) > /x(c) A zz(ra). Since ( H F ) O R ) H (/z| M )(i?) £ {0,1}, we either have fi(x) = 1 = fi(c) A /z(ra) or /z(c) ^ /i(m). We have in either case that fi(x) = fi(c) A /z(ra). Thus /i = /zl^ + /I\M- B Example 6.11.3 Lei: L = [0,1]. R is a complete local ring with unique maxi­ mal ideal M and /z is a L-subring of R such that /i* = R. ]3 a coefficient field F of R such that zz = /Z|F + /Z|M • Let R denote the tensor product Kp % ^ p > * where K = P(x), P is a perfect field of characteristic p > 0, and x is transcen­ dental over P. Then M = (xp ® 1 — 1 ® xp \ is the unique maximal ideal of R. In fact, R is a complete local ring since M is nilpotent. Now R has a coefficient field F = Kp (g) 1. Define the L-subset fi of R as follows: /z(z) = 1 if z e 1<8> Kp~l and /JL(Z) = \ if z G R\l ®KP~ . Then \x is an L-subring of R.

244

Now R = F + M, y\F is \ on F\K and 0 on R\F, / i | M is \ on M\{0} and 0 on R\M. Hence MIF + MIM is less than 1 on \®KP \K. Thus fi ^ /XIF + H M We now show that there does not exist any coefficient field F of R such that /J, = II\F + LL\M- For any such F, F n ( l 0 K p _ 1 ) = K else F D l ^ F " 1 since [Kp~l : K}=p and so 3 z G F such that zp = 1 ® xp~l. Thus (xp~2 0 1 - z) p2 = 0 ^ (xp~2 0 1 - z) p . However, this is impossible since Mp = (0). Hence JI\F is ^ on F\K and as —2

—1

above \i ^ y\p + /i|M- (We note K ( x p 0 1 — 1 0 x p ) is o coefficient field of R and so coefficient fields of R are not unique.) Lemma 6.11.4 Let /i and v be L-subrings of R. Then (1) for all a G L, /i a + va C (/i + i/) 0 ; f£,) z/ /i and z/ /lave t/ie snp property, then V a G L, /i a + va = (/i + ^) a Proof. x G /i a + Va <=> 3 ?/ G /xa, ^ G z^a such that x = y + z => W{fi(y) A v(z) \ x = y + z} > a <£> (/i + ^)(#) > o, <5> x G (/i + v)a- If M a n d v have the sup property, then the implication in the previous sentence becomes an equivalence. ■ Proposition 6.11.5 Let /i be an L-subring of R. Let S and T be subrings of R. (1) If jia = (n\s)a + 0*|r)a V a£ A W> then fj, = fi\s + n\T(2) Suppose that \i has the sup property. If \i — \±\s + /i|r> then /ia = (Mls)a + (fi\T)a Vae L\{0}. Proof.

(1) By Lemma 6.11.4, Ma 2 (Mis + A*|r)a 2 (Mls)o + (MITJO = Ma

and so /i a = (fi\s + ^|T)O V a G A W - Thus fi = JA\S + /i|r(2) Since /i has the sup property, fi\s and fi\r have the sup property. Thus by Lemma 6.11.4, (/i|s)a + (Hr)a = (M|S + H r ) a V a E A W - Hence /i a = M s + M|T)« = Ms)a + (Mlr)a V a G A W - ■ Lemma 6.11.6 Let fj, be an L-subring of R. Let S and T be subrings of R. Let g be a homomorphism of R into a ring Rf. Then g(ii\s+H\T) = <7(MIS)+#(M|T)-

245

Proof.

Let y € g(R). Then

gfals + »\T)(y)

= = =

V{(v.\s + l*\T)(x)\g(x)=y} V{V{(/x|5)(a:i) A (v\T)(x2) \ x = xx + x2}\g(x) = y} V{V{(Hs)(ari) A (H\T)(X2) \ g{xx) = yu g(x2) = y2} 127 = 2/1+2/2} V{(V{(M| s )(xi)|p(a;i)=yi})A

=

( V { ( M I T ) ( Z 2 ) I #(z 2 ) = 2/2}) I 2/ = 2/1 + 2/2}

=

V{#(/z|s)(2/i) A ^(/X|T)(2/2) I 2/ = 2/1 + 2/2}

=

(<7(/x|s)+p(Hr))(2/)

since p(x) = #(>i + x 2 ) = g(x{) + ^(x 2 ) =2/1+2/2=2/- The case for y G # ' , 2/ ^ #(-R) is trivial. ■ Let I be an ideal of R and g the natural homomorphism of R onto R/I. Let 5 be a subring of R. Then i? = S + J <^>
g{v\s + v\i) =g(v\s)Proof. By Lemma 6.11.6, it suffices to show g(ii\i) C 0M. Suppose that 2/ e R/I, y ^ 0. Then g(^i\i)(y) = V{(A*|/)(ar) | g{x) = y} = 0 since g{x) = y implies x £ I. ■ Corollary 6.11.8 Let /x be an L-subring of R. Let S be a subring of R and I an ideal of R. Let g be the natural homomorphism of R onto R/I. If n = / i | s + / i | / , iheng(fi) = g(p\s)Proof.

g(fj) = g(fi\s + fi\i) = g(p\s) by Corollary 6.11.7. ■

Theorem 6.11.9 Let fj, be an L-subring of R such that /z* = R. Let S be a subring of R and I an ideal of R. Let g be the natural homomorphism of R onto R/I. Suppose that R = S + 1 and S D I = {0}. / / A{xi(m) I m e 1} > V{/x(c) | c e 5 } , then g(fj) = g(n\s) implies /z = fi\s + / i | j . Proof.

Let y G R/I. Then 9(t*)(y)

= =

V{Kx)\g(x)=y} V{/d(c + m) I g(c + m) = y,ce S,m 6 1}

246

gMs)(y)

= =

=

V{(t*\s)(c)\g(c)=y} (»\s)(c)

Mc)

since 3 unique c G S such that g{c) = y. That is, g(fj)(y) = V{y/(c + m) \ m G I}, g{c) = y; g{u\s)(y) = Mc)> g(c) = y. Now /i(c + m)

>

11(c) A fi(rn)

=

(/x|s)(c)A(H/)M

=

(^| 5 + M|/)(c + m),

where this latter equahty holds since (/i|s)* = 5, (fi>\i)* = I, and c -h m is an unique expression as an element from S plus an element from / . By hypothesis, fi(c) = //(c) A /x(m). Thus if g(n) = g(^\s), then

Kc) > Kc + m ) > (Hs + A*|/)(c + m ) = Mc) and so fi(c -\-m)

= (fi\s

+ M|/)(C + m).

■

The following example illustrates an interesting complication in the fuzzification of the concept of a coefficient field. E x a m p l e 6.11.10 Let L = [0,1]. fi is an L-subring of R, u* = R, F is a coefficient field of R w.r.t. M, fi = fx\jp + A*|M, and ( H F ) H ( H M ) = #/x, but }I\F is not an L-subfield in R and /JL\M is not an L-ideal of R : Let R denote the polynomial ring Q[x], where Q is the field of rationals. Let F = Q and M = (x). Clearly, F is a coefficient field of R w.r.t. M. Define the L-subset fi ofR by fi(r) = 1 ifre Z, fi(r) = § ifx G Z[x]\Z, and /x(r) = § if x G R\Z[x]. Then fi is an L-subring of R such that fi* = R. Now (/x|/r)(r) = 1 if r G Z, (/x| F )(r) = i if r G Q \ Z, and (fJ,\F)(r) = 0 if r e R\Q. Thus fi\F is an L-subring of R, but not an L-subfield in R. Now

b*\M)(r)

1 I2 I 0

ifr = 0 ifr G {xr(x) \ r(x) G Z[x]}\{0} ifr G {xq(x) \ q(x) G Q[ar]\Z[x]} otherwise.

Then fi\M is an L-subring of R, but not an L-ideal of R. Let r G R. Then r has a unique expression, r = c + ra, where c G F and m G M. Thus 2 u(r)

ifceZ,

= u(c + m) > /J,(C) A fi(m)

2

= 1A - = - =

n(r)

mexZ[x]\{0}, /x(r) = u(c + m) > fi(c) A /j,(m) = l A - = - = ^(r)

247

if c G Z, m G

xQ(x)\xZ[x],

1 2 1 fi(r) = /x(c + m) > /x(c) A ;u(ra) = - A - = - = /x(r) i / c G Q\Z, m G xZ[x]\{0}, /z(r) .= fi(c + m) > /x(c) A /x(m) = - A - = - = fi{r) ifce

Q\Z, m G xQ[x]\xZ[x], /x(r) = /x(c + m ) > /x(c) A ^ ( m ) = 1 A 1 = 1 = jiz(r)

i / c G Z , m = 0, /x(r) = /x(c + m) > /x(c) A /x(m) = - A 1 = - = /x(r) ifceQ\Z, ra = 0. Thus fjb(r) — n(c) A fi(m). Now (V\F + V>\M)(T)

=

(V>\F + H\M)(C + ™)

= =

V{Ox|F)(u) A ( M | M ) ( ^ ) I r = u + v} V{(/xjjr)(u) A (M|M)(V) | r = U + V,

= =

(MIF)(C) A ( H M ) M n(c) A/z(m).

77ms /x = /x| F + /X|M. Clearly, (/i| F ) D Qz|M) = #• VKe aZso noie £/m£ ^(/i) zs not an L-sub field of R/M, where g is the natural homomorphism of R onto R/M. We may identify R/M with Q. Now g(/j,)(z) = \/{fJi(y)\ | g{y) = z}. If g{y) = z, then y = z + xr{x) for r{x) G Q[x]. IfzeZ, then g(fi)(z) = 1. If z e Q\Z, then y G Q[x}\ Z[x] and so g{^){z) = ±. Hence g(/j,) is not an L-subfield in R/M. Theorem 6.11.11 Suppose that F is a coefficient field of R w.r.t. M. Let fi be an L-ideal of R such that /i* = R. Then [JL\F is an L-subfield in R and /X|M is an L-ideal of R. Proof. Now ( / / | F ) * = F and IAF is a constant on -F\{0} by (1) of Propo­ sition 6.5.4. Also, H\F is 0 on R\F. Thus ji{x) = /x(x _1 ) for every unit x G R. Hence p\p is an L-subfield in R. We show that /X|M is an L-ideal of R by show­ ing that all the level subsets of /JL\M are ideals of R. Let a G (/X|M)(-R)> a > 0. Let x G fianM. Then OX|M )(#) — M( X ) since x G M. Thus x G (/i|M)a- Hence

248

/ i a H M C (/i| M )a- Clearly, (//| M )a Q Va H M since a > 0 and (/i| M )* = Af. Thus (fJ>\M)a = Ha^M which is an ideal of R. ■ Let fi, be an L-subring of R. By Proposition 6.5.4, V y e R, y a. unit, and V x G R, fi(xy~l) > fjb(x) A fi(y) if and only if fi(y) = ^(y-1). In either case, ^(1) > >L/,(y) V y E R, y & unit. Let i? be a quasi-local ring and let F be a field such that F C R and the identities of F and R coincide. Let /J, be quasi-local L-subring of R. Then /Z|F is an L-subfield in R. Proposition 6.11.12 Suppose that F is a coefficient field of R with respect to M. Let fi be an L-subring of R such that /x* = R. If K is a subfield of F such that F/K is algebraic and fi D 1\KI then fi\p is an L-subfield in R. Proof. For all a e L, F D ( ^ | F ) 0 2 K- Since (II\F)CL is a subring of F containing K and F/K is algebraic, (/z|ir)a is a field. That is, all the level subsets of fi\p are fields. Thus ii\p is an L-subfield of F and so (/J,\F)(C) = (^| i r)(c~ 1 ) V c 6 F, c ^ 0. If u e R is a unit and w ^ F, then (fi\F)(u) — 0 = (/i|^)(i/ - 1 ). Thus /LA|J? is an L-subfield in R. ■

6.12

E x i s t e n c e of L-Coefficient Fields

Suppose that L is a chain. The purpose of this section is to continue the examination of the existence of L-coefficient fields in L-subrings because of the structural properties they give to L-subrings. For example, in the crisp case, a complete regular local equi-characteristic ring is isomorphic to a formal power series ring. We give results which yield the existence of coefficient fields in an L-subring. We show how the existence of an L-coefficient field in an L-subring yields structure results for the L-subring. Throughout this section, R is assumed to have prime characteristic p and as usual M is a maximal ideal of R. It is shown in Example 6.11.10 that R may have a coefficient field with respect to M and // = / / | F + A*|M» v e t A*IF need not be an L-subfield in R and H\M may not be an L-ideal of R. In the following two theorems, we assume that R contains a field K with the same identity as R. We identify g(K) with K in R/M. Theorem 6.12.1 Suppose that M is nil and that R contains a field K. If R/M has separating transcendence basis over K, then R has a coefficient field which contains K. If R/M is separable algebraic over K, then R has a unique coefficient field containing K.

249 Proof. The first part of the result follows from [181, Theorem 2, p. 50]. Suppose that R/M is separable algebraic over K. Let F\ and F2 be coefficient fields of R which contain K. Let 0i E F\ and 02 E F2 be such that g(9i) = g{62). Then 0i - 02 E M. Since M is nil, 3 a positive integer e such that (0X - 02)pC = 0, where we recall that p is the characteristic of R. Hence 0 ^ = 0jf. Thus tf(0f) = ^ (^f) and so if (0X) = K (0 2 ). Since ^ | K ( ^ ) is one-to-one, 0X = 0 2 . Hence Fi = F2. ■ Lemma 6.12.2 Suppose that M is nil and R has a coefficient field F. Let fi be an L-subring of R such that V a, b E /x(iJ) ti^ft a > 6,

( ^ n F ) p n ( / i f l n F ) = (MflnF)p. T/ien /x = /X|F + M|MProof. Let c £ F. Suppose 3 a positive integer e such that n(cpe) > /x(c). Let /x(c) = b and /x(cp ) = a. Then c ^ naC\ F and c p E jia H JF. Thus since c E /i& PI F, we have that (/^ fi F)p fl (/i a f l F j D (/xtt D .F) p , a contradiction. Since n(cpe) > /x(c) in general, we have that fJ,(cpe) = /x(c). Let m e M. There exists a positive integer e such that mp = 0. Thus fi(c) = fi(cpG) = jz((c + m)pe) > fi(c + m). Also /i(ra) = ^ ( - c + c + m) > fJi(—c) A /i(c + m) = /x(c) A ^x(c + m) = /x(c + ra). Hence JU(C), /i(m) > /x(c + m) > /i(c) A /z(ra).

Thus yix(c + m)

=

/x(c) A /i(m)

= (HF)(C) A(/x| M )M =

=

V

{ ( H F ) ( C 7 ) A(/z| M )(ra')|c + ra = c' + m ' , c '

G

^

m

'

G

M

l

(since c + m has a unique expression as an element from F plus an element from M) (HF + HAfXc + m).

Hence /x = fi\p + M|M- B Theorem 6.12.3 Suppose that M is nil, R contains a field K, and R/M is separable algebraic over K. Let fi be an L-subring of R such that /x D IK- Then H = n\F + /j,\M and fi\p is an L-subfield in R, where F is the coefficient field containing K.

250

Proof. That F exists follows from Theorem 6.12.1 and that fi\p is an Lsubfield in R follows from Proposition 6.11.12. Now V a, b G fi(R) with a > 6, (fib fl F)p D (/j,a C\F) = (fia n F)p since JJL^IK and i?/M is separable algebraic over if. Thus the desired result follows from Lemma 6.12.2. ■ We use the notation n 3 to signify that a local ring is not necessarily Noetherian. Theorem 6.12.4 Suppose that (R, M) is a complete local ring (n 3 ) and R' is a subring of R such that M' = R' D M, where M' is a maximal ideal of R!. If R/M is separable over R'/M', then every coefficient field of Rf with respect to M' is extendable to one of R. Proof. If Rf has no coefficient fields with respect to M', then the result holds vacuously. Suppose that R' has a coefficient field K w.r.t. M'. Let G be a p-basis of K. Let Go = g(G), KQ = g(K), and Fo = R/M. Let BQ be a relative p-basis of FQ/KQ. Then Go U BQ is a p-basis of Fo since FQ/KQ is separable. Let B C R be a set of representatives of JE?O, i.e., g maps B one-to-one onto Bo- Now Rf [G, B] D KP3 [G] = K, j = 1,2,.... Let F = Hf^c^Rf [67, B]), where cl denotes closure in R with respect to the M-adic topology. Then F is a coefficient field of R and F D K. ■ Lemma 6.12.5 Suppose that R has a coefficient field F with respect to M such that fia = (fjLa fl F) + (fia H M) V a G ^(R). Then fx = /i-|jr + /X|MProof. Let c e F and m e M. Then as in the proof of Lemma 6.12.2, (II\F + M|M)(C + m) = /x(c) A fi(m). Suppose that /x(c + m) = a. Then c-f- m = d + m!, where c' G fia fl F and m7 G /za D M by hypothesis. Thus c = c' and m = m! and so //(c), /x(m) > a. Not both /x(c) > a and /i(m) > a else c + m G /xt for some b G /i(i?) such that b > a since /z(c + m)> /x(c) A /x(m) in general. Hence either /z(c) = a and /x(m) > a or //(c) > a and //(m) = a. Thus /x(c) A /i(m) = a. Hence \i = JI\F + /X|M- B Theorem 6.12.6 Suppose that /J, has the sup property. Suppose further that (iJ, M) anrf (/za, M fl /xa) are complete local rings (n 3 ) V a G /x(ii). If R/M is separable over /x a /(M fl /za) V a G //(i?), then R has a coefficient field F such that fi = fi\F + II\M> Proof. Let ao denote the maximal element in /^(R). Let Fao denote a coefficient field of fiao. Let b,a £ /i(R) be such that b is the immediate successor of a under > (a > b). Let Fa be a coefficient field of \xa. Now [ibj{M fl fib) is separable over fia/(M D /z a ). By Theorem 6.12.4, 3 a coefficient field Fb of /i& such that F a C F^. Consider r G /x(i?), where r has no immediate predecessor

251 under > and in fi(R) (the limiting case). Suppose we have a collection of fields {Fa\a G fjL(R), a > r} such that V 6, a G fx(iJ), a > b > r, that ft, Fh are coemcient fields of /z a , /z6, respectively, with F a C ft. Let ft = U a > r f t . Since Mr = U a>T /x a and

M n /xT = M n u a>T ^a = u a > r ( M n /xa), ft is a coemcient field of / i r . Hence by transfinite induction, we have a col­ lection of fields {Fa\a G fi{R)} in R such that V b, a e fi(R) with a > b that ft, ft are coemcient fields of \xa, //&, respectively, such that ft C ft. Let F = U aG/z ( i? )ft. Then F is a coemcient field of i? = Uaefl(R)/ia and ft = Ffl/i a V a G /z(#). Hence /i a = F n ^ a + M n / i a V a G fi(R). Thus by Lemma 6.12.5, ^ = M|F + / J | M . ■ HR/M is separable algebraic over / i a o / ( M n/i a o ) in the above theorem, then i?/M is separable over fxa/(Mr\fia) V a e n(R). If ^ is quasi-local in Theorems 6.12.6 and 6.12.7, then fi\p is an L-coefficient field of fi by comments preceding Proposition 6.11.12. Theorem 6.12.7 Let fj, be an L-subring of R. Suppose that fi(R) = {ao, a i , . . . , a n } , where ao > ai > . . . > a n . Suppose further that (R, M) and (/xa, M Pi /i a ) are complete local rings \f a E l-i(R) such that RP C fiao. Then R has a coefficient field F such that /x = /JL\F + M|M> where /JL\F is an L-subfield in R. Proof. Let W» = ^ai/{M n /i a i ), * = 0 , 1 , . . . ,n. Let C* be a relative pbasis of Wi/Wi-i, i = 1 , . . . , n. Then Cf is a relative p-basis of Wf /Wf^, i = 1 , . . . , n. Let G be a relative p-basis of W0/W%. Then C n U . . . U d U G is a p-basis of Wn = R/M and, in general, d U ... U d U G U C£ U ... U Cf +1 is a p-basis of W^, i = 0 , 1 , . . . , n. Let Bi and iJ be sets of representatives in R of d and G, respectively, i = 1 , . . . , n. Let Fi = rif^dirf

[Bu . . . , S i , H, Bl,...,

B? +1 ]),

z = 0 , 1 , . . . , n. Then ft is a coefficient field of /xa., z = 0 , . . . , n and ft 2 z = 1 , . . . ,n. Hence ^ = ( F n / x a J + (Mn/xaJ,

ft_i,

i = 0 , 1 , . . . , n, where F = ft. Thus ix = fi\F + H M by Lemma 6.12.5. Now F0 = (fi\F)a0 C . . . C (/x| F ) an = ft = F and Fp C ft. Hence F/ftj is algebraic and so 0 , 1 , . . . , n. Thus /x|i? is an L-subfield in R. ■

(JJLIF)^

is a field for i =

252

6.13

Structure Results

Let F/K be a field extension, let L b e a chain, and let £, 7 b e L-subfields of F such t h a t 7, C 2 I K - If C a n d 7 are linearly disjoint over 1 ^ , t h e n we write £ 7 = £ (g) 7, where £7 is t h e composite of £ a n d 7. T h e following two results give t h e s t r u c t u r e of the L-field extension /JL\F over /J,\K, where K C ji^y P r o p o s i t i o n 6 . 1 3 . 1 Let F/K be a field extension. Suppose that /z is an Lsubfield of F such that /z 2 IK and that /i is finite-valued, say /x(i?)\{0} = {ao, a i , . . . , a n } , w/iere a o > a i > . . . > a n . 1 / 3 an intermediate field Hi of\iajK such that /j,ai = /x ai _i 0 K -Hi for i — !>•••> n> ^ e n 3 L-subfields fii of F , z = 0 , 1 , . . . , n, sizc/i £/ia£ /z = /io 0 yL^i 0 . . . 0 /z n fewer 1/c), i f C /z ao = ^Q and /xj = # i , i = l , . . . , n . Proof.

Since /z a i = /xa*_i 0/<: fli for z = 1 , . . . , n, clearly M* = V>an = Ma0 0 X # 1 ®K • • • 0 K # n -

Define the L-subset ^ of F by /Zi(x) = 1 if x G K, /z;(x) = a^ if x G Hi\K, and /z;(x) = 0 otherwise, z = 0 , 1 , . . . , n , where # 0 = AW T h e n fii is an L-subfield of F, \±i D \K, a n d /z* = Hi, i = 0 , 1 , . . . , n. T h u s /i* = /XQ 0 K • • • ®K Mn a n ( ^ fiid fij = IK for i, J = 0 , 1 , . . . , n ; z ^ j . T h u s it follows t h a t the composite /io . . . // n = ^ 0 0 - • .0Mn- Clearly, /z D /z 0 • •. Mn- Let x 6 //*. Now (/z0 . . . Mn)* = fiQ . . . /z*. Suppose t h a t /z(x) = a;. T h e n x G /zai\A*ai_i • Now (/i 0 . . . /X n )fa)

= =

(^0 . • • A**)(A**+1 • • • Mn)0*0 V{(/i 0 • • • m){xi) A . . . A (/i 0 . • . /Z*)(x r ) A ( / x i + i . . . n„)(yi) A . . . A ( / z i + 1 . . . fin)(yr)}\ x = E j = i xjVjiXj e ( / i 0 . . . /x»)*, % e ( / i i + i . . . / i n ) * , jf' = l , . . . , r ; r G N } .

(6.1)

Now (/i 0 • • • / ^ ) * = f4 • • • /** = HQ ®K . . . 0 K Hi = / i a . and ( / i i + i . . . / i n ) * = / i * + 1 . . . /x* = # ; + i 0 x - . - % # n Since x G M a A / V - i , 3 a representation of x, x = J ] j = i xj2/j> s u c n t n a t % £ # and Xj e fia., j = l,...,r. For this representation, we obtain for the minimum in (6.1) above (/x0 • •. fii)(xi)

A . . . A (/i 0 • • • fJ>i)(xr)

which is > ai since / V = H0 0 X Hi 0 ^ . . . 0

K

fli

253 and Hj C \xa., j = 0, 1,...,i. Thus At < (MO • • • t*n)(x)

< fl(x) = CLi.

Hence \i = /xo . . . /xn • B Theorem 6.13.2 Let F / i ^ be a /le/d extension with characteristic p > 0. Suppose that Fp C if. Let /x be an L-subfield of F such that /x* = F and /x D IK- Suppose that /x is finite-valued, say /x(i?)\{0} = {a 0 , a i , . . . , a n } 5 where a$ > a\ > . . . > an. Then 3 L-subfields \±i of F, i = 0, 1 , . . . , n, swcft t/iat /i = /io ® Mi ® • • • ® Mn- Furthermore, V i = 0 , 1 , . . . , n, 3 set X{ of Lsubfields of F such that V v 6 Xj, z/ 3 liC> v*/K is simple, and [i% = ^v^Ziv (over IK)Proof. We have that u* = /xan 2 Ma0 2 -K" and /x*p C if. Let Bi be a relative p-basis of / x ^ / z i ^ ^ , B$ a relative p-basis of /x ao /if, and iLj = K(Bi), i — 0 , 1 , . . . , n. Then jiai = HCH-^K Hi, i = 1 , . . . , n, since /x*p C if. Let the L-subfields /x; of F be defined as in Proposition 6.13.1, i = 0, 1 , . . . , n. Then /JL = /io ... ®/xn. Since iff C K, Hi = ( g ^ ^ i f (&i), i = 0 , 1 , . . . ,n. Define the L-subsets i/^ of F, 6; E Bi, i = 0 , 1 , . . . , n, by vn^x) = Hi(x) if x G K(bi) and ^ ( x ) = 0 otherwise. Then i/^ is an L-subfield of F, ( ^ J * = K(bi), and /x* = <S)bieBiK(bi), i = 0 , 1 , . . . ,n. Now /x; = ^bieB^ibi since /i; is constant on ifi\if, z = 0 , l , . . . , n . ■ Theorem 6.13.2 says that /x is the tensor product over IK of L-subfields v of F whose supports are simple field extensions of K. In fact, for i/* = K(b), it follows that z/ = l x ( ^ ^ ) , i.e., z/ itself is a simple L-field extension of 1 ^ . We apply Theorem 6.13.2 to Theorem 6.12.7 as follows: In Theorem 6.12.7, 3 coefficient fields F of R and K of /xao such that F ^ K ^> Fp, where /x is an L-subring of R such that /x D lK. Then Theorem 6.13.2 gives a structure result for JI\F over I K In the remainder of this section, S denotes a commutative ring with identity. Definition 6.13.3 Let R = 5 [ # i , . . . , xn] be a polynomial ring in the indeterminates x\,..., xn over S. Let /x be an L-subring of R. If mn

in=0

where

mi

21=0

Cilmmmin e 5 ,

l_i(r) = A{A{/x(cil...in) | ij = 0 , l , . . . , m i ; j = 1 , . . . , n } , A{a i | j = l , . . . , n } } , t/ien /x is called a polynomial L-subring of R, where aj — /X(XJ) if Xj appears nontrivially in the above representation of r and aj = 1 otherwise, j = 1,..., n.

254

Definition 6.13.4 Let R = S[[xi,..., xn]] be a power series ring in the indeterminates X\,..., xp over S. Let \i be an L-subring of R. If V r = ZTn=o "" E™=o Cii-inXl1 • • •
Proof. Suppose that /z is a polynomial L-subring of R. Then V r G i?, r / 0 , where r is written as in Definition 6.13.3, ^(r) = AJAJ/^c^ A ) | ^ = 0, 1 , . . . ,rrij; j = 1 , . . . , n}, A{a i |i = 1 , . . . , n}} = /i(co...o)Aju(r-co...0) = r MIS(CO...O)A/X|M(7—co...o). Hence /* = ^ | S + M | M - Clearly, ^ ( ^ i ) = Kxj)AKr)> j = 1 , . . . , n. Conversely, suppose that // = ^ | s + H M and V r G ii, r ^ 0, and V Xj, j = 1 , . . . , n, fi(xjr) = fi(xj) A /x(r). The proof is by induction on n. Suppose that n = 1. Then /i(]C£Lo CiX*) = / i ( c o) A M^ ]C£li cixl~1) = Kco) A (/x(ar) A M 5 X i c*^" 1 ) = . . . = A{^(co), /i(x), /x(ci),..., /x(x), Ax(cm)} = (A^ 1 /i(ci)) A //(#). Thus /x is a polynomial L-subring of R. Assume the result is true n — 1 > 1 (the induction hypothesis). Let r G R. Then 3 po, Pi, • • •, Pmn e S[xly... ,x n _i] such that r = £™!Lo P*n < n • T n u s ^x(r) = (^(p 0 ) A ufa) A . . . A /z(p m J) A fi(xn) by the n = 1 case. The desired result now follows easily from the induction hypothesis. ■ Theorem 6.13.6 Let R = S[[xi,..., xn]] be a power series ring in the inde­ terminates # 1 , . . . , xn over S and M = {xi,...,xn). Let \i be an L-subring of R. (1) If fi is a power series L-subring of R, then fj, = JJ,\S + P\M and V r G i?, r ^ 0, and V Xj, j = 1 , . . . , n, fi(xjr) = fi(xj) A ^(r). (%) Suppose that ji\M is an L-ideal in R. If // = //| 5 + /z| M and V r e R, r ^ 0, and V a^-, jf = 1 , . . . , n, fi>(xjr) = /X(XJ) A A*(r), £/ien /1 is a power series L-subring of R.

Proof.

(1) The proof here is the same as in Theorem 6.13.5.

255

(2) The proof is by induction on n. Let n = 1. Let d = M ( E ^ O QX*). Then

d = /i(co) = =

A^xY^Li0^'1)

//(coJA^A^E^i^-1)

A{^(c 0 ),..., /x(cfc_i), /x(x), fi(J2iLk

°ixi~k)},

k = 0 , 1 , . . . . Thus fi{ck) > d, /x(x) > d, and / * ( £ £ * <***"*) > d, k = 0 , 1 , . . . . If either fi(ci) = d for some i or ii(x) = d, then d=(A{Mci)|i = 0,l,...})A/i(x), the desired result. Suppose that jj,(ci) > d, z = 0 , 1 , 2 , . . . and /x(x) > d. Then M Z £ U * * ' " * ) = d, fc = 0 , 1 , 2 , . . . . Thus ^ ) A M E ^ ^ " f c ) = / * ( * £ £ * * * ' - * ) = H M ^ I S * ***-*) > H M W V AilAfd^fcCiX*-*) (since xt k /X|M is an .L-ideal of ij) > /X|M(^) = A*(x) > d. Hence ^(ESfc °i ~ ) > d, a contradiction. Thus the n = 1 case holds. Since //|(Xn) and / i | ^ 1 ) i > ) X n _ ^ are L-ideals of R, the result now follows in a manner similar to that of Theorem 6.13.5. ■

Example 6.13.7 L = [0,1]. Let R denote the polynomial ring Q[x], where Q is the field of rational numbers. Let F = Q and M = (x). Clearly, F is a coefficient field of R with respect to M. Define the L-subset fi of R by /x(r) = 1 ifreZ (the integers), fi(r) = | ifrE Z[x)\Z, fi(r) = ^ ifrE R\Z[x]. Then fi is an L-subring of R, \i — fi\p + V\M, IAF ^S n°t an L-subfield in ii, and /X|M is not an L-ideal of R by Example 6.11.10. Now V r G iJ, r / 0, fi(xr) = | if r e Z[x] and (i(xr) = \ if r £ Z[x]. Also fj,(x)Afi(r) = § Afi(r) = | ifre Z[x] and = ^ if r £ Z[x]. Hence fi(xr) = fi{x) A fi(r). Thus fi is a polynomial L-subring of R by Theorem 6.13.5.

Theorem 6.13.8 Suppose that R = S[x] (S[[x]\) is a polynomial (power se­ ries) ring over S and that /x is a polynomial (power series) L-subring of R. If [I\M is an L-ideal of R, then ji is constant on M\ (0), where M = (x) .

Proof. Let m e M, m / 0. Then m — xr for some r G iJ, r / 0. Hence /i(ra) = ji(xr) = fj,(x) A fi(r). Now r = xks for some nonnegative integer A: and

256

some s G R such that s £ M. Thus fji^xr)

=

/z(x fc+1 s)

= >

M|M(^+15) HM(^+1)VHM(S)

=

fz(xfc+1)V0

=

/i(x f c + 1 )

= = =

ii(x) A /x(xfc) ( by Theorems 6.13.5,6.13.6) /z(x) A . . . A JA(X) v(x).

Hence fi(x) A /z(r) > //(x) and so /z(x) A ii(r) = fi(x). That is, fi(m) = /i(x). ■

6.14

Completions

We let L be a chain in this section. In previous sections, we gave existence results of L-coefficient fields in L-subrings and showed how these results led to structural results for L-subrings. In this section, our purpose is to describe L-subrings which are complete with respect to a given pseudo-metric. We define, independent of pseudo-metrics, the completion of a fuzzy subring. We show that this definition agrees with that in Section 6.10. We then give a natural definition of the completion of a polynomial L-subring, i.e., one where the completion is a power series L-subring. We show that this definition agrees with the general definition of the completion of a finite-valued L-subring. We also give structural results for polynomial L-subrings and for power series Lsubrings. Throughout this section, S denotes a commutative ring with identity. Definition 6.14.1 Let m be a function from R # x R # into R # , where R # denotes the nonnegative real numbers. If m satisfies the following properties, then m is called a metricizer: (1) V x j G R # , ra(x, y) = 0 <£> x = 0 and y = 0; (2) Vx, y, z e R#, x < y => ra(x, z) < m(y, z) and ra(z, x) < m(z, y)\ (3) Vx, y, z, w E R#, ra(x + y, z + w) < ra(x, z) + m(y, w). Proposition 6.14.2 (1) Suppose that Vx, y G R # , ra(x, y) = x V y. Then m is a metricizer. (2) Let k\ > 0 and &2 > 0. Suppose that Vx, y G R # , m(x, y) = k\x + /c22/. Then m is a metricizer. (3) Suppose that for all x, y G R # , m(x, y) = >/x 2 +y2. Then m is a metricizer. ■

257

Theorem 6.14.3 Let (5i, p{) and (5 2 , p 2 ) be metric spaces. Suppose that m is a metricizer. Define p from 5i x 5 2 x 5i x 5 2 into R # as follows: V (x, £, 2/, 5) G 5i x 5 2 x 5i x 5 2 , p{x,t,y,s)

= ( m o ( p 1 xp 2 ))(x,i/,t,s).

Then p is a metric on 5i x 5 2 . Proof. Now p((x, t), (2/, 5)) = 0 <* m(pi(x, 2/), p2(£, s)) = 0 «» pi(x,y) = 0 and p2(£, 5) = 0 <^> x = y and t = 5 <^> (x,£) = (2/, 5). Also, p((x,t),(2/,s))

= =

rn(pi(x,y),p2(t,s)) rn(p1(y,x),p2(s,t))

=

p((y,s),(x,t)).

Now i = rn(pi(x,y),p2(t,s)) < m(p1(x,z) + p1(z,y),p2(t,r)+p2{r,s)) < m(pi(x, z),p2(t, r)) + m(pi(z, y), p 2 (r, 5)) - p((x,t),(z,r)) + p((2:,r),(2/,5)).B Now L-singletons have the property that xa = yt, <& x = y and a = b when a > 0 < b. Thus we can apply the above result to our fuzzy situation, e.g., S\

p(0M),(2/,*))

= RsmdS2 = A M Throughout this section, (R, p*) is a metric space, where R is a commuta­ tive ring with identity and p* is a metric on R. Let Rc denote the completion of R with respect to p*. We assume throughout that R C Rc. Theorem 6.14.4 Let p be an L-subring of R. Define the fuzzy subset pc of Rc as follows: V x G Rc, ^ c (x) = \f{A{p(xn) c

I n = 1,2,...} \ xn-> x,xn e R}. c

Then p is an L-subring of R and pc(0) = p(0). Proof. Let x, y e Rc. Then pc(x - y) = V{A{^(2n) I n = 1,2,...} | z n G R, zn -> x - 2/} > V{A{//(xn - 2/n) I n = 1,2,...} | xn,yn G R, xn —► x, y n -» 2/} > V{A{^(x n ) A /x(2/n) I n = 1,2,...} | x n , yn G ii, x n -> x, y n -> 2/} = V{A{/x(xn) I n = 1,2,...} I x n G /?, x n -> x}A V{A{/z(2/n) | n = 1, 2 , . . . } | 2/n e R, yn -► 2/} = pc(x)Apc(y). Similarly, /ic(x2/) > /xc(x) A pc(y). Clearly, c M (0) = MO). ■ We call /zc in Theorem 6.14.4, the completion of p in Rc. Proposition 6.14.5 Let p be an L-subring of R. Then (pc)* C (p,*)c. If A{p(x) I x G /x*} > 0, then (pc)* = (p*)c.

258

Proof. x G (pc)* <& pc(x) > 0 => 3{xn\n = 1,2,...} C p* such that xn —> a: (by definition of /xc) <^> a: G (^*) c . The implication becomes equivalence under the added hypothesis. ■ P r o p o s i t i o n 6.14.6 1RC =

(1R)C.

Proof. Let x G Rc. Then (1R)C(X) = 1 <^ V{A{l/i(x n ) | n = 1,2,...} | xn G i?, n = 1,2,..., xn -+ x} = 1 and the latter statement always holds. ■ P r o p o s i t i o n 6.14.7 Let p be an L-subring of R. For alia G p(R)\{0}, (pc)a- If p is finite-valued, then (pa)c = (v>c)a V a G /z(i?)\{0}.

(pa)c ^

Proof. x G (/xa)c «=> 3 {x n | n = 1,2,...} C pa, xn —► x => V{A{/x(xn) | n = 1,2,...} | xn G i?, n = 1,2,..., x n —> x} > a X G (Mc)a- The imphcation becomes an equivalence if p is finite-valued. ■ Let L = [0,1]. Let (R, p*) be a metric space and (i? c , p*c) be its completion. Let p be a fuzzy subring of R. Let p be a pseudo-metric on p. Let p c be a pseudo-metric on pc such that pc = p on *S(/i) x <S(/x), where S(p) = {xa \ p(x) < a). Consider S(\R) as R x (0, 1] for xa G 5(1,R) with a > 0. (Recall that XQ = yo even if x ^ y.) We assume in the following result that p is defined by m o (p* x d) on i£ x (0,1] and pc is denned by m o (p*c x d), where m is a metricizer and d is a metric on [0,1], e.g., absolute value. We assume that m has the property that a sequence {(xn)an \ n = 1,2,...} in p converges to xa in 1R (a > 0) if and only if xn converges to x in R and an converges to a in (0,1]. T h e o r e m 6.14.8 Let L = [0,1]. Let p be a fuzzy subring of R and let p be a pseudo-metric on p. Suppose that for every Cauchy sequence {(xn)an \ n — 1,2,...} in pc there is a Cauchy sequence in p which converges to the same element in S(1RC) which {(xn)an \ n = 1,2,...} converges to. Then pc is complete with respect to pc. Proof. Suppose that {(xn)an \ n = 1,2,...} is a Cauchy sequence in pc which converges to xa C l # c . Let {(yn)sn \ n = 1,2,...} be a Cauchy sequence in p which converges to xa. Then {yn | n = 1,2,...} converges to x and {sn | n = 1,2,...} converges to a. Thus pc{x) > \/{A{p(yn) \ n = 1 , 2 , . . . } , . . . , A{p(yn) | n = k, k + 1,...},...} > V{A{*n | n = 1 , 2 , . . . } , . . . , A{sn \ n = k, k + 1 , . . . } , . . . } — a. Hence xa C pc. Thus pc is complete. ■ Throughout the remainder of the chapter, F denotes a field. Let R = F[xi,..., xn] be a polynomial ring over F. Then Rc — F[[xi,..., xn]]. That is, the power series ring, F[[xi,..., xn]], is the completion of R with respect to the M-adic topology of R, where M = (xi,...,xn). Let pbe a, polynomial

259

L-subring of R. Then Definitions 6.13.3 and 6.13.4 motivate the following def­ inition of /j,e. Theorem 6.14.9 Let R = F[x\,..., xn]. Let ji be a polynomial L-subring of R. Define the L-subset /j,e of Rc as follows: oo

oo

/ A E ••• E ci,..ina;i1...4«)=A{A{M(ci,..iJ|ij=0,l,...;j = l,... ) ii=0

in=0

n}, a\ A ... Aan}, where aj = v{xj) if xj appears nontrivially and a,j = 1 otherwise, j = 1 , . . . , n. Then \ie — \x onF and \ie is a power series L-subring ofRc. Proof.

Ckarly, M* = n on F. Now M e ( £ ~ = 0 •.. E ~ = o

^..AA1

■ • •4" "

1

E ~ = o • • • E ^ U ^ i - i X • • • 4 " ) = MA{M(cii...i„ - *i...*„) \*j = 0,1, • ■ ■ ;j = l , . . . , n } , a i A . , . A ( i B } > A{A{/u(cil...in) A M(^«i...i„) I *j = 0 , 1 , . . . ; j = l , . . . , t i } , a i A . . . A o „ } = A{A{A{^(c il ... in ) | ij = 0 , 1 , . . . , ; j = l , . . . , n } , ox A . . . Aan}, A l A f / i ^ , . . . ^ ) | ij = 0 , 1 , . . . ; j = 1 , . . . , n } , ai A . . . Ao„}}

> A{^(E°T=0 • • • E~=o^...<»^ • • • 4"), Me( E~=o • • • E ^ n . . . ^ ! 1 • • • 4*)}Also, Me( E~=o ■■■EZ=o^-iA1 ■■■<" E~=o •••E~=o^,..i^i 1 ••• X

nn/

x n

n )

= =

M VZ^fci=0 ' * • L^kn=0 vZ^ty1=0 * • • tLiwn=0

C

'Wi...'wridkl-w1,...,kn-wri)Xi

A{A{/i( Z^tui=0 ' • • Z^wn=0 S - W n ^ i - w i , . . . , ^ - ^ )

•• •

I &/i — 0 , 1 , . . . ;

h = l , . . . , n } , a i A . . . A a n } > A{A{A{^(cWl...Wndfcl_u,li...|fcn_t£,n) | ^ ^ = 0 , 1 , . . . , A^; h = 1 , . . . , n } | A;^ = 0 , 1 , . . . ; /i = 1,..., n } , a x A . . . A a n } > A{A{A{A{/x(cWl...ll,n), fJ>(dkl-Wl,...,kn-wn)} | i«fc = 0 , 1 , . . . , ^ ; ft = 0 , l , . . . , n } | *;& = 0 , 1 , . . . ; h = 1,..., n}, ax A.. .Aa n } = A{A{A{^(c il ... in ) | ^ = 0 , 1 , . . . , ; j = l,...,n},aiA...Aa n },A{A{/L6(di 1 ...i n ) | ij = 0 , 1 , . . . ; j = 1,... , n } , ar A.. .A

«n} > A{Me( E~U • • • £~=o ^ . . . ^ i 1 . . . zjr), /xe( E~ = 0 • • • E~ = 0 d^A1 . . . x^)}. Hence \xe is an L-subring of R. By definition, /ze is a power series L-subring of Rc. ■ For a G ^(-R), (// a ) c denotes the completion of \xa in i£c with respect to the metric on /i a induced by that of R. Recall that we assume R C Rc. If R is & polynomial ring, the metric on R is the one induced by the M-adic topology. In view of Theorem 6.14.9, \xe and fic should be closely related, if not equal.

Theorem 6.14.10 Let R be the polynomial ring F[xi,..., xn] and let fi be a polynomial L-subring of R. Then (fJ>e)a = (Ma)c V a € n(R) and fj,e C fic. If/i is finite-valued, then [ie = /u,c. Proof. Let y denote E2°?=o • • • E ^ o ^ i . - . ^ i 1 .. .x^ e Rc. Let a e L, a < /x(0). Then y G (/xe)a <* fJ>e(y) > a <£> A J A I ^ C ^ . . . ^ ) | ij = 0 , 1 , . . . ; j = 1,.. . , n } , a i A . . . A a n } > a <3> c^...^ G ^ a , «j = 0 , 1 , . . . , and a^- > a, j =

260

1 , . . . , n => y G (fia)c. Let y G (/xa)c. Then 3z m = £ ^ 0 • • ■ E ^ o di^.iA1 ••• x^ G /xa, m = 0 , 1 , . . . , such that z m -* y. Thus given j > 0, 3 fc such that V m > fc, Zm - y G ( x i , . . . , x n ) J . Thus c^...^ G /i 0 V z x . . . i n such that 0 < i\ -f . . . + in < some qj, where qj -+ 00 as j —> 00. Hence => (above) becomes «=> and so (/i c ) a 2 (/^a)c = ( ^ ) a by Proposition 6.14.7. Thus /i c 2 Me s i n c e /x«(0) = JJ,(0) = /i c (0). If \i is finite-valued, then (// c ) a = (// a ) c = (^ e ) a and so fie = /zc. ■ Theorem 6.14.11 .Le£ 1/ 6e an L-subring of the polynomial ring R = S[xi,..., xn] such that v* C 5. Then v[(x\)ai, • • •, (£n)aj ^ a polynomial L-subring of R, where aj > 0, j = 1,..., n. Let y - £™L 0 ••■ TZ=oci,-iAl

Proof.

• • • 4 n - By Theorem 5.1.8,

( l / K x i J a , , . . . , ( * n ) a j ) ( ! / ) = V{( E . " i o • ■ ■ E ^ ( * i » - n k

1

...*

n

((^ijaj*1 ■ • •

in

((^n)a n ) )(2/) Mil...^ e i / * , ^ . . . ^ = ^ ( ^ 1 . . . i n ) , 2 j = 0 , l , . . . , m i , m j e N , j = l , . . . , n } = A{A{i/(ci1...»n) | i, = 0 , 1 , . . . , mj, m^ E N, j = 1 , . . . ,n},6i A . . . A 6 n } (since y has an unique expression of the form Y^Lo • • • X ^ = o c ii--in^i 1 • • • x% for distinct ( i i , . . . , i„)) = A{A{(i/[(xi) a i ,..., (>n)aj)(c» 1 ...t n ) | ^ = 0 , 1 , . . . , raj, mj G N, j = 1 , . . . , n } , A{6j | j = 1 , . . . ,n}}, where the latter equality holds by Theorem 5.1.8 and where the bj = aj if Xj appears nontrivially in the expression for y and bj = 1 otherwise. ■ Theorem 6.14.12 Let ji be a polynomial L-subring of R = S[xi,..., Then /x = / x | s [ ( x i ) a i , . . . , ( x n ) a J , where /J,(XJ) = a,j > 0, j = 1,... ,n.

xn],

Proof. Let y = ZT=o • • • Z?n=o *!...<„*? • • • 4 " € A Then ((/*|s)[(*i) ai , • • •»(*n)aj)(l/) = A i A l ^ l s ) ^ . . . ^ ) I ij = 0 , 1 , . . . , m d , m,- G N, j = 1 , . . . , n } , ai A . . . A an} (as in the proof of Theorem 6.14.11) = A{A{/i(ci1<>>in) | ij = 0 , l , . . . , m i , mj G N, j = l , . . . , n } , ^ ( x i ) A . . . A/x(arn)} = M l / ) - ■ The following definitions concerning power series are given in a manner so that they are similar to the corresponding ones for polynomials. Definition 6.14.13 Let R = S[[a?i,..., xn]] be a power series ring. Define the L-subset TT = £ " = 0 . . . E£=o(4i....iJ«< 1 ...i B ((*i)-i) < 1 • • • ( ( * » ) - . > of R as follows, where the Uilm„in and the a,j are in L\{0} : V y G R, 7r(y) = *{*{vii.-.in I h = 0 , 1 , . . •; j = 1 , . . . , n } , 6i A . . . A 6 n } */ 2/ = E ~ = 0 • • • TZ=odii--iux\l • • • < " , ^ e n e tfie ^ . . . ^ = i ^ . . . ^ if dh^An ^ 0 and vh„An = 1 otherwise and where the bj = aj if Xj appears nontrivially and bj = 1 otherwise; 7r(y) — 0 otherwise. Definition 6.14.14 Le£ z/ 6e an L-subring of the power series ring R = 5[[xi, . . . ,x n ]] suc/i tta* i/* C 5. £>e/me tte L-subset i/[[(xi) f l l ,..., (x n ) a n ]] o/i? as

261

follows, where ad > 0, j = 1,..., n : Vy = E ^ = o • ■ ■ TZ=o ^i-iA1 « , M K x x J a , , . . . , (*n)aJ])G/) = V{(( E ^ = 0 • • • E ^ 0 ( 4 n

1

...-J,,.,

n

• • • x'n € ((Xljaj*1

d

. . . ((z n )aj* )(2/) I di!...»n ei/*,Ut!...i n = K i i . . . i n ) » ^ = 0 , l . . . ; j = l , . . . , n } . Theorem 6.14.15 Let v be an L-subring of the power series ring R = 5[[xi, . . . , xn]] such v* C 5. Then v[[(xi)ai, • • •, (#n)aj] is a power series L-subring of R, where aj > 0, j = 1,..., n. Proof.

Let y = £ £ U ■ • • £ ~

= 0

^h-iA

■ ■ ■ 4 n € A Then (i/[[(*i) a i ,...,

(*n)«J])(l/) = V { ( E ^ 0 •• • E~=o(*i...-ink1...*n ( O n U * G

dil....tn

1

•••( K ) a J ^ ) ( y ) |

d

"*,V>il...in = K H...in)> *j = 0, 1, . . . J j = 1, . . . , 7l} = A{ A ^ C ^ .. .*„ )

| ij = 0 , 1 , . . . ; j = 1 , . . . , n}, 6i A . . . A 6 n } (since y has an unique expres­ sion of the form E ^ = o • ■ • E ^ = o ^ l - i n ^ i 1 • ■ • x n for distinct ( i i , . . . , i„)) = A{A{(i/[[(xi) a i ,..., (^n)a n ]])(Q 1 ...i n ) I ^ = 0 , 1 , . . . ; j = l , . . . , n } , A { 6 i | j — 1 , . . . , n}}. That ^[[(^i)ai, • • •, ( x n)a n ]] is an L-subring of R follows by an argument similar to that in the proof of Theorem 6.14.11. ■ Theorem 6.14.16 Let fi be a power series L-subring of R = S [ [ x i , . . . , xn]]. Then /J, = fJ,\s[[(xi)ai,. • •, (xn)an]], where fi(xj) = a.j > 0, j = 1 , . . . ,n. Proof. Let y = Y™=o-•-TZ^^A • ■-^ € # . Then ((Ai|s)[[(*i) 0l , • • • >(*n)aj])(2/) = A{A{(^| s )(ci 1 ...i n ) | ij = 0 , 1 , . . . ; j = 1 , . . . , n}, a1 A.. .Aa n } (as in the proof of Theorem 6.14.15) = A{A{//(ci1...in) | ij = 0 , 1 , . . . ; j = 1 , . . . , n}, /x(xi) A . . . A /x(x n )} = j/(y). ■

6.15

EXERCISES

1. (104) Let X be the set of all £-prime L-ideals ir of R such that 7r(0) = 1. Let V(n) = {TT G £ I ^ C TT} if /x G L H . Let /i, v G L H . Prove the following statements: (a) If /i C i/, then V » C V(fi). (b) l / ( / i ) U ^ ) C VX/ifli/). (c) V(U) U V ( l j ) = V(lInj),

where J and J are ideals of R.

(d) V(p) = V ( 0 = W O ) , where £ = ( ^ . (e) If {m\i el}

C LR, then V(UieIfii)

= ni€/V(/^).

2. (104) Let Z denote the ring of integers and let L = {0,a, 6,1} be the Boolean algebra of four elements. Define fi G LR as follows: fi(x) = 1 if x G (6), fi(x) = a if x G (2) \ (6), /i(x) = 6 if x G (3) \ (6), and fi(x) = 0 otherwise. Define v G LR as follows: i/(x) = b if /x(:r) = a, u(x) = a if

262

^(x) = 6, and v{x) = fi(x) everywhere else. Show that V(fi) U V{y) C FOxni/). 3. (104) For xeR, denote V(l{x}) by V(ar). Then V(x) = {ir G £|TT(0) = 1}. Prove the following statements. (a)V(0)=3E. (b)V(l)=0. (c) V(x)UV r (j/) = Vr(x2/). (d) ^ ( l s ) = n x G j E ;y(x), where E C R. 4. (104) Let X(x) = X\V(x) V ar G R and B = (X(ar) | a; G iJ}. Let Spec(i?) denote the spectrum of R in the usual sense. Prove the following statements. (a) B constitutes a base for a topology on X. (b) X is a compact space. (c) Spec(i?) is a subspace of X. (d) Spec(i?) is dense in X. 5. (104) Denote X by Fspec(i2). Fspec(i2) is called the fuzzy p r i m e spect r u m of R. Prove the following statements. (a) If R is a principal ideal domain, then B is the actual topology on Fspec(i?). (b) A subset T of Fspec(i?) is closed <^ there exists a subset F of R such that F = V(1F). (c) If T is a subset of X, then the closure of T, T = V ( l p ) , where P = nnejr7Ti and where -K\ = {x G i?|7r(x) = 1}. (d) If 7T, f G Fspec(i?), then {¥} = {£} <£> TTI = £ x . Conclude that Fspec(i^) (unhke Spec(iJ)) is not a To-space. 6. (104) Let / : R —» R' be a homomorphism of rings. Prove the following statements. (a) / induces a continuous function / * : Fspec(it / ) —> Fspec(ii) denned

by f"W) = / " V )

(b) If / is surjective, then / * is injective and Fspec(iJ') is homeomorphic to the closed subset V(IKerf)(c) If / is an isomorphism, then / * is a homeomorphism.

263

7. (104) Let / : R —► R' be an epimorphism of rings. Prove the following statements. (a) If each £-prime L-ideal of R is constant on Ker / , then Fspec(i?) is homeomorphic to Fspec(i? / ). (b) Fspec(i?) and Fspec(R/lZ(R))

are homeomorphic.

8. (65) Let L = [0,1]. Suppose that R has a composition series of length n. Let [i be a fuzzy ideal of R. Prove that if /z = f i n . . . n fm is an irredundant ^-primary //-representation of /x, then |Im(/z)| - 1 < m < (|Im(^)|-l)(n-l). Let R denote an integral domain and K its quotient field. Let L = [0,1] and let XR D e the fuzzy subset of K defined by XR (X) = 1 if X e R and XR (X) = aif x e K\R, where a G [0,1). Let /? be a fuzzy i?-submodule of if. Then (3 is called a fractionary fuzzy ideal of R if there exists d G R,d ^ 0, such that d\ o f) C. XR for some a G [0,1). Let (3 be a fractionary fuzzy ideal of R. Then /? is said to be fuzzy invertible if there exists a fractionary fuzzy ideal (3f of R such that f3r (3 = XR f° r some a G [0,1). If there exists a fractionary fuzzy ideal /?' of i? such that (3'P = XR , then (3 is said to be invertible . 9. (113) Let (3 be a fractionary fuzzy ideal of R with d\o (3 C. XR\ where d G i?, d 7^ 0, and a G [0,1). Prove that there exists a smallest b G [0,1) such that d\ o /? C x # • 10. (113) Let /? be a fractionary fuzzy ideal of R and d G i£, d 7^ 0, and 6 G [0,1). Prove that d(3a C i? Va G [6,1) if and only if di o /? C X{R • 11. (113) Let /? be a fractionary fuzzy ideal of R with di o ft C xR , where d e R,d ^ 0, and 6 G [0,1) is smallest. Prove that (di o /?) a is an ideal of i? for Va G (6,1]. Prove also that (di o /3)a is an ideal of R for Va G Im(di o 0)\{b} if and only if Im(di o 0)\{b} C (6,1]. 12. (113) Let (3 be a finite-valued fuzzy .R-submodule of K. Prove that (3 is a fractionary fuzzy ideal of i? with di o (3 C x # , where d G # , d ^ 0, and 6 G [0,1) is smallest if and only if /?a is a fractionary fuzzy ideal of R for every a G (6,1] and (3a' is not a fractionary ideal of R for every a! < b. 13. (113) Let (3 be a fractionary fuzzy ideal of # with d\ of) C x # , where d G iJ, d ^ 0, and 6 G [0,1) is smallest. Suppose that V{0(x)\x G iA/3*} < L If /?* is maximal such that d/3* C R, prove that 6 = V{(3(x)\x G lf\/?*}.

264

14. (113) Let (3 be a fractionary fuzzy ideal of R. Suppose that V{(3(x)\x G K\/3*} exists. Prove that (3 is fuzzy invertible if and only if /?* is an invertible fractionary ideal of R. 15. (113) Let (3 be a fractionary fuzzy ideal of R. Prove that if (3 is invertible, thenIm(/3) = {0,1}. 16. (113) Let /?,/?' be fractionary fuzzy ideals of R such that /?/?' = XR for some a G [0,1]. Let Im(/?') = {6o, &i,..., &™} with bo < b\ < ... < 6 m = 1 and Im(/3) = {ao,ai, • • • >an} with ao < ai < . . . < an = 1. If /3' 6 . and /? ai are fractionary ideals of R for j = 1 , . . . ,m and i = 1, ...,n, prove that either /? or /?' is 2-valued and also that either a = ao or a = bo. 17. (114) Let (3 be a fractionary fuzzy ideal of R such that Im(/3) = {a 0 , a i , . . . , a n } with ao < ai < . . . < a n . Prove that if (3ai is invertible for i = 1, . . . , n, then (3 has a finite minimal generating set (as a fuzzy i£-submodule). Let (3 be a fractionary fuzzy ideal of R. Then /3|# is a fuzzy ideal of R. If /3\R is a prime (maximal) fuzzy ideal of J?, then (3 is called a prime (maximal) fractionary fuzzy ideal of R. If Vfc G if\i£, /3(fc) = 0, then (3 is called an integral fractionary fuzzy ideal of R. 18. (114) Let (3 be a fractionary fuzzy ideal of R. If /3 is a product of fuzzy invertible prime integral fractionary fuzzy ideals, prove that the factor­ ization is unique. 19. (114) Let (3 be a fuzzy invertible fractionary fuzzy ideal of R such that Im(/3) = { a o , a i , . . . , a n } with ao < a\ < ... < an = 1 and such that (3ai is a fractionary ideal of R for i = 1 , . . . , n. Prove that there exists an a, a > a n _ i , and a unique fractionary fuzzy ideal (3' of R such that f3(3r = 20. (114) Prove that the following conditions are equivalent on an integral domain R. (a) R is a Dedekind domain. (b) Every integral fractionary fuzzy ideal (3 of R such that V{f3{x)\x G K\(3*} exists is fuzzy invertible. (c) Every fractionary fuzzy ideal (3 of R such that V{(3(x)\x G K\(3*} exists is fuzzy invertible. (d) Every finite-valued fuzzy ideal [i of R with /z(0) = 1 is uniquely expressible as a product of a finite number of maximal fuzzy ideals. Let (3 be a fuzzy ideal of R with /3(0) = 1. Define the fuzzy subset (3e of K by (3e(x) = (3(x) VxeR and (3e(x) =a0Vx e K\R, where a0
265

Im(/3). Then (3e is a fractionary fuzzy ideal of R, called an extended integral fractionary fuzzy ideal of (3. 21. (114) Let R be a Dedekind domain. Let (3 be a finite-valued extended integral fractionary fuzzy ideal of R. Prove that (3 is a product of a finite number of extended integral maximal fractionary fuzzy ideals. 22. (114) Let R be a Dedekind domain. Let (3 be a finite-valued fractionary fuzzy ideal of R. Prove that (3 is expressible as a product of a finite number of extended integral maximal fractionary fuzzy ideals and a 2valued fractionary fuzzy ideal.

266

Chapter 7

ALGEBRAIC L-VARIETIES AND INTERSECTION EQUATIONS Unless otherwise specified, L is a chain throughout this chapter. The concept of an algebraic L-variety is introduced in order to bring the current knowledge of fuzzy commutative ring theory to bear on the solution of nonhnear systems of intersection equations of L-singletons. It is shown for every finite-valued L-ideal fi of a polynomial ring in several indeterminates over a field with /J,(0) = 1 that the algebraic L-variety of /J, can be expressed as a union of irreducible algebraic L-varieties, no one of which is contained in the union of the others. The purpose of this chapter is to give some meaning to fuzzy commutative ring theory developed to this point and to put some direction to its further study. We bring fuzzy commutative ring theory to bear on a natural application area, namely, the solution of nonlinear systems of intersection equations of Lsingletons. Let R denote the polynomial ring F[xi,..., xn], where F is a field and X\,..., Xfi a r e algebraically independent indeterminates over F. Let C be a field containing F. C may be taken to be the algebraic closure of F or an algebraically closed field with infinite transcendence degree over F. Let Ck denote the set of all ordered A>tuples with entries from C, k a positive integer. Our approach is to consider those L-ideals \x of R which are finite-valued and are such that fi(0) = 1 since these are precisely the L-ideals of R which have 7Zprimary L-representations. We define the algebraic L-variety Ai(fx) of /i and show that from an irredundant 7^-primary L-representation of /i, M(fi) is a

267

finite union of irreducible algebraic L-varieties, no one of which is contained in the union of the others. We then apply this result to the solution of a nonlinear system of intersection equations of L-singletons. We show that there exists an L-ideal \i of R which represents this system and the irredundant 7£-primary L-representation of y/Ji displays the solution of the system in a manner similar to that of the crisp situation. The results of this chapter are taken from [16, 17, 121, 166, 167, 175, 178].

7.1

Algebraic L-Varieties

If / is an ideal of R, we let M(I) denote the algebraic variety of / , [154, p. 203]. If Z is a subset of Ck, we let I(Z) denote the set of all / e R which vanish at all points of Z. Then T(Z) is an ideal of R, [154, p. 203]. We now give definitions for the fuzzy counterparts of Ai and X. Let c be a strictly decreasing function of L into itself such that c(0) = 1, c(l) = 0, and V a £ L, c(c(a)) = a. The following approach has the advantage that c may be changed to fit the application. Definition 7.1.1 Letx be a finite-valued L-subset ofCk, say, Im(x) — {ao,«i, . . . , a n } , where a$ < a\ < ... < an. Define the L-subset X(x) of R as follows:

I(x)(f)={

( c(an) c(ai) [ c(a0)

if f e R\l(xan); iffE J(Xa i+1 )\2:(XaJ,i = l , . . . , n - l ; iffelixaj.

If n = 0, then we define Z(x)(0) = 1. Definition 7.1.2 Let fi be a finite-valued L-ideal of R, say Im(fi) = {&o> ^i 5 • • •» 6 m }, where bo < 6i < . . . < frm- Define the L-subset M(/J) of Ck as follows: M(fi)(z)=<

( c(bm) c(bi) [ c(b0)

if z e Ck\M(fibmy, ifze M(fJLbi+1)\M(iibi),i ifze -M(/X6i).

jVt(^) is called an algebraic L-variety

= l,...,rn-

1;

(of n).

In Definition 7.1.1, it is possible for I ( x a i + 1 ) = 2"(XaJ or R = I(xa„)- In this case c(ai) £ Im(I(x)), i = l , . . . , n . Similarly, it is possible for c(6») ^ lm(M(fJL)) for some i = 1 , . . . , m. P r o p o s i t i o n 7.1.3 Let x be defined as in Definition 7.1.1. Then (1) (2) (3) (4)

I(x)c(ai) = I(Xa i + 1 ) for i = 0 , 1 , . . . , n - 1; if 0 < 6 < c(a n ), then I(x)fc = i?; if c(a»+i) < 6 < c(a»), then I(x)& = 2"(Xc(6)) for z = 0 , 1 , . . . , n - 1; if c(a0) < b < 1, then I(x)& = 0-

268

Proof. (1) / G I(x)c(aO ^ Ax)U) > c(a{) & f G J(xa i + 1 ) by Definition 7.1.1. (2) l(x)b = R since c(an) < the smallest element in Im(X(x))(3) / G I(X)b <* Iix)U) > b ** * ( * ) ( / ) > c(ai) ^ / G X(x)c(aO & f £ Z(Xai+1) by (1) <^ / G l(Xc(b)) since Xai+1 = Xc(by (4) T{x)b = 0 since c(a 0 ) is the largest element in Im(X(x))- H For x as defined in Definition 7.1.1, X(x)c(a*) = 2"(Xai+i) i s an ideal of R for z = 0 , 1 , . . . , n - 1. Thus since Im(x) C {c(a») | z = 0 , 1 , . . . , n}, X(x) is an L-ideal of R, Theorem 3.1.9. Proposition 7.1.4 Let JJL be defined as in Definition 7.1.2. Then (1) M(fi)c{bi) = M{fibi+1) for i = 0 , 1 , . . . ,ra - 1 (2) ifO < a < c(bm), then M{v)a = Ck; (3) ifc(bi+1) < a < c(6i), then M(fi)a = M(fjbc(a)) for i =0,1,... (4) */c(&o) < a <1, then M(fi)a = 0-

,m-l;

Proof. (1) z G M(fi)c(bi) <^ M(fi)(z) > c(bi) <£> z G M(/J,bi+l) by Definition 7.1.2. (2) M.(fJ,)a = Cfc since c(6 m ) < the smallest element in Im(M(/i)). (3) 2 G ^l(/x)a ^ M M ( ^ ) > a <£> -M(/x)(z) > cfa) & z G .M(M)C(60 <^> z G M(fibi+1) by ( 1 ) ^ 2 6 A4(>c(a)) since /x6.+1 = /z c(a) . (4) Al(//) a = 0 since c(&o) is the largest element in Im(.M(/i)). ■ Proposition 7.1.5 Let x a^d ^ be as defined on Definitions 7.1.1 and 7.1.2, respectively. Then (1) \Im(M(l(X)))\ = |/m(I( x ))|; ^ \Im(l(M(»)))\ = \MM(fi))\. Proof. (1) Suppose that Im(Z(x)) = {b0, &i,..., bh} with b0 < h < ... < bh. Then l(x)bi+1 C X(x)fci and so Al(Z(x)& i+1 ) D A*( Z C x k ) from the crisp case and since for b e Im(T(x)), I(x)& = I(Xd) for some d 6 Im(x). Hence Jm{M{Hx))) = {c(bo),c(bi),...Abh)}. (2) Suppose that lm(M(/j,)) = {ao,ai, • • • ,a,j} with a 0 < ai < . . . < a,j. Then M(n)at+1 C A4(/0a« and so J(M(fi)ai+1) D I ( X ( M ) 0 i ) . Hence Im(l(M(»)))

= {c(a0),c(ai),..

.,c(aj)}.

■

Proposition 7.1.6 Let x and \i be defined as in Definitions 7.1.1 and 7.1.2, respectively. Then (l)Vae L, M(l(X))a = M(l(Xa)); (2)Vbe L, l{M{ii))b = I(M(j»)).

269

Proof. (1) For i = 0,1,... ,n - 1, M(I(Xai+1)) = M(l{x)c{ai)) by Propo­ sition 7.1.3(1). Suppose that c(a{) E Im(X(x)). Let bn_i = c(ai) for i = 0 , 1 , . . . , n . Then MpUUat))

= M(I(x)^)

= M{X{x)Ubn^)

=

M(l(x))ai+1-

Suppose that c(di) £ Im(X(x)). Then by Proposition 7.1.4(3), M(I(x)c(ai)) = X(X(x)) c (c(a i )) = M(I(x))ai = M(I(x))ai+1, where the latter equality holds an since c(a,i) £ Im(X(x)) implies a* = c(c(a,i)) £ lm(M(l(x))) d so M(l(x))(z) k >ai& M(I(X))(z) > a i + 1 . For 0 < a < a 0 , M(I(Xa)) = M(I(C )) = M « 0 » = Ck = A4(X(x))a since ao is the smallest value in Im(%). Suppose that an < a < 1. Then M(X(xa)) = M(I(Q)) = M{R) = 0 = M(X(x))a since an is the largest value in Im(x). For any other a, M(l(Xa)) = -M(X(x)c(a)) = M(I(x))a. (2) For i = 0 , 1 , . . . ,ra - 1, I(M(/ibi+1)) = X(M(fjL)c^.)) by Proposition 7.1.4(1). Suppose that c(b{) E Im(M(/J>)). Let a m _i = c(bi) for z = 0 , 1 , . . . ,ra. T h e n I ( M ( M ) c ( 6 i ) ) = X ( M ( M ) a m _ i ) = I ( ^ ( ^ ) ) c ( a m _ i _ l ) = I ( M ( M ) ) b i + 1 . Suppose that c(a^) ^ Im(.M(/i)). Then by Proposition 7.1.3(3), l(M(fi)c^i)) — X(jM(//))c(c(6i)) = T(M.(ii))bi = T(M(/J,))bi+1, where the latter equality holds since c(pi) ^ Im(.M(/i)) imphes bi = c(c(6^) ^ Im(X(.M(/i)) and soX(.M(^))(2:) >bi& X(M(fi)){z) > 6 i + i. For 0 < 6 < 60, X( -M(^)) = X(M(R)) = X(0) = i? = T(M.(/J,))b since 6o is the smallest value in Im(^). Suppose that 6 m < b < 1. Then X(M(fib)) = X(M(®)) = X(Ck) = 0 = X(M(fi))b since 6 m is the largest value in Im(^). For any other 6, l(Ai(fJ,b)) = T(M(/J,)c(b)) = X(M.(/Jb))b- M Proposition 7.1.7 Xe£ x and /i be defined as in Definitions 7.1.1 and 7.1.2, respectively. Then (i)l(M(l(x)))=l(x); (2) .M(I(.M(M))) = M M ) Proof. By Propositions 7.1.5 and 7.1.6 and the crisp case, we have the following arguments. (1) J(M(I(X)))c{ai) = l(M(X(xUai))) = 1(M( I(Xai+1))) = X(Xai+1) = 2"(x)c(o4). where i = 0 , l , . . . , n - l . If c(a i + i) < b < c(at), then l(M(l(x)))b = l{M{l{X)b)) = l(M(l(Xc(b)))) = X(Xc(6)) = 2fo)6, where * = 0 , 1 , . . . , n - 1 . For 0 < 6 < c(a n ), I ( M ( J ( x ) ) ) b = i? = I(x)fc- For c(ao) < 6 < 1,1(M(l(x)))b = 0 = l(X)b. Thus I(A^(I(x))) 6 = I(x)fc V 6 6 L. (2) M(I(M(n)))c(bi) = M{ l(M(ric{bi))) = M(l{M(t*bi+1))) = M(»bi+1) — M(fi)c(bi), where i = 0 , 1 , . . . , m - l . If c(bi+1)
270

T h e o r e m 7.1.8 Let a be an L-subset of C . Then a is an algebraic L-variety if and only if a is finite-valued and V a G Im(a), aa is an algebraic variety. Proof. Suppose that a is an algebraic L-variety. Then a = M (//) for some finite-valued L-ideal fi of R. Hence Mae. Im(a), 3 b G Im(/i) such that a = c(6). Thus either aa = M(/a)c(b) = -M(iJLb'), where b' is the successor of b in Im(/x) or M(ii)c(b) = Ck = M((0)). Since M{nb') and M((0}) are algebraic varieties, aa is an algebraic variety. Conversely, suppose that a is finite-valued and aa is an algebraic variety V a 6 Im(a). Then aa = M(I^) for some ideal /( a ) of R, a G Im(a). Now M(I^) = M(I(M(I^))). Thus a a = M(J^) for some ideal J^ of i? such that if a, a' G Im(a) with a < a', then j( a ) C J(a'\ namely j( a > = T(.M(j( a ))). Let Im(a) = { a 0 , a i , . . . , a n } , where a 0 < ai < . . . < an. Define the L-subset \x of R by //(/) = c(a n ) if / G R\j(an\ fi(f) = c(ai) if / G j( o *+i)\j("0, and ^ ( / ) = c(a0) if / G J ^ . Then /i is an L-ideal of R. Now -M(/z)a. = ^ ( ^ ) c ( c ( a i ) ) = Mifx^a.^)) = M{J^ai)) = a a i and so a = -M(/x). Thus a is an algebraic L-variety. ■ P r o p o s i t i o n 7.1.9 If n is a nonconstant prime L-ideal of R, then /i = I(M(/i)). Proof. Now Im(/i) = {^,1}, where b < 1 and /ii is a prime ideal of i?, Theorem 3.5.5. Suppose that \ix ^ (0). Then Im(I(Al(/z))) = I m ( » . By Proposition 7.1.6, X(.M(ju))i = Z(.M(/zi)) = /ii since /ii is a prime ideal of R. Now I(M(l(fi)))b = R = y>b- Hence l{M{fi)) = //. Suppose that HI — (0). Then M(fj)(z) = c(l) = 0 if z G Ck\M(iibl) = Ck\M({0)) = 0 and k M{y){z) = c{b) if z G M(fibl) = C . Now I(M(fi))(f) = c(c(b)) = b if / G iJ\I(M(/x) c ( 6 ) ) - J2\ (0) and I(M(0) = 1. ■ Theorem 7.1.10 Suppose that /i z's defined as in Definition 7.1.2 and yLx(0) - 1. TTien M(fi) = M(J}J) = M(7l(fi)). Proof. By Theorem 6.1.10, U(fi) = y/jl. By Definition 3.8.1 and Theorem 6.1.10 and the fact that \i is finite-valued, Im(v//Z) C Im(/x). Now, by Lemma 6.1.8 and Theorem 6.1.10, M{p)a = M(fic{a)) = M(^JJ^j) = M{{^1 ) c ( a ) ) = M(yfjl)a if c(&i+i) < a < c(6i) for z = 1 , . . . ,ra — 1 since In^A^^/jiJ)) C Im(A^(^)), where the latter inclusion follows since M(fibi+1) = -M(^bi) O M((y/jl)hi+1) = M((y/jl)bi) and so c(bi) $ Im(M(/i)) => c(b{) £ hn(M(rfi)). By Proposition 7.1.4, M(fi)c{bi) = M(fibi+1) = M>/P&^7) = M((^JI ) 6 i + 1 ). Let j be the largest nonnegative integer such that j < z, 6j G In^yZ/I) and let z* be the smallest nonnegative integer such that i* > i and 6i*+i G Irc^y/I). Then •M( v /p£^7) - M(^jl^) = M(y/JI)cibj) = M(JjI)c{bi), where the

271

latter equality holds since if b{ <£ lm(y/ji), then c(bi) £ lm(M(y//I)). Hence ■^(M)C(60 = M(y/jl)c(bi). If j doesn't exist and i* does, then (y/JI)b*+1 = R. Thus 0 = M(y/jIS£) = M(jibim+l) D M^bi+1) = M^)c{hi) D M(y/jl)c{bi) D 0. Hence M(fi)c(b.) = M(^/JI)c^bi). Suppose that j exists and i* does not. Then i = m and so .M(/z) c(6m) = Ck = M{y/Jl)c{bm) . Hence M(fi) = M(y/Ji). ■ Corollary 7.1.11 Lei L be a dense chain. If IT is a 'prime L-ideal of R be­ longing to the primary L-ideal ip of R, then A4(ip) = M.{ir). Proof.

i / 0 = 7T by Theorem 3.7.11. ■

Lemma 7.1.12 Suppose that \i and v are L-ideals of R such that Im{fi) = {6 0 ,61,..., frm}, where bo < b\ < . . . < 6 m = 1 and Im{y) = {&,1} where b < 1. T/ien A*(/x fl 1/) = A4(M) U . M ( I / ) . Proof. (1) Suppose that bi < b < &i+1. Then {6 0 ,61,..., bi, 1} C Im(/x fl ^) C Im(^) U Im(i/). (1.1) We first show that M(fi H i/) c(6i) = (M(fi) U . M ^ ) )<:(&,•) V fy G Im(/x). Now M(/i n i/) c(1) = Cfc - Cfc U C* = M(M)C(I) U M(U)C(1) = (M( M ) ^M)c(i). (1.1.1) Suppose that 1 > fy > 6 i + i. (1.1.1.1) Suppose that bj £ Im(/xn ^). Then (/x n !/)&,, = (/x fl ^)b i+1 since bj+i is the next largest possible element in Im(/x D v). Thus A<(/x fl ^)c(6j) — M((fi n i/) 6i ) (Proposition 7.1.4(3)) = M((n n i/) 6i+1 ) = - M ( w i t i n ^ + 1 ) = M(fJLbj+1) U -M(i^ i+1 ) (crisp case) = -M(/z)<.(&,.) U -M(i/i) (Proposition 7.1.4(1) and bj+1 > b) = M(n)c(bj) U .M(i/)c(6) = M(/x) c(6 .) U M{y)c{bj) (since c ^ )

>O) = ( ^ ) U M M ) C ( W . (1.1.1.2) Suppose that bj G Im(/x fl v). Then A4(/x fl v)^) = M((fi D ^)t * + i) 5 where j * is the smallest integer > j such that frj*+i G Im(^ fl 1/). Now (/i fl i/)& r+1 = (/i n u)bj+1. Hence A4(M n i/)c(fej.) = M((fi fl i/)6i+1) = / (M(/J) U A ((^))C(6J.) as just argued above. (1.1.2) Suppose now that bj < bi. Then M(fi C\u)c^b.) = M((fi fl v)bj+1) (since bd, bj+1 G Im(/x fl 1/)) = -M(/i&i+1 n i^ i+1 ) = A4(/X6i+1) U M ^ + i ) = M(/J)c{bj) U A4(/i) = M(fi)c{bj) U 0 = ^ ( M ) ^ ) U - M M C ^ . ) = ( M / ^ ) U M{v))c{bj) since c(fy) > c(6) and c(b) is the largest element in Im(z/). (1.1.3) We now consider the case bj = bi. Then fy G Im(/x fl 1/). (1.1.3.1) Suppose that bi < b < bi+i. (1.1.3.1.1) Suppose that b <£ Im(/x fl v). Then c(6) £ Im(.M(/i n v)). Thus

A4(M ni/)c(6.) = M(fi n i/)c(6) = M({fi n i/)6) = Af(/i6 n uh) = M{^b) uM(^) = M{fJLbi+1) U Ai(i^) = M(fi)c{bi) U M(iJ) = M(v)c{bi) = (M(fJL)\JM(v))c{bi).

A^(M)C(60

U 0 = M(M) c(fei) U

272 (1.1.3.1.2) Suppose that b € lm(fx n v). Then M(\x D i/) c(6i) = .M((/u D

v)b) = X(^6 n i*) = M(w>i+1 n ii) = -M(w>i+1) u M{R) = Mirier)

u0

= M M ) C ( 6 0 U - M ^ W ) = (M(ji) U .MC))c(6«)(1.1.3.2) Suppose that 6j = b. Then for i* the smallest integer > i such that

fei.+i e im(/i n «/), M(/i n i/)c(6i) = .M((^ n v)bt.+1) = M ( M n v)bi+1) = M([ibi+1 n i/6i+1) = M{nbi+l) U A^(^fci+1) = M(fi)c{bi) U M ( i ^ ) = A^(M)C(6*) U M(v)c(b) = M(n)c(bi) U X(i/) c ( b i ) = ( X ( ^ ) U A^(^))c(6i)- Hence we conclude that M(fi n v)c{bj) = (M(fi) U M{v))c(bj) V 6,- € Im(^). (1.2) We consider b. (1.2.1) Suppose that b s Im(/j, n ^). Then for i* > i smallest such that 6,. + i

6 im(/in i/), M(/i n u)c{b) = M{(n n v)bi.+l) = A^((M n i/)6i+1) = M(nbi+1 n «Vn) = A4(A*6«+I) U -M(l/&i+1) = X(M)c(60 U M(UI) = M(n)c{b) U M(u)c{b) = (M(ji) U M("))c(&). (1.2.2) Suppose that b £ Im(/x n v). Then 6, < b < bi+1 and {nf\v)b = (yuPl

j/) 6i+1 . Thus x(/x n i/)c(6) = A<((/i n v)b) = M{{n n u)bi+1) = .M(/x&i+1 n * 0 = M(fxbi+1) U A4(iVn) = M(fi)c{bi) U .M(^) = -M(/*)c(6) U M(u)c(b) = (M(ji) U M{v))c{b). Hence V d G Im(/x)Ulm(i/), Al(/x fl ^)c(d) = (-M(M) U -M(/x))c(d) under the assumption that bi < b < fei+1. (2) Now assume that 6 < bo- Then 6 G Im(/x D i/). (2.1) Let fy G Im(jtx), fy < 1. (2.1.1) Suppose that bj £ Im(/xD v). Then (/i D v)b. = ( / i f l ^ K + 1 since 6j + i is the next possible largest value in Im(/i fl v). Thus M(/JL H ^) C (6J) —

M((t* n i/)6i) = M((n n i/)6i+1) = -M(/x6i+1 n ubj+l) = M / ^ + i ) u A^K + 1 ) = A^(M6i+1) U Ai(^i) = M(fi)c(bj) U X(^) C (6) =; M(fi)c{bj) U M(^)C(6J.) = (M(M)UMM) C ( 6 J .). (2.1.2) Suppose that bj G Im(/i fl ^). Let j * be the smallest integer > j such that 6j* + i G Im(/ifl ^). Then (// D ^)&i+1 = (/i fl ^ ) 6 r + 1 . Thus JM(/Z fl

v)^) = M((n n v)bj*+1) = M((n n ^)6,-+1) = X(/^J+i n ^.+1) = M(fibj+1) U A^K + 1 ) = M(/i6i+1) U .M(i/i) = yW(/i)c(6.) U M{y)c{h) = M(ii)c{bj) U MMc(6j)^(M(M)U^(,))c(M. (2.2) We now consider b. Let j > 0 be the smallest integer such that bj G Im(^ H i/). Then (/x H i/) 6i = (/x n i/)^- Thus M(/i H i/) c(6) = ,M((^ fl v)bj) = M((ii H r/)6o) = M(iib0 n ^ 0 ) - M ( i J n i^) = M ( ^ ) = M{y)c{h) = 0 U A%)c(6) = -M(/i)c(6) U A<(i/)c(6) = (M(IJ) U M(i/)) c( 6). Thus V d G Im(/x) Ulm(i/), X ( / i H i/) c(d) = (M(/x) U M(v))c{d). Now suppose that d £ Im(/i) U Im(i/). Then d £ Im(/i D v) and so c(d) ^ lmM(fJL n i/). Hence A^(/i fl i/)c(d) = A^((/x fl v)d) = Al(/id D ^ ) = M(fid) U M(i/ d ) = M(/x) c ( d ) U -M(i/)c(d) - ( M M U M(u))c{d). Thus V cf G L, M(/z H ^)c(d) = ( A<(M) U ^ M ) c ( d ) - Let a G L\c(L). Since c(L) D c(Im(/x) U Im(z/)), a ^ c(Im(/i) U Im(z/)) and so c(a) ^ Im(/x) U Im(^). Hence c(a) £ Im(/i n i/). Thus M(fx D i/) a = M((fi D v)c{a)) = M(v>c{a) H^c(a)) = M(iic(a))

273

U M(vc{a)) = M(fji)a U M(v)a M(/i) U M(v). M

= (M(fi) U M(u))a.

Therefore, M(fx n i/) =

Theorem 7.1.13 If /i and v are finite-valued L-ideals of R such that /x(0) = i/(0) = 1, then M(fjL Hu) = MQx) U .M(i/). Proof. Suppose that Im(i/) = {&o,&i, • • • A l > where 60 < h < . . . < bq. Define the L-subsets i/W of i j by i/W(a;) = 6; if x G iJ \ i ^ . + 1 and i/W(x) = 1 if £ G i^. +1 for z = 0 , 1 , . . . , q - 1. Then i/ = n?~0V*) and each i/W is two-valued. Hence M(/i n z/) = .M^x n i/°> n . . . n i/(9-i)) = M ^ \JM(VW) U...U 1 A^(^-i)) = M(fi) U M(i/(°) H . . . H i/te" )) = X ( / i ) U A4(i/) by Lemma 7.1.12. ■ Lemma 7.1.14 Suppose L is dense at 1, z.e., Va G £\{1}, 3b £ L such that a < b < 1. Le£ x and V be finite-valued L-subsets of a set S. If Xb — Vb V b G [0,1), thenxi = ViProof. 1 G Im(x) and 1 £ 1111(77) is impossible since if a is the largest element in 1111(77), then a < 1 and so Xb ^ 0 — ^6 for some 6 G L such that a < b < 1, a contradiction. Suppose that 1 ^ Im(x) and 1 ^ 1111(77). Then xi = 0 = 771. Suppose that 1 G Im(x) and 1 G Im(r7). If |Im(x)| = 1 = |Im(r7)|, then the result is immediate. Suppose that |Im(x)| > 2. Since Im(x) and 1111(77) a r e finite, 3 b G Im(x), b < 1 such that xi = Xb = Vb = Vi- " We need not assume L is dense at 1 in the remainder of the section since proofs of the next two results exist which are similar to those of Lemma 7.1.12 and Theorem 7.1.13. However, we make this assumption in order to present a slightly different type of proof. Lemma 7.1.15 Suppose that a and (3 are algebraic L-varieties such that Im(a) = {ao, «i) • • •, a n } , where 0 = ao < a\ < ... < an and Im((3) = {0,a}, where 0 < a < an. Thenl(aUp) = 1(a) Hl(/3). Proof. We note that 0 = ao < a so there always exists largest ai < a. Suppose that ai < a < ai+\. Then Im(a U /?) 2 {a a n } (1) We first show that l(a U 0)c{aj) = (1(a) fU(/3))c{aj) V a5 G Im(a). (1.1) Suppose that 0 < aj < ai. (1.1.1) Suppose that ad £ lm(a U (3). Then (a U /?)0j. = (a U /?)a , , since aj + i is the next largest possible element in Im(a U /3). Thus X(a U /3)c(aj) = I ( ( a U /?) a ,) (Proposition 7.1.3(3)) = J ( ( a U /?)„ i+1 ) = J K i + 1 U paj+l) = X(a„ i + 1 ) n X(A, i+1 ) (crisp case) = I ( a ) c ( o i ) n I(fia) (Proposition 7.1.3(1) and 0 < aj+1 < a) = l(a)c(aj) n X(/?)c(o) = I(a)c( 0 i ) n X(/3)cK) = (Z(«) nl(/3))c(a,)-

274

(1.1.2) Suppose that a6 e Im(a U 0). ThenX(a U 0)c(aj) = 1((<* U 0)aj.+1), where j * is the smallest integer > j such that a^.+i € lm(a U 0). Then (aU /3) a .. + 1 = (aU /J) 0 i + 1 . Hence J ( a U 0)c{aj) = X((a U /?) a .. + 1 ) = X((a U 0) O i + 1 ) = X(a a . + 1 U 0 Oi+1 ) - X(a„ i + 1 ) n I(/?„ i + i) = X(a) c ( o . } D X(/?) c(a .) (as just argued) = ( X ( a ) n X ( / ? ) ) c ( 0 j ) . (1.2) Suppose now that a,j > Oj+i. (1.2.1) Suppose that j < n. Then X(a U 0)c(aj) = X((a U P)aj+1) (since a.,-, a j + 1 € Im(a U 0)) = X(a a , + 1 U /3 a , +1 ) = X(a a . + 1 ) n X(A, i+1 ) = X(a) c ( a .) n 1(0) = I ( a ) c ( a . ) n i? = I ( a ) c ( 0 i ) n I(/3) c( a) = X(a) c ( a .) n X(/3)c(a.) = (1(a) ft !(/?))<:(«,)• (1.2.2) Suppose that j = n. Then l(a U /?)c(a„) = -R = (X(a) n X(/3)) c(a „). (1.3) We now consider the case aj = a^+i. Then a,- € Im(a U /3). (1.3.1) Suppose that a i + 1 < a n Then X(a U /?) c ( 0j ) = X((a U /?)a i+ i) =

I K + 1 u /J„i+1) - x(aa.+1) n i(/3a,+1) - x(a) c(a . } n J(0) = i(a) c(oj) n i? - J ( a ) c ( o i ) n I(/3) c ( a ) = X(a) c ( a . } n X(/?) c(a . } = (X(a) D X(/3)) c(o .). (1.3.2) Suppose that a i + i = a n . Then X(a U 0)c(aj) = R = R n R = J ( a ) c ( a i ) n J(/3) c ( a i ) - (1(a) n I(/?)) c( a,)(1.4) We consider the case aj = a*. (1.4.1) Suppose that a* £ Im(a U /?). Then (a U /?) ai = (a U /3) a . Thus X(a U 0)c(ai) = X((a U 0)ai) = I ( ( a U 0) o ) - X(a a U /3a) = X(a a ) n !(/?„) = X(a a i + 1 ) n X(/?a) = I ( a ) c ( a i ) n X(/?) c(0) = X(a) c ( o i ) D X(/3) c(Qi) - (1(a) n X(/?)) c(ai) . (1.4.2) Suppose that a^ e Im(aU 0). (1.4.2.1) Suppose that a £ Im(a U /?). Then a,i < a < ai+i. Thus X(a U 0)c{ai) = J ( ( a U /?) 0i+1 ) - I ( ( o U £)„) = X(a a U /3„) - J(a„ 1 + 1 U 0a) = X(a a i + 1 ) D !(/?„) = J ( a ) c ( o i ) n I(/3) c( o) = X(a) c ( a i ) n X(/?) c(0i) = (1(a) n X(/3)) c(ai) . (1.4.2.2) Suppose that a e Im(a U /?). Then X(a U /3) c(oi) = X((a U /?)«) = X(a a U /?a) = I ( a a ) n l(0a) = X(a a i + 1 ) n T(/3) c(0) = X(a) c ( a i ) n X(/3) c(ai)

= (X(a)nx(/?))c(ai). (2) We consider a. (2.1) Assume that a; < a < a i + i . (2.1.1) Suppose that a € Im(a U 0). Then X(a U /3) c(o) = X((a U /3) 0i+1 ) - I ( o 0 4 + 1 U /3 0i+1 ) = J ( a Q i + 1 ) n X(/? ai+l ) = X(a) c ( a i ) n 1(0) = X(a) c ( a ) n i i = X(a) c ( a ) n X(/?) c(a) - (1(a) n X(/3)) c(a) . (2.1.2) Suppose that a £ Im(aU 0). Since aj < a < a i + i , (a U 0)a = (a U /3) 0i+1 . Thus I ( a U /3) c(a) = X((a U /3)a) = X((a U 0)ai+l) = I(aai+1 U 0ai+1) = I ( a a i + 1 ) n I(/3a i + l ) = I ( a ) c ( o i ) n 1(0) = I ( a ) c ( o i ) n i? = T(a) c ( a ) ni(/?) c ( „) = ( J ( a ) n i ( / ? ) ) c ( o ) . (2.2) Suppose that a = Oi+i. (2.2.1) Suppose that ai+1 < an. Then I ( a U 0)
275

l(<*)c(a) n J(/3) c ( o ) - (1(a) n T(/?)) c( a). (2.2.2) Suppose that a = ai+1 = an. Then I ( a U /3) c(a) = R = Rn R = l(*)c(a) fl Z(/J) c(fl) = (1(a) H T(/J)) c(a ). Thus V d G Im(a) U Im(/3) \ { 0 } , I ( a U / 3 ) c M = ( I ( a ) n l ( / 3 ) ) c ( d ) . Now suppose that d £ Im(a)Ulm(/3). Then d £ Im(a U 0) and so c(d) i Im(l(a U /?)). Hence l(a U /3) c(d) = l((a U /J) d ) = J ( a d U & ) = I ( a d ) n l(0d) = I ( a ) c ( d ) n l ( / 3 ) c ( d ) = (Z(a) nZ(/3)) c ( d ) . Let 6 € L\c(L). Then b $ c(Im(a)Ulm(0)) and so c(b) £ lm(a) U 1(0). Hence c(6) £ Im(a U 0). Thus X(aU/?) 6 =I((aU/3) c ( 6 ) ) = J K ( f c ) U / 3 c ( 6 ) ) - J(a c ( 6 ) )nJ(/? c ( f e ) ) = T ( a ) 6 n J ( / ? ) 6 = (J(a)nl(/3))6. Finally, we consider the case c(0) = 1. We have shown V 6 G L\{1} that I(aU0)b = (l(a)Dl(0))b. That I(aU0)1 = (T(a)nX(/?))i foUows by Lemma 7.1.14. ■ Theorem 7.1.16 If a and 0 are algebraic L-varieties such that 0 G Im(a)C\ lm(0), then l(a U 0) = 1(a) n T(/3). Proof. Suppose that Im(oj) = {ao, a i , . . . , a n } , where ao < ai < . . . < an and lm(0) = { 6 0 , 6 i , . . . , 6 m }, where b0 < bi < . . . < 6 m . There is no loss in generality in assuming that bm < an. Define the fuzzy subsets /?W of Ck by /?« (z) = 0 if z G Cfc \ 0bi+l and /jW (*) = bi+1 if z G /? 6 , +1 for i = 0 , 1 , . . . , m - 1 . Then /? = U ^ 1 /? (i) a n d eac *i /? (i) is two-valued with 0 G Im(/?«). Each /3« is an algebraic L-variety by Theorem 7.1.8. Hence by repeated use of Lemma 7.1.15, l(a U 0) = l(a U /?(°) U . . . U pi™'1)) = l(a U /?(°> U . . . U ft™-2)) n

j(/?(^-i)) = ... = i(a) n z(/?(°)) n... n 1(0^-^) = x(a) n z(/?(°>) n... n J(/j(™-3))

7.2

n

J(0(m-2)

y

^(m-l))

=

j^

R

j ^ ( 0 ) y _ y ^(m-l))

=

J( a )

R

Irreducible Algebraic L-Varieties

Definition 7.2.1 Let a be an algebraic L-variety. Then a is irreducible if V algebraic L-varieties £ and £ such that a = ^ U ( either a = £ or a = £; otherwise a is called reducible. Theorem 7.2.2 Let a be an algebraic L-variety. Then a is irreducible and nonconstant if and only if Im(a) = {0,a}, 0 < a, and aa is irreducible. Proof. Suppose that a is irreducible. Let Im(a) = {a 0 ,ai, • • • ,«n} 5 where ao < ai < ... < an and suppose that n > 2. Define the L-subsets £ and C of Cfc by £(z) = a n if z G a 0 n , £(*) = a n -2 i f 2 e ^ . A ^ , £(2) = a(z) otherwise and £(z) = a n _! if z G a a n , £(z) = 01(2:) otherwise. Then £ and £ are

276

algebraic L-varieties by Theorem 7.1.8. Let z G Ck. If z G a n , then (f U C)(^) = Z(z) v CM = a n V a n _i = a n = a{z). If z G a n - i \ a n , then (£ U C)(*) = £(z) V C(z) = a n _ 2 V a n _i = a n _i = a(z). If z <£ a n _ i , then (f U C)(z) = £(z) V C(z) = OL(Z) V a(z) = a(z). Thus a = f U C and a D £ and a D (. Hence a is not irreducible, a contradiction. Thus n — \ and Im(a) = {ao, a i } . Suppose that 0 < a 0 . Define the L-subsets £ and C of Ck by £(z) = ai if z G a a i , £(z) = 0 otherwise and £(z) = a0 Vz e Ck. Then £ and C are algebraic L-varieties by Theorem 7.1.8. Let z G Ck. If z G a 0 l , then (f U C)(z) = £(z) V £(z) = ai V a 0 = ax - a(z). If z £ a 0 l , then (£ U C)(*) = £(*) V £(*) — 0 V ao = ao = a(z). Now a = £ U C and a D £ and a D ( . Hence a is not irreducible, a contradiction. Thus ao = 0 and Im(a) = {0,a\). Suppose that aai = G U H, where G and H are algebraic varieties such that aai D G and aai 0> H. Define the L-subsets £ and C of Ck by £(z) = ai if z G G, £(z) = 0 otherwise and £(z) = ai if z G J7, CC2) — 0 otherwise. Then £ and £ are algebraic L-varieties and a = £UC. However, this contradicts the irreducibility of a since a D £ and Q D ( . Thus a a i is irreducible. Conversely, suppose that Im(a) = {0, a}, 0 < a, and a a is irreducible. Suppose that a = £ U £, where £ and C are algebraic L-varieties. Then aa = (^ U ( ) a = ( a U ( a and £ a , Ca are algebraic varieties by Theorem 7.1.8. Since aa is irreducible, either £a = aa or £a = a a , say £ a = aa. Now a is the largest value a takes on and a D £. Thus V 2 G £a, £(*) = a = a(z). Let z G G f c \a a = G fe \£ a . Then 0 = a(z) > £(z) > 0. Hence a = £. Thus a is irreducible. ■ Example 7.2.3 Le£ L = {l,a,6,0}, where 1 > a, 6 > 0 anrf a ana7 6 are not comparable. Let G be an irreducible variety. Define the L-subset a of Ck by a(z) = 1 if z G G and a(z) = 0 otherwise. Define the L-subsets £ and £ of Ck by £(z) =aifzeG, £(z) = 0 if z £ G; £(*) = b if z G G, C(*) = 0 z/z £ G. Le* z eCk. If z e G, tten (£ U £)(*) = £(*) V £(*) = a V b = 1 = a{z). If z£G, then (£ U C)(z) = £(2) V CO*) = 0 V 0 = 0 = a(z). #ence a = £U( and a D £ and a D ( . Hence we see that Theorem 7.2.2 does na£ aave an immediate generalization to the case where L is not a chain. Theorem 7.2.4 Let fi be a nonconstant finite-valued L-ideal of R. Then A4(fi) is irreducible if and only if T(M.(n)) is prime. Proof. Suppose that l(M([i)) is prime. Then lm(T(M(n))) = {6,1}, where b < 1. Hence lm(M(fi)) = {0,c(6)} since M(I(M(fi))) = M(fi). Now M(I(M(ij)))c(b) = M(I(M(IL))X) by Proposition 7.1.4. Since I(M(fj))i is a prime ideal of R, A4(/x)c(6) = X(X(X(/z))) c ( 6 ) is irreducible. Thus M(fi) is irreducible by Theorem 7.2.2. Conversely, suppose that M{y) is irreducible. Then Tm(M(n)) = {0,a}, where 0 < a. Hence lm(l(M(/i))) = {c(a), 1}. Now M{fj)a = M(l(M(/j,)))c{c{a)) = ,M(I(M(Ai))i). Since M(/i)a is irreducible, X(M(/J))I is prime. Thus X(M(fj)) is prime. ■

277

Theorem 7.2.5 Let \i be a finite-valued L-ideal of R with ^(0) = 1. Then X(M(JJ)) = v? = n(fi). Proof. By Theorem 6.1.10, ll(fi) = yfjx. V b G [0,1], (y/Ji)b = y/fib = I(M(nb)) = I(M(n))b by Proposition 7.1.6. ■ Since this section deals with L-ideals \i which are finite-valued and have the property that /x(0) = 1, it is evident by now that the results in this section hold interchangeably with y/Jl and TZ(/JL). Corollary 7.2.6 Let /i and v be finite-valued L-ideals of R such that /i(0) = ^(0) = 1. Then M(fi) C M.{y) if and only if yffl D y/v. Proof. By Theorem 7.2.5, M(fi) C M(v) <£> y/JI = X(M(fi)) D X(M{v)) = y/v, where strict containment is preserved by Proposition 7.1.7. ■ Theorem 7.2.7 There exists a one-to-one correspondence between algebraic L-varieties a with 0 G Im{a) and radical L-ideals. Proof. Let a be an algebraic L-variety. Then 3 an L-ideal ^ of R such that a = M(fi). Then /z(0) = 1. Consider the correspondence a = A4(fi) H-» y/JI. By Corollary 7.2.6, M(fi) = M{y) <£> y/JI = y/v. Hence the correspon­ dence is single-valued and one-to-one. Given y/JI, M(fJ,) »—» y/JI and so the correspondence is onto. ■ Theorem 7.2.8 Every algebraic L-variety a with 0 G Im(a) can be uniquely expressed as the union of a finite number of irreducible algebraic L-varieties no one of which is contained in the union of the others. Proof. Now a — M.(ii) for some finite-valued L-ideal fi of R with /x(0) = 1. Now X(AA(fi)) = y/JI and y/JI has an unique irredundant 7^-primary Lrepresentation X(M.(/J,)) = y/JI = -K\ f l . . . H 7i>, where TVi is a prime L-ideal of R, i = 1 , . . . , r. Thus by Theorem 7.1.13, M(fi) = M(I(M(H)))

= Mfa

fl . . . n 7Tr) = M(7T!) U . . . U M(7Tr),

where Aifa) is irreducible, i = 1 , . . . , r. TfMfc) = U{M(TTJ) \ j = 1 , . . . , r; j ^ i}, then in = X( Mfc)) = X(U{M(
278

Given a finite-valued L-ideal fi of R with /z(0) = 1. Let yJJL = TTI f i . . . fl 7rr be an irredundant 7£-primary L-representation. Then the TTI are the minimal prime L-ideals belonging to /i by Theorem 6.2.20. We thus may obtain M{ji) as the union of algebraic L-varieties of the minimal prime L-ideals among the prime L-ideals belonging to the 7^-primary L-ideals in an irredundant 1Zprimary L-representation of fi. E x a m p l e 7.2.9 Let L = [0,1]. Let R = F[x, y) z], where F is the field of complex numbers and x,y, z are algebraically independent indeterminates over F. Define the fuzzy subset /J, of R by

1 y

'

iff = 0

) I

,x2z )\(x2z) (x2+y2-l,x-

if

iffeR\(x2+y2-\,x2z).

[ 0

Then fi is a fuzzy ideal of R. Now ^f]x is such that

' i i >/?(/) = { 1 0

iff = o iffe(xz)\(0) if(x2+y2-l,xz)\(xz) iffeR\(x2+y2-l,xz).

Hence

Mo = = »i = — Mi

n

R (x2 + y2 (x2 *> (0)

l,x2z)

^JJiI =

(x2 + y2

l,xz)

xz

V^

=

( )

^/M^

-

(0).

Fk = M((0)), MQM)

H--=

f

4)

c(0) = l

if w € M((0))\M((xz)) if w€ M((xz))\M((x2 + y 2 - l , xz)) 2 2 ifw€M((x +y -l,xz)).

Consider the fuzzy subsets ui, \, V of R defined by

«(/) = (

1

\ 0

X(f)

iff

e(x2+y2-l,x2z)

otherwise

iff£(x2z) otherwise

iffe(0) otherwise

279 Then u, \ , V are fuzzy ideals of R and \x = u D x n n. Define the fuzzy subset V>« ofR,i=l,...,6by

V>(1)(/) = { 01

i//6<x2,y-l) otherwise

,2/ + i) ^ (2) (/) = { 0l i/7e(x otherwise + y -l,z) v>(3)(/) = { I iff£(x otherwise 2

2

i//e<x2)

f l

^(4)(/) = ^ ^

J

2

otherwise

1

^(5)(/) = [ ^

otherwise

V><«)(/) = f 1


(^ ^

otherwise.

Then ^W is a /uzz?/ ideaZ 0/ iJ, z = 1 , . . . , 6, suc/i £/m£ a; = i/A1) Pi i / ^ H ?/>(3) since (x2 + y2 - l , x 2 z ) = (x 2 ,y - 1) H (x 2 ,y + 1) H (x 2 + y 2 - l , z ) , and % = V>(4) H V>(5) «ircce (x2z) = (x2) D (z) . Afoo 77 = i/>(6)- Thus \x — nf=1i/j^ and in fact this is an irredundant fuzzy primary representation of fi. Now iffe(x,y-l) otherwise VW w ) " | Q otherwise iffe(x,y + l) otherwise W \J) — ^ Q otherwise

W(f) =■{1 WKf) =■{i

V^w = V><3> 1

VWKf) = {2

iff€(x) otherwise

(5) i>W v/^w == V v/^w / 6) sftpid ^(6)" = V ((6) -. V

Hence ^//I = nf=17rW, where 7rW = y/ipW is a fuzzy prime ideal of R, i = 1 , . . . , 6. We have the following fuzzy algebraic varieties:

M{«W)(w) = | J IifweM((x,y-l)) therwise

280

v

/v

'

I U

otherwise

€M M*v (3))(«0 ^ /v ' = \1 0\ *" otherwise

M{*W)(w) = l
0 v y = ( n M-v ( 5 ) /)H [0

M(7r^)(w)

=

+ y2 1

~ ' *»

if^M((x))

otherwise €

()

^ ^ otherwise

c{l-)VweCk.

Then M.(fi) = Uf=1.M(7rW) and in fact M.(TT^) is irreducible and no M.(jr^) is contained in the union of the others, i = 1 , . . . ,6. Consider the nonlinear system of equations of fuzzy singletons, (* 6 ) 2 + ( 2 / a ) 2 - U (xb)2zu

= =

0l 0±.

Then a solution is given by a > j and b A u = ^ and the solution of x2 -f- y2 — 1 = 0 and x2z = 0. Note also that fi = ((x2 + y2 — l ) i , (x2z)i \ by Corollary 3.1.29. If we let c(0) = l , c ( i ) = i > c ( i ) = i , c ( l ) = 0 J then the above representation of M(fi) seems to better represent the solution of the above nonlinear system of equations of fuzzy singletons. The M.(ir^) for i = 1,2,3, yield the crisp part of the solution while the M.(TT^) for i = 4, 5,6 yield the fuzzy part. In Example 7.2.9, it was shown how a solution of a system of fuzzy inter­ section equations could be displayed by a primary representation of the L-ideal generated by the defining polynomials of the intersection equations. We now show this holds in general. Theorem 7.2.10 Let \i = ( ( / i ) 0 l , • • ■, (fq)aq) U 1 {0 }, where A , . . . , fq G R, 1 > ai > . . . > aq > 0 and a ^ aq. Suppose that ( / i , . . . , / g ) i=- R. Let { a ^ , . . . , airn } = { a i , . . . , aq} be such that a^ > . . . > a^m. Let Farn-u-y

= {fkWk >

airn_u},

281

u = 0 , l , . . . , r a - l , andletTairrx 0/ i? as follows:

= {/i,..,/ g }. Define the L-subsets u, uu ... ,cj m

w(r) = =- ;

i1 tfre^o ^€^0

f 1

ilre(F ifre(T ai ai

, ,

10 VrtfraJ)

w«W = ^F 1( aim_u+1

\

x

m —u

)

ifr
uu — 1 , . . . ,m — 1. "mi?) = W m ^ - \ 1 a«,i l

t/re(0) z/r£(0)

T/ien a;, a ; i , . . . , a;m are L-ideals of R and /z = a; fl a;i f l . . . fl a;m. Proof. Note that TaiQ = 0- That a;, CJI, . . . ,a; m are L-ideals of R follows from the fact that their level sets are ideals of R. Let r E R. Then

[' 00 = <| ( wa ,nnwWi fl in. . . •. nn w^m) )((rr)) =

aaim-3^

a*, Oil [, 1 1

i f rifreR\(F e u \ < ^ a 4 mai>J

ifi rf e^ (( ^^ )^\ }( \^ ^. 1^ )) iiffrr€6<<^^ iali)l \>(\0( 0> ) e (0) (0).• if rr €

By Corollary 3.1.29, M W = V{((r 1 ) 1 (/ 1 ) 6 l + . . . + ( r , ) i ( / , ) 6 , ) ( r ) \neR,bi = a, if rj 7^ 0 and bi = 1 otherwise, i = 1 , . . . , q}. Now r € \^airn_ ) \ (^oi ._t ) if and only if 3 a representation r i / i + . . . 4- rqfq of r such that some /& has rk ^ 0, where ak = i D . . . D w m . ■ Theorem 7.2.11 Lei /i = ( ( / i ) 0 l , . . . , (/„)„,) U l{o}, w/iere / 1 ? . . . , fq 6 R, I > ai > ... > aq > 0 and a\ ^ aq. Suppose that ( / i , . . . , fq) ^ R. Let u, c^i,... ,a; m 6e defined as in Theorem 7.2.10. Let lu; = Qoi Qok0 lu, Ooi nn..... • n Qofco and i Wu = Qu\ Qui nn. ..•. • n QQnuku fcu lu^

282

be IZ-primary representations of 1^ and lu,u, respectively, u = 1 , . . . , ra. For each u = 0 , 1 , . . . , ra, define the L-subsets fiui,...,/iuku of R as follows: V reR, , v [ 1 iif/ rr e QOJ Moj(r) ~i / r ■ i Qoj

■{i

j = l,...,fc0. fiuj(r)

= <

f 1

{ aim_u+l

ifrE

Quj

ifr£Quj

j = = l,...,/cu; u u = = l,...,ra.

T/ien the £/ie following assertions hold: TAera (1) l^ui» (1) l^ui 5 • • •»/^ufc • • ? ^uk L-ideals L-ideals of R, of ui?,=u 0=, 1 0, .,.1. ,,ra; . m; .. u are u are (%) w uu = = // i uu i n . . . f l /x u ib u , ix u = 00 ,, 11 ,, .. .. .. ,, ra; ra; n n (^nn ... n^i ) n... n (/x ■n... ^ M = (/i fjLoko) fcm) m 25) n . • n / i mn/w 0i n... (3; (/xoi n... n n/zoo (win ... n/j,lkl) fcl n. • • n (/Ami mi

w

an IZ-primary L-representation of \i.

Proof. If u>u = IK? then ku = 1 and Q u i = i£. (In this case, we may simply drop uu.) Assertion (1) holds since the level sets of the fj'uj &re ideals of R. Now let r 6 R. Then for j = 1 , . . . , ku and u = 1 , . . . , ra, /xujO") = ai m _ tt+ i if and only if r ^ QUJ-. Hence for u = 1 , . . . , m, (xxul fl . . . D /J,ukj)(r) = aim_u+1 <^ r £ Quj for some j <£> r £ lWu = {Faim_u y «* w u (r) = a i m _ u + 1 . Similarly, (/xoiH.. .n/io/fej)(r) = 0 if and only if c*;(r) = 0. Thus it follows that assertion (2) holds. The representation of \x in (3) follows by the appropriate substitution. That the /iuj are 7^-primary L-ideals follows from Theorem 3.9.4. ■ Let R denote the polynomial ring F [ x i , . . . , xn] in n indeterminates over the field F. Then every ideal of R has a primary representation. Let feu-

fcn,j

(7.1) (7-i) 2 • • • E K -. .ww O Mi C^^ ii)^^ )r* . • • • ((*»)*„,)*- = ( i)*,>j = i,• • •.«. E11

*i=i

*„=i

r

••((*n)fcni)in=(rj)ai,j = l>..

■ ,q,

denote g nonhnear equations in the L-singletons (xi)bx,..., (#n)6„> where 6^= 6f if Xi appears in equation j and 1 otherwise, i — 1 , . . . , n; j = 1 , . . . , q and where the (^...1^)1 and the (rj)aj are L-singletons and the rj and the r ^ . ^ are in F. Let fclj

Knj

1 in ,j =j = l , . 1, ...,g. i ) ' 1 • •••(a;n) £ = E ' ■•'• E w^W • • (*n)S • • • ,9. Eri^ i - i. -. W 21=1

in = l

Then the system of equations (7.1) is equivalent to the following two systems of equations:

283

fj=rj,

j = l,...,g

(7.2)

and hj A . . . A bnj =dj,

j = 1 , . . . , q.

(7.3)

Let fi = < ( / i ) a i , • •., (fq)aq) U 1 {0 }. It is clear that in (3) of Theorem 7.2.11, y/JJ^ H . . . n ^/Mofco gives, via unions of the corresponding irreducible algebraic L-varieties, the crisp part, (7.2), of the solution to the L-intersection equations, (7.1), while

(vT^TTn... n y/ji^) n... n (V/wn... n vTwT) gives the fuzzy part, (7.3).

7.3

Localized L-Subrings

The notion of algebraic L-varieties was introduced in order to use primary representation theory of L-ideals to examine the solution of fuzzy intersec­ tion equations. The concepts of quasi-local L-subrings and complete local L-subrings were developed in order to lay the ground work for the examina­ tion of fuzzy intersection equations locally. In this section, we characterize local rings in terms of certain L-ideals. We also characterize rings of fractions at a prime ideal in terms of L-ideals. We apply our results to fuzzy intersection equations. In particular, we show that the L-ideal which represents a system of fuzzy intersection equations in a polynomial ring is such that its extension in a ring of fractions represents the same system of fuzzy intersection equations. Throughout this section, R denotes a commutative ring with identity and L is a dense chain. Recall that fi* = {x G R\/JL(X) = ^(0)}. Define /i# = {x G R\fj,(x) > ^(1)}. Then JA* and fi# are ideals of R. Let S be a set of L-singletons of R such that if x a , x\, £ 5 , then a = b > 0. Recall that foot (S) = {x\xa G S}. If fi is an L-ideal of R such that fi = (S) U 0^(0) for some <S, then S is called a g e n e r a t i n g set for \x. If S is a generating set for /z, and (5\{x a }) U 0M(0) C [i Vxa G 5 , then S is called a minimal generating set for \x. If 5 is a subset i?, we let (S) denote the ideal of R generated by S. A commutative ring with identity, but not necessarily Noetherian, is said to be local if it has a unique maximal ideal. (Such a ring is called quasi-local in Section 6.5). In Section 6.5, the definition of a quasi-local L-subring of R was given when R was assumed to be local. That is, an L-subring fi of a local ring R was called quasi-local if ji(x) = /x(x _1 ) for all units x of R. If [i is an

284

L-ideal of R, then fi(x) = iz(l) for all units x of R. Hence if R is a local ring and fi is an L-ideal of R, then /i is a quasi-local L-subring of R. We also know that if fi is an L-ideal of R, then fi(y) > fi(l) \A/ G R. If // is a nonconstant L-ideal of # , then ^(0) > ii(l). Definition 7.3.1 An L-ideal n of R is called local if\/x G R, fJ,(x) = ^(1) is equivalent to x being a unit in R. Note that if \x is an L-ideal of R which is local, then /i is not constant since 0 is not a unit of R. Let R denote the polynomial ring F[x] over the field F. Define the L-subring /x of R by fj,(z) = 1 if z = 0, /i(z) = ± if z G F \ { 0 } , and ^(z) = I if z G i?\F. Then ^ is an L-subring of R. Also, /x(z) = /i(l) if and only if z is a unit. However, ji is not an L-ideal of R. We also note that R is not a local ring. Lemma 7.3.2 Let [i he a nonconstant L-ideal of R. Then fi is local if and only if /z# is the unique maximal ideal of R. Proof. \x is local <^> Va: G R, [i{x) > zz(l) is equivalent to being a non-unit in R <£> /A# contains all the non-units of R o fi# is the unique maximal ideal of R. * Example 7.3.3 Let L = [0,1]. Let ji he the fuzzy suhset of the ring ZQ defined as follows, where 0 < a$ < a\ < a,2 < 1 : r a0

rtx)=

\

a

i

[ a2

ifxeZ9\(3)

*/*e(3)\{0} if x = 0.

Then \i is an L-ideal of R. Suppose that ao < a\. Then fi# = (3) is the unique maximal ideal of Zg and so [i is local. Suppose that ao = a\. Then \i is not local even though JJ, is a quasi-local L-subring of R. The ring ZQ has no L-ideals which are local since ZQ is not local. Recall that an L-ideal /i of R is a generalized maximal L-ideal if /x is not constant and for any L-ideal v of i?, if \x C v, then either /z* = v* or v = 1R. Then an L-ideal fi of R is generalized maximal if and only if |Im(/i)| = 2, ^(0) = 1, and n* is a maximal ideal of R. We see that the L-ideal fi of Example 7.3.3 with a 0 < a\ and a2 = 1 is not maximal even though fi is local. Let fi and A be L-ideals of R. Then fi and A are said to be equivalent if {/ia | a G lm(fi)} = {Aa | a G Im(A)}.

285

Theorem 7.3.4 The following (1) R is local; (2) R has an L-ideal which (3) all generalized maximal (4) all generalized maximal

conditions are equivalent: is local; L-ideals of R are local; L-ideals of R are equivalent.

Proof. (1) o (2): Suppose that R is local and M is the unique maximal ideal of R. Then \M, the characteristic function of M in i?, is an L-ideal of R which is local. The converse follows by Lemma 7.3.2. (2) => (3): Let A be a generalized maximal L-ideal of R. Then A* is the unique maximal ideal of R. Since |Im(A)| = 2, A* = A#. Thus A is local by Lemma 7.3.2. (3)=>(4): Suppose ji and A are generalized maximal L-ideals of R. Then fi and A are local. Hence /x* = /i# = A# = A*. Therefore, \i and A have the same chain of level ideals. (4) =» (1): Trivial. ■ If R is Artinian, we say that an L-ideal \x of R is of maximal chain if the level ideals of fi form a composition series. Theorem 7.3.5 Let R be Artinian. of R of maximal chain is local.

Then R is local if and only if every L-ideal

Proof. Suppose that R is local with unique maximal ideal M. Let fi be an Lideal of R of maximal chain. Since R is Artinian, Im(/x) is finite. Suppose that H# 7^ M. Let Im(/i) = {ao, a i , . . . , a n } , where ao < a\ < . . . < a n . Then \iax = fjL#. Let ao < a < ai. Define the L-subset A of R by X(x) = a if x G M\fi# and A(x) = fi(x) otherwise. Then A is an L-ideal of R such that its level ideals form the chain R = /xao D M = \ia D \iax D Ma2 D • • • D fJ>an = {0}. However, this contradicts the assumption that \i is of maximal chain. Therefore, M = fi# and hence \x is local by Lemma 7.3.2. The converse follows from Theorem 7.3.4 and the fact that there exist Lideals \x of maximal chain. ■ An L-ideal \i of R is called normalized if /x(0) = 1. Theorem 7.3.6 R is a field if and only if the set of all normalized L-ideals of R which are local coincides with the set of all generalized maximal L-ideals

ofR. Proof. Suppose the condition concerning local and generalized maximal Lideals holds. Since R has generalized maximal L-ideals, R has L-ideals which are local. Hence R is local. Let M denote the unique maximal ideal of R. Suppose M ^ (0). Define the L-subset /i of R by /i(0) = 1, fi(x) — a\ if

286

x £ M \ { 0 } , and fj.(x) = a0 if x £ R\M, where 0 < a 0 < ai < 1. Then fi is an L-ideal of R which is local. However, \x is not a generalized maximal L-ideal, a contradiction. Thus M = {0}. Hence R is a field. If R is a field, the desired result follows easily. ■ Throughout the remainder of the section, 5 denotes a closed multiplicative system in R such that 0 ^ 5 and which is saturated, i.e., Vx,y £ R,xy £ S implies x,y £ S [31]. Let RS"1 denote the corresponding ring of fractions. Then RS"1 = {f(r)/f(s) \ r £ R, s £ 5 } , where / is a homomorphism of R into RS"1 such that Ker / = {x £ R\ xs = 0 for some 5 in 5 } and the elements of f(S) are units in RS"1, [261, p. 222]. If I is an ideal of R, we use the notation IS"1 for the ideal of RS"1 generated by / ( / ) . Definition 7.3.7 Assume \x and ji' are L-ideals of R and RS"1, respectively. Then \J is called the localized L-subring of \i in RS"1 if Im(fi) = Im{^') and y!a = \iaS~1 Va elm(fi). In the following example, we show that not every L-ideal of R has a localized L-subring in RS"1. Example 7.3.8 Consider the ring of integers Z. Let L — [0,1]. Let fi = 6 i , 2 ± \ . Then Im(fi) = {0,^,1}, fii = (6), and fii = (2). Let S denote the complement of (2) in Z. Now 2 = 6 • 3 - 1 £ (6) S"1 and so fiiS"1 = (6) S"1 = (2) S"1 = fxiS"1. Thus \i does not have a localized fuzzy subring fi' in XS"1 since if JJ! is a fuzzy ideal of XS"1 such that \Ja = jiaS"1 Va £ Im(fi), then Im{fir) = {1,0} ^ 7m(/x). Example 7.3.9 Let L = [0,1]. Let Z(p) be the localized ring of the ring of integers Z for a prime number p. Let fi and \i' be fuzzy subsets of R = Z and RS"1 = Z( p ), respectively, defined as follows: fi(x) =a,i ifxe H'(x) = aiifxe

^Z\pi+1Z, p%v)\pi+1Z{p),

where i = 0 , 1 , . . . and 0 < a 0 < a\ < ... < 1. Then clearly ji, ^ are fuzzy ideals ofL, Z( p ), respectively, and / / is a localized fuzzy subring of fi in Z( p ). We say that the ring of fractions RS"1 is a localized ring of R at a prime ideal, if there exists a prime ideal P of R such that S — cP, the complement of P in R. Theorem 7.3.10 The ring of fractions RS"1 is a localized ring of R at a prime ideal of R if and only if there exists an L-ideal /i of R which has a localized L-subring \i' in RS"1 and S D # \ M # - ^n suc^ a case> M# ^s a prime ideal of R and RS"1 is a localized ring of R at fj,#.

287

Proof. Suppose RS~* is a localized ring of R at a prime ideal P of R. By Theorem 7.3.4, there exists a local L-ideal / / of RS"1. For all a G Imfju'), let Ia be the contracted ideal of fi'a (i.e., J*c = 7 a ). Then Ia are the counter images of [i!a in i? which are prime to 5, [261, Theorem 15, p. 223], and so \Ja = IaS~x. Also J& C i a if and only if /i£ C \Ja. Let /x be the L-subset of R defined by /JL(X) = \/{a\x G Ia). Then obviously \i is an L-ideal of R. Now x e fib <=> b < W{a\x G Ia} = V{a|/(x) G /x^} = /*'(/(*)) & f(x) G ^ . Hence f(x)/f(m) G / i b S - 1 ^ x G /i&, m G 5 <& f{x) e n'b, m e S & f(x)/f(m) G fi!h, where the latter equivalence is a consequence of the following argument, //(/(a;) = Ai'(/(m)/(x)//(m)) > / / ( / ( m ) ) V , / ( / ( z ) / / ( m ) ) = M ' ( / ( x ) / / ( m ) ) (since / ( m ) is a unit and \J is an L-ideal) > / / ( / ( # ) ) and so / / ( / ( # ) ) = fi{f{x)/f{m))). Thus Im(/i) = Im(/z') and /i^ = fiaS~x. Hence / / is the locahzed L-subring of fi in RS~X. Since / / is a local L-ideal of RS~X and J L ^ = / z # S _ 1 , /i# = P and hence 5 = R\fi#. Conversely, assume fi is an L-ideal of R which has a locahzed L-subring / / in P S - 1 and S 2 R\fi#. Since yu'a = /XaS'-1 for aU a Glm(/z'), /x'(/(r)//(s)) > fi(r) Vr/s G P 5 " 1 and jz(l) = / / ( l ) . Thus jx'(l) = fi'(f (s) / f (s)) > fi(s) > /i(l) = fi\l) and so /z(s) = /x(l) Vs G 5. Therefore, S C ii\/x # . Thus 5 = R\fi# and hence /z# is a prime ideal of R. Therefore, RS_1 is a locahzed ring at fi#. ■ Let S be a set of L-singletons. Define the L-subset as of R by Vx G i£, 0\s(£) = V{a | x a G 5 } . Recall that foot(5) = {x | x a G 5 } . If r G i£ and xa is an L-singleton, we let rxa denote the L-singleton (rx)a. The following result is an easy consequence of Corollary 3.1.29. T h e o r e m 7.3.11 Let S be a set of L-singletons of R. Let a be the L-subset of R defined by Vx G i?, k

^(z) = V { ( ^ r i ( x O a J ( z ) \rieR,xaieS,i

=

l,...,k]keN}.

i=l

Then a — (S), where (S) = (as) • ■ L e m m a 7.3.12 Suppose that fi is an L-ideal of R such that fi has the sup property. Let S — UaeL«Sa, where Sa Q {xa \ x G R, fi(x) = a} if a G Im(fi) and Sa = 0 if a G L\Im(fi). Then fi = (S) U O^o) if and only if fia = (foot(\Jb>aSb)) Va G Im(fi). Proof. Suppose that fi = (S) U 0M(0). Clearly, fia 2 (foot( Ub>a56)). Let y G fl, y + 0, be such that /i(y) = a. Then a = (S) (y) = V{(£* = 1 ^(^)a«)(y)

288

I Ti G R, xai G S,i = l , . . . , / c ; k G N} by Theorem 7.3.11. Thus y G (foot( Ub>aSb)) since fi has the sup property. Hence it follows that fia = (foot(U6>a<S&)). Conversely, suppose that \ia = (foot(U6>a £&)) Va Glm(/z). Let y e R, y ^ 0, be such that ^(2/) = a. Then y G (foot( U&>0S6)). Hence y = X)»=i r ^ i for some r^ e R and (x;) a i G <Sai such that a; > a, 2 = 1 , . . . , k. Thus fi(y) > ((S) U 0 /x(0) )(y) >a = fi(y). Hence /x = (S) U O^o). ■ Example 7.3.13 Let L — [0,1]. Let R = Z. Define the fuzzy subset ^ of R by ( 1 M(r)={ i [ 0

ifre (4) t/r€(2>\(4> ifreR\(2).

Then fi is a fuzzy ideal of R, S = {4i, 2 i } is a minimal generating set for //, HY = (4), and /ii = (4,2) = (2). 77ms {4,2} is not a minimal generating set for n\. However, fi\ has a minimal generating set, namely {2}. Proposition 7.3.14 Suppose [i is an L-ideal of R such that \i has the sup property. Let S = UaeL«Sa, where Sa C { xa \ x G R, fi(x) = a} if a G Im(fi) and Sa = 0 if a G L\Im(fi). If foot(Ub>a(Sb) is a minimal generating set for \ia Va G Im(/i), then S is a minimal generating set for fi. Proof. By Lemma 7.3.12, S is a generating set for (i. Suppose that <S is not a minimal generating set for //. Then 3xc G S such that ji = (S\{xc}). Thus \±a = (foot(( Ub>aSb)\{xc})) Va G Im(/i) by Lemma 7.3.12. Hence it is not the case that foot( Ub>aSb) is a minimal generating set for fia Va G Im(/x). ■ Example 7.3.15 Let L = [0,1]. Consider the polynomial ring 7L\x\ over the ring of integers Z. Let \x = ( 6 i , 2 i , a ; i ) . Le£ 5 denote the complement of (2) m Z[x]. 7%en ira(fz) = {0,^,1}, /xi = (6) , and /ii = (2,x). Now 2 = 6 - 3 - 1 G ( B ^ " 1 andso^S-1 = (6) S " 1 = (2) S " 1 and j ^ S " 1 = {6)x)S~1 = (2,x) S"1. 77ms /z /ms a /lizzy localized subring /jf in Z [ x ] 5 - 1 . The level ide­ als of /z' are fi[ = )Ui5 - 1 and /x^ = fiiS"1. Now { 6 i , 2 i , x i } is a minimal 2

2

2

2

generating set for /J,, but {6i,2i,a;i} zs no£ a minimal generating set for n'. {61,2:1} ana7 { 2 i , # i } are minimal generating sets for / / . We note that (2) z's the contracted ideal of [i!x and not ji\ — (6). In the proof of Theorem 7.3.10, we have I\ = (2) and Ii = (2, x). Definition 7.3.16 Let S denote a set of L-singletons such that if xa and Xb G 5 , then a = b > 0. Let [i be an L-ideal of R. Then S is called an Sminimal generating set for /x if \i = (S) U 0^(0) and V# Efoot(S), there does not exist s G S such that sx G (foot(S)\{x}).

289

Proposition 7.3.17 Let S denote a set of L-singletons such that if xa and x\> G 5 , then a = b > 0. Let fi be an L-ideal of R such that fi has the sup property. If S is an S-minimal generating set for \±, then S is a minimal generating set for /J,. Proof. By definition, S is a generating set for [i. Suppose \i = (S\{xa}) U 0^(0) for some xa G S. Then ( ( 5 \ { x a » U 0 M ( O ) ) ( X ) = a and so 3 yi,y2,. • • ,yk e foot(5\{x a }) such that x = Ylt=i riVi^ where r» G R, i = 1 , . . . , fc, by Theorem 7.3.11. Thus Vs G S,sx = £ * L i ( 5 r ^ G (foot(5)\{x». Hence S is not an 5-minimal generating set for /x. ■ If x a is an L-singleton of R, then f(xa) = f{x)a. Let /z and / / be L-ideals of i? and i ? 5 _ 1 , respectively, such that / / is a localized L-subring of ^ in RS~1. If <S is a set of L-singletons which generate /x, then {f(x)a \ xa G S} generates / / and we say that /i and y! have the same set of generators and we write f{S) for {f(x)a

| ^a € 5 } .

Theorem 7.3.18 Let A, A' be L-ideals of R, RS-1, respectively, such that A has the sup property. If X has an S-minimal generating set and X' is a localized L-subring of X in RS_1, then X and X' have the same minimal generating sets and Im(X) = Im(X'). Conversely, if RS~X is a localized ring at a prime ideal of R, X and Xf have the same minimal generating sets, and Im(X) = Im(Xf), then A7 is a localized L-subring of X in RS~X. Proof. Suppose that S is an 5-minimal generating set for A and A' is a localized L-subring of A in i J 5 _ 1 . Then A(0) = A'(0) and S is a minimal generating set for A by Proposition 7.3.17. Also Im(A) = Im(A') and X'a = XaS~x for all a G Im(A) and so A' has the sup property. Clearly, f(S) generates A'. Suppose f(S) is not a minimal generating set for A7. Then 3 xa G S such that A' = UOA(o) in RS'1. Thus A' = ( £ y b € T ( / ( ^ » U 0 A ( 0 ) , where T = 5 \ { x a } and (f(y)b) is the L-ideal in RS_1 generated by f(y)bHence A'a = ( E (f(y)b))a = E (f(y)b)a 2/b G T

ybeT

(since A' has the sup property)

=

E (f(v)b)a Vb€S"

=

{ioot{S')\{x})S-\

where S' = {yb € S\b > a}, S" = S'\{xa}, and (foot(<S')\M) is the ideal in R generated by foot(S')\{*}- Now Aa = (foot(«S')\{*}) + (x) and in RS'1, ( f o o t ^ O U * } ^ - 1 = A'a = A Q 5- J = (foot(,S / )\{a;})5- 1 + (x)S~1. Hence 3s € S such that f(x) = / ( ( E J U ^ S K ) * - 1 ) aad s o sx ~ E i U r * W e K e r /> where the ^ 6 foot(5')\{x} and the rt € R. Thus there exists s' £ S such

290

that (sfs)x = Yli=i(s'ri)yi- Hence S is not an 5-minimal generating set for A, a contradiction. Thus f(S) is a minimal generating set for A'. Conversely, assume that A and A' have the same minimal generating sets and Im(A) = Im(A'). Also assume \x is the L-ideal of R denned in the proof of the first part of Theorem 7.3.10. Then A' is a localized L-subring of fi in RS'1. Hence Im(A') = Im(/z) and A'a = naS~l Va e Im(A'). Now from the proof above, fi and A' have the same generating sets. Thus [i and A have the same generating sets. Therefore, X = fi and subsequently A' is a localized L-subring of A in RS~X. I We now apply our results in the following example. Example 7.3.19 Let L = [0,1]. Let R denote the polynomial ring M[x,y,z] in the algebraically independent indeterminates x,y,z over the field of HI of real numbers. Then the ideal (x2 — y,x2z} represents the nonlinear system of equations x2 - y = 0 x2z = 0 and has the reduced primary

representation

(x2 - y, x2z) = (x2 -y,z)n

(x2,y)

.

Hence yj{x2 -y,x2z)

= (x2 -y,z)n

(x, y)

2

and the prime ideals (x — y,z) and (x, y) display the solution of the nonlinear system of equations via their corresponding irreducible affine varieties. Now consider the following nonlinear system of fuzzy intersection equations (xb)2 - Va = {xh)2zu =

0i 0i.

Then this system is represented by the fuzzy ideal fi = ({x2 — y)±, (x2z)± \ and S = {(x2 — y)i, {x2z)i} is a minimal generating set for \x. In order to examine the system locally, we consider either of the prime ideals (x2 — y,z) and (x,y), say, P = (x,y) , and we form the quotient ring Rp. Then in Rp, the extended ideal [261] of (x2 — y,z) is (x2 — y, x2z)e = (x2,y)e . Hence the corresponding nonlinear system of fuzzy intersection equations is Va = (xb)2 =

0i 0,.

This system is represented by the fuzzy ideal v — (yi, (x2)± \ in R. Now S = {y±,(x2)i} 4

is a minimal generating set for v. By Theorem 7.3.18, we have 2

291

that S is a minimal generating set for the fuzzy localized subring v' of v in RP. Hence v' represents the same system of fuzzy intersection equations as u does. If we consider the prime ideal N = (x2 -y,z) , then in RN (x2 - y, x2z)e = 2 (x -y,z) . Hence the corresponding nonlinear system of fuzzy intersection equations is {xb)2 - ya

=

01

zu

=

Oi.

4 2

This system is represented by the fuzzy ideal X = /(x 2 — y) i, z\ \ in R. We have that {(x2 — y)±,zi} is a minimal generating set for X and also for the fuzzy localized subring X' of X in Rjy.

7.4

Local Examination

In this section, L denotes a dense chain and R a commutative ring with iden­ tity. The notion of algebraic L-varieties was introduced in order to use pri­ mary representation theory of L-ideals to examine the solution of L-intersection equations. Local concepts of subrings were developed in order to lay the ground work for the examination of L-intersection equations locally. In this section, we carry out a local examination of L-intersection equations. We show that a system of L-intersection equations can be examined locally to obtain the general solution to the crisp part of the system. Let M be a multiplicative system in R [261, p. 46]. Let N = {x £ R \ mx = 0 for some m G M}. Then N is an ideal of R. If N = {0}, then M is said to be regular. Let h be the natural homomorphism of R onto R/N C RM , the quotient ring of R with respect to M.. If 7 is an ideal of R, then the ideal in RM generated by h{I) is called the extended ideal of I in Rj^ and is denoted by h(I)e. If J is an ideal of RM, then h~x(J) is called the contracted ideal of J in R. Let /i be an L-ideal of R. Define the L-subset h{^)e of RM by Vy G RM, h(fi)e(y) = V{a G L | y G (h(n))aM}- Then h(/u,)e is an L-ideal of RM- Let a G L. Now y G h(fi)a <^ h(fi)(y) > a <^> V{/x(x) | h(x) = y} > a <= 3x G fia such that h(x) = y <£> y G h(fia)y where the "<= " becomes "<=>" if ji has the sup property. Hence if \i has the sup property, then h(fi)a — h(fia) and so (h(/j,))aM = h(fia)M- We use the notation fie for h(fi)e at times. If / is an ideal of i£, we sometimes use the notation Ie for h{I)e. If (3 is an L-ideal of RM, then we use the notation /?c for h~1(/3) at times. If J is an ideal of RM, we sometimes use the notation Jc for / i - 1 ( J ) . Suppose that \i has the sup property. Then (/xe)(y) = a ^ h(fi)e(y) = a O V{b\y G h(fj,)bM — K^M} = a <* a is maximal in L such that y G h(fi)aM =

292

K^aJM = K^aY \faeL.

(since ji has the sup property) = fiae. Hence (/^e)a = V>ae

Theorem 7.4.1 Let p be a primary L-ideal of RM- Then y/ffi = (y/]3 ) c . Proof. We have that (yflF )* = V W * (Corollary 3.6.6) = v W c (Propo­ sition 6.8.1) = (V3T ) c ([261, Theorem 16, p. 224]) = ((v^)*) c = ({yffl ) c )* (Proposition 6.8.1). Now (y/]3 ) c is a prime L-ideal of R by the proof of Theo­ rem 6.8.4 since y//3 is a prime L-ideal of RM. Hence it suffices to show I m ( v ^ ) = Im((-v//?)c). By the definition of contraction, the proofs of Proposition 6.8.1 and Theorem 3.7.11, we have that Tm(y/jF) = Im(/?c) = Im(/?) = Im(V^) =

Im((V^)c). ■

Theorem 7.4.2 Let \xbe a primary L-ideal of R such that //* is disjoint from M. Then (1) fj, = fj,ec and y/JL = (y/fl)ec. (2) fie is primary and yfjii? = (y/Jf)e. Proof. (1) Since fi* is disjoint from M., M. is prime to //* by the proof of [261, Theorem 16, p. 224]. By [261, Theorem 16, p. 224], //* = (/u*)cc. Also (n*)ec = (/J>ec)* by Proposition 6.8.2. Hence in order to show that \x = // ec , it suffices to show that Im(/x) = Im(/zec) since |Im(/i)| = 2 by Theorem 3.5.5. Let x e R. Then (/i cc )(x) = J r " 1 ^ 6 ) ^ ) = h(/j,)e(h(x)) = V{a e L \ h(x) € (h(fi))ajy} = V{a E L | fc(z) G / i ^ a M € Im(/i). Now RM ^ ^M since /i* is disjoint from jM. Thus Im(/i ec ) = Im(/z). By [261, Theorem 16, p. 224], ^ = ( y ^ ) 6 0 . Now ^ = (y//I)* by Corollary 3.6.6. By Proposition 6.8.1, (y/£)* = ((y/£)*) ec = ((y^) 6 0 )*. N o w Im(y//I) = lm((y/p)ec) by the argument in the previous paragraph since y/JI being prime must be primary. Hence y//J = (y/Jl)ec. (2) By the definition of the extension of an L-ideal, we have (/ie)* = (/i*)e and the latter ideal is primary in RM- NOW Im(/xe) = Im(/z). Hence // e is primary in i ? M . Thus ( ( ^ ) e ) * = ((>/£)*)* = ( v 7 ^ ) e = V&P = >/S^ = ( V / ^ )*> where the third equality holds from [261, Theorem 16, p. 224] since (y/ji)* is a prime ideal in R. Since fie is a primary L-ideal of RM, y/J^ is a prime L-ideal of RM. Since y/Jl is a prime L-ideal of R, (y/]T)e is a prime Lideal of RM. Hence it suffices to show that Im( v / /F) = Im((y//I) e ). However, this is now immediate. ■ Lemma 7.4.3 Let /i and v be L-ideals of RM.

Then (/x fl v)c = fic D vc.

293

Proof. Let r e R. Then (/x n u)c{r) = h'1^ n i/)(r) = (/* CI J>)(M*0) = /x(/i(r)) A i/(ft(r)) = / i " 1 ^ ) ^ ) A h-\v)(r) = ( J r 1 ^ ) n / i - » ) ( r ) = (MC n i^)(r).B Theorem 7.4.4 Let /x 6e an L-ideal of R such that \x has a reduced primary representation /x = n™=1/i;. Suppose that for 1 a = Q\Qa, and J a = X>a fl {/xi,... ,//&}. Since xx has a primary representation, Im(xx) is finite and so /i has the sup property. Thus by the arguments opening the section, (xxe)a = (xxa)e = ((n? =1 /x;) a ) e = (n? =1 (/x;) a ) e - (n M i 6 1 , f l (^) a ) c = n A i i e I a ({v>i)a)e (by [261, Theorem 17, p. 225] since (/x;)a = (x^)* is a primary ideal of R, \±i € Va) = n^L 1 ((/x i ) a ) e (since xx* ^ Xa and z G { 1 , . . . , k} implies (lH)a = R) = nti((Mi) e )a = (nti(Mi) e )a. Thus /xe - n t i / ^ e . In order to show that xxe = n f L ^ 6 is reduced, we apply Theorems 7.4.1 and 7.4.2 as follows: y/JIf = y/jl* => ( ^ / 7 ^ ) c = ( \ / ^ 7 ) c => v ^ ? 3 = y 7 / ^ => V ^ — x/W =^ * — 3 s m c e ^ — ^i=i^i is reduced. Also /Xje D nk=l^jfiie => H = Vjec 2 {r\ki=1^^ie)c = C\ki=liilLjmec = n^^fii which is impossible since /x = H™=1xxi is reduced. By Lemma 7.4.3 and Theorem 7.4.2(1), we have ec M

= (nJU/*e)c - nJUw" = n t i w - ■

Example 7.4.5 Let L = [0,1]. Let R denote the polynomial ring M[x,y,z] in algebraically independent indeterminates x, y, z over the field R of real numbers. Then the ideal (x2 — y,x2z) has the reduced primary representation (x2 - y, x2z) = (x2 -y,z)n

(x2,y) .

We also have y/(x2 -y,x2z) = (x2 -y,z)n (x, y). Now consider the nonlinear system of fuzzy singletons (xb)2-ya 2

(xb) zu

= 0{ = Ox.

(7

'4j

The solution to system (7.4) is {(0,0,r) | r e R } U { ( s , s 2 , 0 ) | s € K } , a = i b Au — 5. Let /J, denote the fuzzy ideal ((x2 — y) j . , (x2z) i \ U 0i. Then ( , ._ I Mr)-< [

1 ifr = 0 \ ifre(x2z)\{0} | i/re<x2-j/,x2z>\(^> 0 ifreR\(x2-y,x2z).

294

Define the fuzzy subset fi{ of i?, i = 1 , . . . , 5, as follows: Hi(r) = 1 if r G (# 2 ,2/) and 0 otherwise; ^ 2 (r) = 1 i / r 6 (x 2 — y, z) and 0 otherwise; fjis(r) = 1 ifre (x2) and \ otherwise; ji^(r) = 1 ifre(z) and | otherwise; li$(r) = 1 ifre(0) and \ otherwise. 5 and fi = nf=1/Z2 is a Then fii is a primary fuzzy ideal of R, i = 1, primary representation of //. Now ^JJiKr) = 1 if r G (x,y) and 0 otherwise; yjji2{r) = 1 ifrE (x2 — y, z) and 0 otherwise; y/JIi{r) = lifre (x) and ± ootherwise; y/JIlir) = lifr€(z) and \4 o\otherwise; ^/J^E(r) = 1 if r G (0) ana7 ^ otherwise. Hence we see that \i — nf = 1 ^ is a reduced primary representation of \x. We see that the crisp part of the solution to system (7.4) is displayed by y/f^i^y/J^2 while the fuzzy part is displayed by ^fjIzC^yf]^LP\^Jjl5. (In order to see this more clearly, one should consider the irreducible fuzzy algebraic varieties corresponding to the y/Jn. Then one would be concerned with c{\) = \ rather than \ and c(^) = \ rather than \.) Consider the quotient ring Rp, where P is the prime ideal (x,y). Since P n cP = 0, we have in RP that (x2 — y,x2z)e = (x2,y)e by [261, Theorem 17, p. 225]. Now

n1*r\cP = (x2,y)ncP

=
//3* n cP = (x2) n cP = 0, /x5* n cP = (o) ncP = ®, while fi2* DcP=

(x2 -y,z)picP^

0,

/z4* n cP = (z) n cP + 0. Thus by Theorem 7AA, we have in RP that /j,e = \if n ^ 3 e fl /x5e and so 0

»e(r) = {

ifreRP\(x2,y)e ifre(x ,y)e\(x2)e ifre(x2)e\{0} ifre{0}. 2

Hence by Theorem 7.4.4,

/x ec (r) = ( ^ 1 n M 3 n / i 5 ) ( r ) = ^

\

ifreR\(x2,y) ifre(x2,y)\(x2) ifre(x2)\{0} ifr e {0}.

295 Consider the nonlinear system of fuzzy singletons ya (xb)2

=

Oi

=

Oi.

(7.5)

Then {(0,0,r) | r 6 1 } , o = | , 6 = ^ is the solution to this system. It is represented by the fuzzy ideal v — ( ( x i ) 2 , 2 / i ) U Oi. Now v — ji\ D /X3 fl /X5 is a reduced primary representation of v. yJJ[\ displays the crisp part of the solution while y/Jii fl y/JI^ displays the fuzzy part. Now consider the prime ideal N = (x2 —y,z). Since NDcN = 0, we have in R^ that (x2 — y, x2z)e = (x2 — y,z) . Now \i2* DcN = (x2 - y, x2z) fl cN = 0,

^4* n cN = (z) n cN = 0, ^5* PicN= (0) fl cN = 0, while liuHcN

^

= (x2,y)

HcN

^Q,

ncN = (x2) n cN ^ 0.

77ms 6y Theorem 7.4.4, we Ziave m i?;v #*a£ Me — M2e H fi4e D /^5e and 50

ifreRN\{x2-y,z)e

[ 0 /z(r) =

i ^eWe\{0} [ 1 tfr€{0}.

Hence by Theorem 7AA,

fiec{r) =

f 0 i

(fi2nf.i4nfi5)(r)

1

ifreR\(x2-y,z) 2 (z) < / r € (x -y,z)\ <>\{0} i/re{0}.

Consider the nonlinear system of fuzzy singletons (x 6 ) 2 - y a

=

«« =

Oi

0

(7.6)

* ■

Then {(s,s 2 ,0)|s G R}, b A a = ^, and u = -| 25 t/ie solution to this system. The system is represented by the fuzzy ideal A = Ux2 — y)i, z\ \ U Oi. Afow A

296

= M2H/X4n^5 is a reduced primary representation of X. y/Ji2 displays the crisp part of the solution while ^/Jil C\ y/jls displays the fuzzy part. We have examined the system (7.4) locally. From the two examinations, we obtain for the crisp part of the solution {(0,0, r) \ r E R} for (7.5) and {(s,5 2 ,0) | s E R} for (7.6). The union of these two gives us the crisp part of the solution to system (7.4). However, the fuzzy solutions to (7.5) and (7.6) are a = ^, b = ^ and b Aa = ^, u = ^, respectively. The fuzzy part of the solution to (7.4) is a = -| and b A u = ^. The two "local" fuzzy solutions do not seem to give us the fuzzy part of the solution to (7.4), at least not immediately. Consider all possible A's of the two fuzzy solutions above 1

1

aAoAo=-A4 4 aAu

=4A2

1

oAoAa =

1

- A 2 4

1

1

2

2

These equations reduce to 1

6Aa = 4

1 a Au = 4 1 6Aa = 4

1 b Au = —. 2 Hence a = j and b A u = ^ which is the solution to the original problem. Problem: Determine a general procedure to find the solution to the fuzzy part of the original problem from the local solutions.

7.5

Fuzzy Intersection Equations

We give necessary and sufficient conditions for the solution of a system of fuzzy intersection equations. We also give an algorithm for the solution of such a system. We apply the results to fuzzy graph theory. We let L = [0,1] in this section.

297 Consider a system of intersection equations of the form exlxi A . . . A elnxn

= bx ;

e

x

ml l

x

A . . . A £mn n

z=

(7.7) ^rm

where e^- G {0,1} and 6;, Xj € L, i = 1 , . . . , m; j = 1 , . . . , n. We show that system (7.7) is equivalent to several independent systems of the type, where bi = ... = bm. Also our proofs concerning the existence of solutions are con­ structive in nature. In fact, we give an algorithm for the solution of a system of intersection equations. We write the system (7.7) in the matrix form Ex = 6, where E — [e^], x = (a?i,..., x n )*, and b = (&i,..., 6m)* and where the superscript t denotes transpose. We assume throughout that V j = 1 , . . . ,n, 3 i such that e^ = 1. We also assume that the equations of (7.7) have been ordered so that bqi+i = ... = bq2 < 6 g2+ i = . . . = bq3 < . . . < bqt+i = ... = bqt+1, where 0 = q\ < #2 < • • • < qt+i = rn. Let Ir = {qr + 1 , . . . , qr+i} for r = 1 , . . . , t. For each j = 1 , . . . , n, let i^ denote the maximum i such that e^- = 1. Let e*hj=0V h e UrsZ\ls and e*hj = ehj Vft6 U3=rIa, where i*. e Ir- Let E] = (e*lj1...,e*rnjy.LetE* = (El...,E*n). Theorem 7.5.1 Ex = b and E*x = b are equivalent systems. Proof. Let i be any row of E and j any column. Suppose that e^ = 1. Let i e Ir- Suppose 3 h e Is, s < r, such that e^ = 1. Let E' be the matrix [e^-], where e'uv = euv if (u, v) ^ (ft, j ) and < v = 0 if (u,v) = (ft, j ) . That is, E' is obtained from E by replacing the ftj-th component of E with 0. It suffices to show that E'x = b and Ex = b are equivalent. Now the ft-th equations of E'x — b and Ex — b are eh\X\ A . . . A OXJ A . . . A ehnxn = bq3+1

(7.8)

ehlxi A . . . A lxj A . . . A e/i„x n = &ga+i,

(7-9)

and respectively. The other equations oi E'x = b and E'x = 6 are identical to each other. Since e^- = 1, we have that x^ > bqr+i > bqa+i. Thus equation (7.8) and the z-th equation are equivalent to equation (7.9) and the i-th equation. Hence the desired result follows. ■

298 Example 7.5.2 Consider the following systems of intersection equations: X 1 A X 3 A £4 X2 A £ 3

_

1

—

I

z i

and X\ A £4 1

X2 A £ 3

1

and x\ A £ 3 x\ A X2 A £ 3

and £1 £ 1 A £2 A £ 3

4 _

1

—

4

z1 _

1

Z I —

4

The first two systems are equivalent while the last two systems are equivalent. The last two systems have no solution. In both pairs of systems, j' = 3 and i* = 2. Theorem 7.5.3 (1) The system Ex = b has a unique solution if and only if V r = 1 , . . . , £, the system Eqr+ix = V+i> • ■ • > E*r+lx

= bQr+1

has a unique solution. (2) The system Ex = b is inconsistent if and only if3i that bi > 0 and < i = . . . = <» = 0.

G { 1 , . . . , m) such

Proof. (1) Suppose that i G Ir and h G Is, where r ^ s. Then e*j = 1 implies e*h- = 0. That is, the t systems E*r+1x = fc9r+1,...,E*r+1x = bQr+l, r = 1 , . . . , t, pairwise involve distinct unknowns. (2) Since the t systems in (1) pairwise involve distinct unknowns, E*x = b is inconsistent <=> one of the t systems is inconsistent. The desired result now follows by applying the condition in (2) individually to the t systems. ■ For the matrix E, let E{ denote the z-th row of E, i = l , . . . , m . We write Eg < Eh & V k = 1 , . . . , n, egk = 1 implies ehk = 1. We write Eg < Eh o Eg < Eh and Eg ^ Eh- The addition of two rows of E is componentwise with 0 + 0 = 0,0 + 1 = 1 + 0 = 1 + 1 = 1. Corollary 7.5.4 The system Ex = b is inconsistent if and only if 3 i, hi,..., hk G { 1 , . . . , m} such that bi > 0, i £ Ir and hu G ISu with r < su for u = 1 , . . . , k and Ei < Ehl + . . . + Ehk.

299 Proof. 3 i E { 1 , . . . , m} such that e*x = . . . = e\n = 0 if and only if 3 i, hi,..., hk E { 1 , . . . ,m} such that i E Ir and hu E ISu with r < su for u = 1,..., k and Ei < Ehl + ...+ Ehk.M We now examine the case where b\ = . . . = bm. Let z E { 1 , . . . , ra}. Suppose that ] ^ i r . . , E i f c . < £ i . If ^ + . . . + £ ifc . < £;, then let c^ = 0 if e i r j = 1 for some r — 1 , . . . , ki and c# = e^ otherwise, j = 1 , . . . , n. If no such Eir exist let ci5 = eih j = 1 , . . . , n. Let C; = (c»i,..., cin) and C = (Ci, • • •, Cm)*(If E1^ + . . . + ^ f c . = Ei, then the z-th equation may be deleted.) Theorem 7.5.5 Suppose that bi = ... = bm = b in system (7.7). Let C denote the matrix defined above. Then the systems Ex = b and Cx R b are equivalent, where R indicates that the relation in the i-th equation is either "=" or ">" depending upon whether d = Ei or d ^ Ei, respectively. Proof. Let 5(0) denote the system Ex = b and let S(i) denote the system obtained from 5(0) by replacing its i-th equation by CiX Ri 6, where Ri denotes either " = " or " > " . Let T(i) be the system C\ x R1 b, ..., dx Ri 6, Ei+1x = 6 , . . . , EmX = b. It is easily seen that E^ x = 6 , . . . , Eik. x = 6, EiX = b and E^ x = 6 , . . . , Eik x = b,d x Rib are equivalent. Thus 5(0) and S(i) are equivalent V i\ = 1 , . . . , m. Now T(l) = 5(1) and so 5(0) and T(l) are equivalent. Assume that 5(0) and T(i) are equivalent (the induction hypothesis). We now show that 5(0) and T{i + 1) are equivalent. As noted above, 5(0) and S(i + 1) are equivalent. Let the (i + l)-st equation of 5(0) (and thus of T(i)) be denoted hyyx A ... Ayh Az1 A...Azk = 6, where y i , . . . ,2/h,^i, ••• ,zk E {xi,... ,xn} and where the (i + l)-st inequahty of S(i + 1) is z\ A ... A zk > b. Now {yi, • • • ,Vh} H {zu..., zk} = 0. Also, y1 A . . . A yh A z1 A ... A zk = b is equivalent to (yi A ... A yh = b and zi A ... A zk > b) oi (yi A ... A yh > b and z\ A ... A zk = b). Since T{i) and S(i +1) are each equivalent to 5(0), T(i) and S(ii + 1) are equivalent. Hence the system T(i) minus the (i + l)-st equation and the system S(i + 1) minus the (i + l)-st inequahty individually imply y\ A . . . A yh = b. Thus we have the equivalence of T(i + 1) and S(i + 1) and thus the equivalence of T(i + 1) and 5(0). Hence the result holds by induction. ■ System (7.7) with b\ — ... — 6 m is consistent <=> V z, 3 j such that e^ = 1. Example 7.5.6 Consider the following system 5(0) : x i A ^2 A X3

=

b

X\ A x2 x i A X2 A X3 A X4

= =

b b

Leti = l. Then E2 < E\. Applying Theorem 7.5.5, we obtain 5(1) : x3 x\ A x2 x\ A x2 A xs A £4

> = =

b b b

300

Let i=2 Applying

in 5 ( 0 ) . Then 5 ( 2 ) = 5 ( 0 ) . Leti = 3 in 5 ( 0 ) . Then E1 + E2 < E3. Theorem 7.5.5, we obtain 5(3) :

Thus T(3) is the

X\ A X2 A x3 x\ A X2

= =

b b

X4

>

b.

system x3 xi A x2

> =

b b

X4

>

b.

T h e o r e m 7 . 5 . 7 Suppose that b\ = . . . — 6 m = b in system (7.7). Let C be the matrix as defined above. Suppose that Chk — Qfe = 1> h ^ z, for some z, /c, where ChX > b and d x = b. Suppose that Ch 2 C%- Let dhk — 0 and duv = cUv tf (u,v) ^ (/i, k). Let D = [dij]. Then ChX > 6, dx = 6 are equivalent to Dh% > b, Di x = b. Proof. Both systems force Xk > b. If Ch < d, t h e n drop t h e h-th equation. In fact, if ChX > 6, Chxx = ... = Chkx = b and Ch < Chy + • • • + Chk , t h e n drop the h-th equation. We also note t h a t x\ A x2 > b is equivalent t o x\ > b and x2 > b. ■ E x a m p l e 7.5.8 The following

systems

are

equivalent:

x

2 A £3 x i AX2

> =

6 6

£3 xi A x 2

> =

b 6.

systems

are

and

# e r e /i = 1 and i = /c = 2. E x a m p l e 7.5.9 Tne following

equivalent:

x

2 A £3

>

6

xi A x2 X1A13

= =

6 6

xi A x 2 X1AX3

= =

6 6.

and

i n the first system,

C\ < C2 +

C3.

301

We may use the following algorithm to solve a general system of intersection equations. We use the notation Ej to denote the complement of Ej. We let 6 denote the zero vector. We also assume that 61 > 0. Algorithm 1. Sort the Ei so that the 6;'s are in nondecreasing order. 2.1. Let Temp and Total each be a row of n zeros 2.2. Let c = bm 2.3. For i — m down to 1 do if hi — c then Temp = Temp + E{ and Ei = Ei NOR Total if c > bi then c = fei, Total = Total + Temp, Temp = Ei and Ei = Ei NOR Total 2.4. J£3i,l
=

^x

4. For each distinct && 4.1. While 3 d and Cj such that (1) ft. = 6^. = 6fc (2) Ri = ' = ' (3) Rj = ' > ' and (4) C; NOR d ^ 0 do Cj = Cj NOR C; 4.2. Let Tfc = J ] ^ V C» such that 6; = 6fc and ft = ' = ' 4.3. If 3 d such that (1) ft = ' > ' and (2) Ci NOR Tfc = 0 then erase d and 6; from matrices C and 6, respectively. The time complexity of the algorithm is 0{m2n). If each row in E and C is denoted as a binary number, then the time complexity becomes 0(ra 2 ). A unique minimal solution can be immediately determined. Let T be the subset of P(V), the power set of V, such that A e T «=> the cardinality of A is 2. A fuzzy graph with underlying set V is an ordered pair

302

(cr, //), where a is a fuzzy subset of V and \x is fuzzy subset of some subset X of T such that V w G l , /x(m>) < a(u) A (j(v). Consider the Cartesian product G = G\ x G 2 = (V, X) of graphs G* = (Vi,Xi), i = 1,2. Then V = Vi x V2 and X = {(u, u2){u, v2) \ u G Vi, u 2 v 2 G X 2 } U {(^i, v)(vi, v ) | v e V^, uivi G X i } . Let fa, fa) be a fuzzy subgraph of Gi, i = 1,2. Define the fuzzy subsets c^ x cr2 of 1/ and /j,ifj,2 of X as follows: V ( u i , u 2 ) G V, ((Ji xcr 2 )(ixi,u 2 ) = C T I ( ^ I ) Acr2(w2); V 7x G Vi, V tx2i;2 G X 2 , VlfJ,2((u,U2)(u,V2))

= (?l(u) A fJ,2(u2V2);

V v G F2, V u i v i G X i , /X1/X2((W1,V)(V1,V)) = <72(v) A ^ i ^ V i ) .

Then (cri x cr2, fJ>i/J>2) is a fuzzy subgraph of G\ x G 2 , [179, Proposition 2.1, p. 160]. We call fa x cr2, /xi/x2) the Cartesian product of fa, ^i) and (cr2, Application [179, Theorem 2.2, p. 162] Suppose that G is the Cartesian product of two graphs G\ and G 2 . Let (cr, /J,) be a fuzzy subgraph of G. Then (
{vu,v12,...,vln}

and V2 = { v 2 i , v 2 2 , . . . , v 2 m } • XiAyj

=
i = l,...,n; j = l,...,m; Xi A Zjk =

fJ,((vii,V2j)(vu,V2k)),

i = 1 , . . . , n and j , k are such that v2jV2k G X 2 ; Vj A wih = /x((vii, v2j)(vlh,

v2j)),

j = 1 , . . . , m and i, fo are such that vuVih G X i . The composition of fuzzy graphs is also defined in [179]. If fa fj) is a fuzzy subgraph of the composition Gi[G 2 ] of graphs Gi and G 2 , then necessary and sufficient conditions are given in [179, Theorem 2.5, p. 164] for fa /x) to be the composition of fuzzy subgraphs of Gi and G 2 in terms of the existence of a solution to a system of fuzzy intersection equations.

303

7.6

//-Intersection Equations

In this section, we consider intersection equations, where L is a completely distributive lattice. A fuzzy relational equation is of the form AX

= B,

(7.10)

where •** ~

B =

Kp'ijJmxn'i

(b1,b2,...,brn)t;

X = (xlyx2,...

,x n )*;

(iij,bi,Xj G L, i = l , . . . , r a , j = 1, . . . , n , £ denotes transpose, and "•" is the max-min composition. In other words, (7.10) is the following system of "linear" equations when the unknowns are xi, x2,..., xn G L : (an A xi) V (a12 A x2) V . . . V ( a i n A x n ) (a 2 i A xi) V (a 22 A x 2 ) V . . . V (a 2 n A xn)

= =

h b2 (7-11)

:

( a m i A xi) V (a m 2 A x 2 ) V . . . V ( a m n A x n )

=

bm

An element a G L is called an V-reduced element if a = 6 V c => a < 6 or a < c. Let M be the set of all V-reduced elements. Lemma 7.6.1 (233) Every element in L can be represented as the union of some elements in M. M First we examine the following system of equations; xi A Xj = aij,l
Q>ij — dji

(7.12)

for all z, ?.

Proposition 7.6.2 The system (7.12) of equations has a solution if and only if aij = V{ais A atj) | s, t = 1,2,..., n}, 1 < z < j < n, (7.13) and if {1.12) has a solution, then (7.12) has a unique least solution Xi = Vs=i}2,...}n,s^iais, i = 1, 2 , . . . , n.

(7-14)

304

Proof. Suppose (7.12) has a solution Xi, i = 1,2,..., n. Then ais < x2- V z, i z£ s, s = 1,2,... ,n. Thus Q>ij S: v s = l , 2 , . . . } n , s ^ i ^ i s S: %i-

Similarly, &ij fly < < Vt=l,2,...,n,t&0>tj

< ^# i -

By (7.12) we have a^&ij

< < <

(V (Vs=l,2,...,n,s^ia>is) {^t=l,2,...,n,t^j^tj) -,n,t&a>tj) 5 = l,2,...,71,3^2^3 ) A (Vt=l,2,.. fcl-"r / \ «X/ 0

' ■ *-*"i7 }

1 < * < j" < n - Hence since L is distributive, aa

iji j

= —

=

(Vs= aiiss ) A (Vt=l,2,...,n ( V t = i ) 2 , . . . , >nt^iat , t ^ j 7^-)) = il>, 2 , . . . , n , s # i «

Aaatjt j | s,t s,£ = 1,2,. 1,2, . . . , n } , V{a iiss A

1 < i < J' < w- Thus (7.13) holds. Conversely, suppose (7.13) holds. As in the above proof it is easy to see that Xi = V^5=1^2,...,n,Sy&iQ'is5 %i ~ s = i j 2 , . .,.,n,s^i^isi

1 < z < n satisfies the system (7.12). Thus (7.12) has a solution. Since *i > V a = i f 2 , . .,.,n,s^iaisi

1 < i < n, we have that (7.14) is the unique solution of the system (7.12) of equations. ■ Remark 7.6.3 By the definition of a solution of a system of equations, we have: If {#i, £ 2 , . . . , x n } , {2/1,2/25 • • • > Vn} are solution sets of (7.12) and xi < 2/i, z = 1,2,... ,n, then {zi, Z2,..., zn} is a solution of (7.12), where X{ < zi < 2/t, i = 1,2, . . . , n . Proposition 7.6.4 TTie system (7.12) of equations has a solution if and only &is a ^ < a^-, flis A Afltj

flij,

(7.15)

w/iere 5 = 1,2,... ,n, 5 ^ z, £ = 1,2,... ,n, £ ^ j . If { x 1 ? x 2 , . . . ,a?n} is a solution of (7.12) and V i < t <<(sS<
305

Proof. Clearly, (7.13) is equivalent to (7.15) over L. Now we only show that solution satisfying (7.16) is unique. Suppose that the system (7.12) of equations has another solution other than the solution (7.14). Since (7.14) is the least solution, by Lemma 7.6.1 there is i (1 < i < n), yi = Xi Va, such that {Xi, X2, . . • , Xi-i, Vi, Xi+i,

. . . , Xn}

is a solution of (7.12), where a € M and a j£ X{ = V s= i ) 2,...,n,s^itti 5 ,

a < Vi
By definition of an V-reduced element, there are t0,so (1 < to < so < n) such that a < a to5o . But t0 i=- i ^ so, since a ^ ais for s = 1,2,..., n, s ^ i. Thus a < Vs=i>2,...,n,s#zat0s = xto> and hence a
ato V a < 7/i A a:to = ato1

contradicting a jtxi = V 5= i )2 ,...,n,5^az5. ■

Remark 7.6.5 The condition (7.15) of Proposition 7.6.4 can 6e stated in the following form. Denote 0

Gl2

«21

0

Gl3 ^23

' • ' •

Gnl

«n2

&n3

'

«i

a2

• o,

where aij — aji for all i,j and the entries of the main diagonal are the least element 0 of L. Then a necessary and sufficient condition for the system (7.12) of equations to have a solution is that the intersection of any two elements selected from A, one from a row (say row i) and one from a column (say column j , i ^ j), is less than or equal to the element aij determined by the intersection of that row and column. We now generalize the above result. Consider the n-system of equations on Asejkxs where I\, I2,...,

= afc, k = 1,2,..., m,

(7.17)

Im Q { 1 , 2 , . . . , n} are index sets.

Example 7.6.6 When m = n^n2 ^ and Ik is the set {(ij) system (7.17) of equations is just (7.12).

\ i < j}, then the

306

Let h = {*i,fc2,...,fc*fc} Q { 1 , 2 , . . . , n } , i.e., the kth equation contains xkl, %k2 > • • • > xkt , & = 1,2,...,TO;and J, = { f c | s G l f c } C { l , 2 , . . . , m } , i.e., the index set of equations containing the unknown element xs in the system (7.17) of equations, 5 = 1,2,..., n. Theorem 7.6.7 The intersection system (7.17) of equations has a solution if and only if ak = V{aZl Aa/ 2 A . . . A ah Jh G JklM

€ Jk2,...,ltk

G J*J,

(7.18)

1 < A; < TO; and z/ (7.18) /io/ds, i/ien £/ie intersection system (7.17) /ms £/ie unique least solution *k = Vjfc€Jsafc, 5 = 1 , 2 , . . . , n

(7.19)

Proof. Suppose (7.17) has a solution x3, s = 1,2, . . . , n . Then ak < xs for s E Ik => Vk£j3ak < xs. Since s 6 4 , we have k G J s . We obtain a^ < Asejk(Wiej3ai) < /\sehxs = ak- By the distributive law, a>k

=

As€/fc(Vi€jaai)

=

\/{ah A a/2 A . . . A a/tfc |ZX G JkxM £ Jfc2,. • •, kfc G Jfctfc },

1 < A; < TO, i.e., (7.18) holds. Conversely, suppose (7.18) holds. Let xk = Vkejsak, s = 1,2,... ,n. Then clearly, (7.17) has a solution {a;i,X2, • • • >xn}> and since for any solution {2/1,2/2, ••• ,2/n} of (7.17) we have Vkejaak < y8, s = 1,2,..., n, it follows that (7.19) is the unique least solution of (7.17). ■ Proposition 7.6.8 The intersection system (7.17) of equations has a solution if and only if for k = 1 , 2 , . . . , 777, aix A aj 2 A . . . A a/tfc < a*,

(7.20)

w/iere Zi G J ^ , Z2 G Jfc 2 ,..., Z*fc G Jfctfc . Proof. That the existence of a solution implies (7.20) follows from (7.18). Conversely, suppose that (7.20) holds. Then V{aZl A a/2 A . . . A aZtfc | Zi G J fcl , Z2 G J f c a ,..., ltk G Jfctfc } < ak.

307 Since Ik = {ku k2,..., flfc =

fc*J,

ak Aak V

A ...

k e Jkl D Jk2D,...,

nJ f c t f c . T h u s

Aak

v

'

tfc times

< <

V{a Zl A a , 2 A . . . A aZtfc | h e JklM afc,

€ Jfc 2 ,...,Z t f c € Jfctfc }

i.e., (7.18) holds. R e m a r k 7.6.9 Generally, 25 notf unique (if it exists).

7.7

the solution satisfying See Example 7.8.4.

xs < V1
Union Equations

In this section, we consider t h e following type of system of equations: Vi V yj = bij, 1 < i < j < n,

(7.21)

where b^ G L. We let b%j = bji when i > j . By t h e duality principal, t h e following propositions are obtained immedi­ ately from Propositions 7.6.2, 7.6.4. P r o p o s i t i o n 7 . 7 . 1 The system

(7.21)of equations

has a solution

if and only

if bij = As,t(bis

V btj), l
If (7.22) holds, then (7.21) has the unique greatest

(7.22) solution

yi = A s = i,2,...,n,S7^z5- ■

(7.23)

P r o p o s i t i o n 7 . 7 . 2 The system (7.21) has a solution if and only z'/V z, j , i ^ j>, bij
(7.24)

w/iere s = 1, 2 , . . . , n, 5 ^ z, £ = 1 , 2 , . . . , n, £ ^ j ; z/ (7.24) ZioZds, tfie solution of {1.21) satisfying Ai
< Vi, i = 1 , 2 , . . . , n,

(7.25)

zs unique. H R e m a r k 7 . 7 . 3 Tfoe condition Let

in (7.24) can 6e stated in the following

I" 1

B =

&12 &13 * * • °in 1

&21

1

&23

' ' * &2n

bni

bn2

bns

•••

1,

form.

308

where b^ = bji, when i ^ j and the entries on the main diagonal are the great­ est element 1 of L. Then a necessary and sufficient condition for the system (7.21) of equations to have a solution is that the union of any two elements se­ lected from B, one from a row (say row i) and one from a column (say column j ? i zfz j)) is greater than or equal to b^ determined by the intersection of that row and column. As in Section 7.6, now we consider the n-union system of equations on L, Vseikys

= 6*,fc= 1,2,... ,m,

(7.26)

where I\, fy, • •., Im Q { 1 , 2 , . . . , n} are index sets. Using the duality principle over L analogues to Theorem 7.6.7 and Proposition 7.6.8, we have the following result. Theorem 7.7.4 For the union system (7.26) of equations, the following con­ ditions are equivalent (1) (7.26) has a solution. (2) For every /c, k = 1,2,..., ra, bk = A{bh Vbh V...V6 Z t f c I heJkl,l2eJk2,...,ltk

eJtkh

(7-27)

where h = {hi,k2,...,ktk}

C {1,2, . . . , n } ,

Js = {k | s 6 / f c } C { l , 2 , . . . , m } . (3) For every fc, k = 1,2,...,m, h G JklM

e Jk2, • • • ,hk € Jtk

bk
bltk.

If the above conditions holds, then Vk = Afc(Ejs&fc,s = 1,2, . . . , m

(7.28)

is the unique greatest solution of (7.26). I Now let #i, #2, • • •, Xn be unknown elements of L. We can form polynomials in the usual manner using A, V, and parentheses, denoted by / ( x i , £2, • • • 5 xn)Examples of lattice polynomials are x\, x^ V#3, {x\ l\x<2)\l {x\l\x?) V (#2 AX3), etc. To solve the polynomial system of equations on the lattice L fk(xi,X2,...,xn)

= bki

(7.29)

fc = 1,2,..., ra, we can use Theorems 7.6.7 and 7.7.4. In fact, if we denote the intersection forms appearing in each form of system (7.29) of equations as Vi =

hseijXs,

Ij C { 1 , 2 , . . . , n}, j = 1,2,... ,p (p < 2n - 1), then (7.29) can be transformed into (7.26). (See Example 7.8.6)

309

7.8

Applications and Examples

We can connect the fuzzy relation equation (7.10) to the system of equations discussed in Theorem 7.7.4: A-X A ■X ===B, B, (7.30) where

A -™-=

KP'ij KP'ij Jmxn'i Jmxn'i

B = )t] = (h., 6(b2u,b. 2. ,...16b7nYt X = {xi , x(x2 1), .x2. ,...,x . , X'n) 7l) ;1

a>ij,bi,Xj G [0,1], i = 1 , . . . , m, j = 1 , . . . , n; and "•" is the max-min composition over the lattice [0,1]. Definition 7.8.1 Suppose a, 6 G [0,1], let aab = b if a > 6, and aocb — \ if a < 6, and assume R = (r n f c ) n X m , Q = (qkv)mxi- Let RQ)Q = S = (suv)nxh where

= Al
Suv

= 1, = 1, n, v = u= l , 2 , . . . ,,n, l , 2 , . . .. , iZ.. Lemma 7.8.2 (219) The fuzzy relational equation (7.30) has a solution if and only ifAtQB is a solution of (7.30) and is the greatest solution, where Af is the transpose of A. I Now, we rewrite the union equation (7.26) as follows: AY A-Y

= B, B,

(7.31)

where i4 ^ = —

y^ksjmXn'i mXni

m; n , /c l , 2 , . . . ,,ra; k= = 1,2,... aks = 1 if 5s G Iifc,a = 00 if s3 ^ IIfc, = 1,2,... 1,2,.. . ,,n, klasksfc = k, s =

•,g';

B = (61,62,.. (b1,b2,...,bm)t; ••,2/n)'. yY = = (2/1,2/2,. (2/i,2/2,...,2/n)*.

Similar to Definition 7.8.1, we have the following on L; aksabk — bk if aks = 1 and aksabk = 1 if aks = 0. If we let A1 O B = ( ^ 1 ^ 2 , •. • , £ n ) S then £ s = Ai
310

Example 7.8.3 Let E = {a,6,c,d}. over L = 2E : X\HX2 XlHx3 X2Hx 3

Consider the following subset equations = = =

{a, b} {b,c,d} {b,d}.

(7.32)

Let A12 = {a, 6} = A2i, A13 = {&, c, d} = A31, A23 = {&, d} = A32. By Remark 7.6.5, since in the matrix 0 A2\ A31 we have A\2 D A 32 = {b,d} £ Aw

A12 0 A32

A13 A23 0

Thus it follows that (7.32) has no solution.

Example 7.8.4 For the lattice L = 2E as above, consider the system of equa­ tions x\f)X2 = {a,b} x2Dx3 = {b,c} (7.33) x3Hx4 — {c, d}. It is clear that n = 4, m = 3, Ix = {1,2}, I2 = {2,3}, I3 = {3,4}, J x = {1}, J 2 = {1,2}, J 3 = {2,3}, J 4 = {3}. Ic* Ax = {a, 6}, A 2 = {6,c}, A3 = {c,d}. It is easy to verify the following:

(Ax n AJ U (Ax n A2) = At (Ai n A2) u (Ai n A3) u (A2 n A2) u (A2 n A3) = A2 (A2 n A3) u (A3 n As) = A3 Thus (7.33) has a solution by Theorem 7.6.7, and the least solution is the following: x\ = {a, 6}, x 2 = {a,b,c}, x3 = {fe,c,d}, x 4 = {c, d}. However, it is easy to see that y\ = {a, 6, d}, y2 = {a, 6, c}, y3 = {&, c, d}, y4 = {a, c, d} is also a solution of (7.33) and satisfies yi < {a, 6, c, d} = Ai U A 2 U A 3 , i = 1, 2,3,4. Example 7.8.5 Consider the following system of equations over L = [0,1] : X\ Ax3 X2 A £4 X3 A £4

= = =

0.5 0.8 0.7.

(7.34)

Here n = 4, m = 3, h = {1,3}, J 2 - {2,4}, J 3 = {3,4}, J x = {1}, J 2 = {2}, J 3 = {1,3}, J 4 = {2, 3}. ZJenote ax = 0.5, a 2 = 0.8, a 3 = 0.7. We have (a x Aai) V(a! Aa 3 ) = ai, (a 2 A a 2 ) V (a 2 A a 3 ) = a 2 , (ai A a 2 ) V (ax A a 3 ) V (a 3 A a 3 ) V (a 3 A a 2 ) = a 3 . By Theorem 7.6.7, (7.34) /ias a solution, and the least solution is the following: xx = 0.5, x2 = 0.8, x3 = 0.7, x 4 = 0.8.

311

Example 7.8.6 Let E be a set and consider the following system of equations over L = [0, l p : (xi A x 2 ) V (xi A x 3 ) = A1, _ 35, (xx A x 3 ) V (x 2 A x 4 ) = A 2 , where Ai, A 2 are some elements ofL, and xi, X2, X3, X4 are unknown fuzzy sets over E. Let Yi

=

XiAx2,

Y2

=

xaAx3,

*3

=

^2 A X 4 .

(7.36)

T/ien the system (7.35) of equations can be transformed as Y1VY2

=

A1;

(

y 2 vy 3 = A 2 .

,

^-67)

Using Theorem 7.7.4, where n = 3, m = 2, Ix = {1,2}, I2 = {2,3}, Ji = {1}, J2 = {1,2}, J 3 = {2}, we have A1 = (Ai V Ax) A (Ai v ^2), M = (A1 V A 2 ) A (A2 V A 2 ). Hence (7.37) /ias a solution, and the greatest solution is F2 Y3

= =

(7.38)

J4IAA2>

A2.

Then from (7.36), (7.38)we obtain the system of equations x\ A X2

=

A\

xi A x 3

=

A1 A A 2

X2 A X4

=

A2

(7.39)

Forn = 4, m = 3, /1 = {1,2}, I 2 - {1,3}, J 3 = {2,4}, J x = {1,2}, J 2 = {1,3}, J 3 = {2}, J 4 = {3}. Hence we have (A1 A Ai) V (Ai A A2) V (Ai A A 2 ) V (A1 A A 2 ) = (Ax A A 2 ) V (Ai A A 2 ) = ( A I A A 2 ) V (A 2 A A2) =

Au AxAA2l A2.

5?/ Theorem 7.6.7, £/ie system (7.39) 0/ equations has a solution and the least solution is x\ = A\ V (Ai A A 2 ) = Ai, x 2 = A\ V A 2 , x 3 = A\ A A 2 , X4 = A 2 . Clearly, this a solution of the system (7.35).

7.9

EXERCISES

1. (166) Let cr be a fuzzy subset of i?. Prove that (
312

2. (166) Let W be a finite subset of cr* such that (W) = (cr*), where a is a fuzzy subset of R. Define the fuzzy subset r of R by r(w) = a{w) if w G W and r(w) = 0 otherwise. Prove that r* = W and (r)* = (r*) = (cr*) = (cr)* . Conclude that b is a zero of (a)* <& b is a zero of (r)* . Let 5 be a set and let x^ Ls. Let .F(x) be the set of all L-subsets C of S such that £ C x- Then T is an L-topology on x if ^ is a collection of L-subsets of S such that V fi G T, // C x, and (1) X , ^ T ; (2) if 5 C T, then fV€<s/i £ ^5 (3) if /i, i/ G T, then /x U i/ G T. If T is an L-topology on x, then (x, T) is called an L-topological space and the elements of T are called closed L-subsets with respect to xLet x,V € -^(SO- Define the L-subset x ~ 77 of 5 by V s G 5,

&-*>(•>-{ ?•> n:*s. Let x and 77 be L-subsets of x- Then (x — v)* = X*\v* '• (X - ri)(s) > 0 & s £ 77* and 5 £ 5\x* O s e x*\v*-

s

£ (x

—

v)* <=>

Let (x, T) be an L-topological space. Let f G .F(x)- Then £ is said to be open in (x, T) if 3 r G T such that f = x — T. Let 77 G ^"(x)- Define the L-subset r/(n) of 5 by V 5 G 5, 77^) (s) = x(s) if 5 G 77* and 77^(5) = 0 otherwise. Then 77^ is called the neutral closure of 77 in x3. (167) Let 77 G JT(x). Prove that

x

- (x ~ v) = V{n) •

4. (167) Let a, /? G .F(x). Prove that (a) (x - a) U ( * - / ? ) = * - ( a n / ? ) ;

(b) (x-a)n(x-/?) = x-(c*u/3). 5. (167) Let (x, T) be an L-topological space. (a) Let £ G ^"(x)- If f is open, then prove that 3 r e T such that x f = r ( n ) . Conversely, if x - £ is closed, prove that £.

313

X is said to be irreducible with respect to T if V OJ, (3 G T, if x = OL U /3, then either x = a or x = PLet (x, T) be an L-topological space. Then T is said to be neutrally closed if Vr G T, r G T. Let (x, T) be an L-topological space. Let 77 G .F(x). Then 77 is said to be dense in x if V r G T such that 77 C r, r ^ = x6. (167) Let (x, T) be an L-topological space such that T is neutrally closed. Prove that conditions (b) and (c) are equivalent and are implied by condition (a). (a) x i s irreducible; (b) V ^ , ( e Fix) iff, C are open and £ ? 0 ^ C, then £ n C / 0(c) V £ G ^"(x) if £ is open and £ =fi , then £ is dense in x7. (167) Let (x, T) be an L-topological space. Let x' G ^"(x) and T ' = {x' H r I r G T } . Prove that (x', T') is an L-topological space. If T is neutrally closed, then prove that T is neutrally closed. 8. (167) Let (x, T) be an L-topological space such that T is neutrally closed. Let x' € -^(x)- Then prove that condition (a) implies condition (b). (a) x' i s irreducible; (b) V £, C G ^ ( x ) iff, C are open and £ n x' ^ <> / ^ C n x', then f n C Let (x, T) be an L-topological space. Then (x, T) is called Noetherian if V descending chains ot\ D a.2 2 • • • of elements of T, 3 positive integer m such that V z > m, cq n ' = o-+i9. (167) Let (x, T) be an L-topological space such that T is neutrally closed. Prove that (x, T) is Noetherian if and only if (x*, T*) is Noetherian. For the remainder of the Exercises, the set S is taken to be Ck. 10. (167) Let a G ^"(x), where x is a algebraic L-variety. If a is an alge­ braic L-variety, then prove that a* is an algebraic variety and ay1' is an algebraic L-variety. 11. (167) Suppose that 3 a G L\{0} such that A{x(z) | * e X*} = a. Let Ta = {T\ T* is an algebraic variety and V z G r*, r(z) > a} U{}. If H is an algebraic variety such that ^ C ^ * , then prove that 3 an algebraic L-variety a E Ta such that a* = H.

314

12. (167) Let x* be an algebraic variety. Suppose that 3 a € £\{0} such that /\{x(z) | z G x*} — a - Prove that (x, 7^) is a Noetherian L-topological space such that Ta is neutrally closed. 13. (167) Let x be an algebraic L-variety such that 0 G Im(x). Prove that x can be represented as a union of a finite number of irreducible algebraic L- varieties.

315

Chapter 8

L-SUBSPACES Part of the purpose of this chapter is to take an axiomatic approach to the generation of L-algebraic structures similar to that for the crisp case in [261, pp. 50, 97, 129]. In Section 8.1, we determine the generating properties which yield the underpinnings for the existence of bases and the uniqueness of their cardinalities for L-algebraic structures. In Section 8.2, we apply the general results of Section 8.1 to the specific cases of L-subspaces in terms of bases, L-field extensions in terms of relative p-bases and transcendence bases, and L-subgroups in terms of p-bases and torsion-free bases. Let G be an (additive) Abelian group and F a field of characteristic p > 0. We show, in particular, that there exists an L-subgroup of G which does not have a p-basis even if G is finite. Let /x, v be L-subfields of F such that / i D i / . We also show that there exists an L-intermediate field of ji/v which does not have a relative p-basis over v even if F has a finite relative p-basis over the support of v. However, if L is a dense chain, then an L-subgroup of G with the sup property has a p-basis and an intermediate L-subfield of fi/v with the sup property has a relative p-basis over i/, if certain compatibility conditions hold. The results of this chapter are taken from [4,127,138,163,164,170,186,188, 189]

8.1

Preliminary Results

Let V denote a nonempty set and let Lv denote the set of all L-subsets of V. Let S(V) denote the collection of all L-singletons of elements of V. Let X € Lv. Let FS(V) denote the set of all sets S of L-singletons of V such that if x 0 , xb e S, then a = b > 0. Let S G FS(V). Then x(5) denotes the L-subset of V defined by \/x e V, x(S)(x) =aifxaeS and x(S)(x) = 0, otherwise. For S e FS(V), we let S* = {x | xa € S}. Throughout, () denotes a function of

316

Lv into itself. Let x G L v . We let (x) denote the image of x- For all x G V, we define the L-subset x ~ z of V by (x —^)(2/) — x(y) '^V ¥" x a n d (x — x )(y) = 0 if y = x V y G V. Let P(V) denote the power set of V. Let s denote a function of P(V) into itself. We assume throughout that 5 satisfies the five following properties: (1) V S,Te P(V), SCT^ s{S) C s(T); (2) V x G V, V S G P(V), x G s(S) => 3 a finite subset S' of 5 such that x G s(S')' (3)V5GP(V),SC«(5);

(4) V 5 e P ( n *(*(£)) = * ( S ) ; (5) V x, y G V, V 5 G P(V), yes(SU {*}), y $ s{S) ^xes(SU {y}). We call (V, s) or simply V a space. The space V has the usual properties of a vector space such as possessing bases of unique cardinality [261, p. 52]. We also assume throughout that

(i;) v x,v e vLv, x c v => (x) c (v); (3') V x e Lv , x c (x); ( 4 ' ) v x e L , « x » = (x>.

Definition 8.1.1 Le£ x £ £ V - TTien x ^s sazrf £o 6e an L-subspace of V if X = (x) • ^ e ^ 5 6 P(V). Then S is said to be a subspace ofV if S = s(5). If xx G L v , we let ^(u) denote the set of all L-subsets x of V such that X C /x. Let PS(xx) = {5 | S G F 5 ( V ) , Vxa G S, x a C xx}. Definition 8.1.2 Let a be an L-subspace of V and let x £ ^""(AO- TAen x is said to be free inuifVxE x*, (x — #) (x) — 0. £e£ 5 G VS(u). Then S is said to be free in /x if x(«S) ^ / ^ e ^n MWe also assume the following property throughout. (6') V X € I V , s(X*) = (x>* • Property (60 is equivalent to the property that V x € £ v , V y G V, y G s(x*) ^ (x> (2/) > 0 (V S G S(V), V i / € F , | / G 5 (5*) *> (S) (y) > 0). Let xx be an L-subspace of V. Then xx* = (xx)* = s(/x*). Hence /x* is a subspace of V. Theorem 8.1.3 Let xx be an L-subspace of V and let x € J~{v)- Then x i>s free in /x if and only if x* is free in u*. Proof. Suppose that x* is not free. Then 3 x G x* such that x G s(x*\{z}). Hence by property (60, (x — x) (x) > 0. Thus x is not free. Conversely, suppose that x is not free. Then 3 x G x* such that (x — #) (#) > 0. Hence by property (60, x e s(x*\{z}). Thus x* is not free. ■

317

Definition 8.1.4 Let /x be an L-subspace of V and let x 6 ^(AO- Then x is said to be maximally free in fi if x is free in [i and there does not exist rj G J-{fj) such that \ C rj and rj is free. Definition 8.1.5 Let /x G Lv and S G VS(fi). Then S is said to be maxi­ mally free in fi if S is free and there does not exist T G VS(fi) such that S C T and T is free. We note that if x(*5) is maximally free in /x, then S is maximally free in /x, but not conversely. It follows that the property that there does not exist T G S(V) such that x(«S) C x(T) C \x and T is free is the property which corresponds to that in Definition 8.1.4. Definition 8.1.6 Let \i be an L-subspace of V and let x € F{ii). Then x is called a basis of /J, if X is free and (x) = /x. Theorem 8.1.7 Let u be an L-subspace of V and let x £ ^(AO- Then x is maximally free in \i if and only ifx* is a basis of u* andM x G x*> x(x) = M( X )Proof. Suppose that is x maximally free in [i. Then x* is free by Theorem 8.1.3. Suppose that x* is not a basis for /x*. Then 3 S such that x* C S C /x* and 5 is free. Define the fuzzy subset 77 of V by 77(2;) = fi(x) if a: G 5 and n(x) = 0 if x £ S. Then 77* = 5 and 77 is free in (j, by Theorem 8.1.3. Since X* C 77*, x C 77, a contradiction. Thus x* is a basis for /i*. Clearly, x(x) = f^(x) V x G x*- Conversely, suppose that x* is a basis for /x* and x0*0 — M x ) V x G x*- Then x is free. Let 77 G ^"(/x) be such that x ^ V a n d f? is free. Then X* = 77* by the maximality of x*- Since /i = x on x* and x* = V* 1 X = V- Thus X is maximal in /x. ■ Theorem 8.1.8 Let a be an L-subspace of V and S G VS(fi). maximally free in /x if and only if <S* is a basis for \i*.

Then S is

Proof. Suppose that S is maximally free in a. Then S* is free in \i*. Suppose that 5* is not a basis for /x*. Then 35, S* C S C /x*, such that 5 is free. Let S' = S U T , where xa G T if and only if x G S\ 5* and a = /x(x). Then 5'* = 5 and so 5 ' is free, a contradiction of the maximality of S. Conversely, suppose that 5* is a basis for /x*. Let S' G 5(/x). Suppose that ScS'. Then 5* C S'* and so <S'* is not free. Hence Sf is not free. Thus S is maximal. ■ Corollary 8.1.9 Let a be an L-subspace ofV. Let S,T G VS(u) be such that T is free and (S) = [i. Then 3 5 ' C 5 such that T\JSf is maximally free in fi

and T n S' = 0.

318

Proof. T* is free and s(S*) = (S)* = p*. Thus 3 S C S* such that T* U 5 is a basis for /x* and T * n 5 = 0, [261, Theorem 23, p. 98]. Let S' = {xa \ x e S and x a G S}. Then T U S' is maximally free in /x and T D S' = 0. ■ The following properties allow us to obtain results concerning the existence of bases for L-subspaces of V. (7') If /JL is an L-subspace of V, then \ia is a subspace of V V a E JJL(V). (8') If fi e Lv and /i a is a subspace of V V a G L, then /x is an L-subspace of V. An illustrative example seems advisable at this point. We show that 3 V, 5, and (), where all enumerated properties so far listed hold except (8'). Example 8.1.10 Let L = [0,1]. Let V = {x, y}. Define s(0) = {y} = s({y}) and s({x}) = V = s(V). For any x £ L v , define (x) by (x) (x) — 1 ifx(x) > 0, (X) (x) = 0 if X{x) = 0, and (X) (y) = 1. Clearly, x C (x). T/iat x C 77 implies (x) C (77) is immediate. It is also clear that ((x)) = (x) • Thus we have properties (1'), (3'), and (4'). Clearly, properties (1), (3), and (4) hold. Property (2) /io/ds since V is finite. Let u, v E V and S Q V. Suppose that u E s(S U {v}) and u £ s(S). Then u = x and either 5 = 0 or S = {y}. Since x E s(S U {v}), i; = x also. Hence property (5) holds. Now s(x*) — {2/} */ x ^ x* and s(x*) = F z/x G x*. 4 b o <X>* = M if x <£ x* and (x)* = V t / x E x * - Hence property (6') /io/ds. Le£ /z be an L-subspace ofV. Then \x = (fi) and so fi(y) = 1 and either /i(x) = 0 or JJL(X) = 1. Hence either Hi = {y} or fii = V. Thus ^1 is a subspace ofV. Hence fia is a subspace ofVMa Elm(fi). That is, property (7') holds. Now let \x G Lv be defined by ji(y) = 1 and /JL(X) = \. Then (\x) (y) = 1 = (/z) (x). Hence // / (/x). 77ms /i is no£ an Lsubspace of V. Now fi\ = {y} and fix = V are subspaces of V. Hence \ia is a subspace of V \f a E [0,1]. That is, property (8') does not hold. We note that {x} is a basis for V with respect to s and {xa} is a basis for ly in terms of () for all a E (0,1]. Proposition 8.1.11 Let {Ui \ i E 1} be a collection of L-subspaces (subspaces) ofV. Then C\iejUi is an L-subspace (subspace) ofV. Proof. For all i E J, Ui = (Ui). Thus (nieIUi) D ni€lUi = DieI (Ui) 2 (r\ieiUi), where the latter containment holds since Ui D DieiUi and so (Ui) 5 Viei. * Proposition 8.1.12 Let {Ui | i E 1} be a chain of subspaces of V. Then Ui^iUi is a subspace of V. Proof. VieI,Ui = s(Ui). Thus UieIUi = Ui€Is(Ui). Let x E s(UieIUi). Then 3 a finite subset U of UieIUi such that x E s(U). Since U is finite,

319 3 j e I such that U C Uj. Hence x G s(Uj). Thus x G U» € /s(l/i). Hence s(UieiUi) C Ui€/s(C7i). For all i G 7, s(C/;) C s(U»€/J7*) and so UieIs(Ui) C 5(U»G/l/»). Thus U*€/£/i = s({JieIUi). ■ Theorem 8.1.13 Suppose that property (8') /lo/cfe. fi^ Le£ V\ C V2 C . . . C Vi C . . . 6e a strictly ascending chain of subspaces ofV. Define the L-subset /A of V by /i(x) = ai if x G Vi\Vi-i, w/iere 1 > a; > a i + i > u > 0 /or z = 1,2,..., and V0 = 0; ^(x) = u i / x G ^ \ U £ x V». Then JJL is an L-subspace of V. (2) Let V = Vo D V\ D ... D V{ D ... be a strictly descending chain of subspaces ofV. Define the L-subset fiofVby /i(x) = a; if x G V£\Vi+i, where 0 < a{ < cii+i < 1 /or z = 0 , 1 , . . . ; /i(x) = 1 if x G H ^ V ; . T/ien /x 25 an L-subspace ofV. Proof. fia. = Vi is a subspace of V V a^ Glm(/x). Now H ^ V * = /x^, where v = A{ai | z = 1, 2,...} in (1) and fl^V^ in (2) are subspaces of V. Hence fia is a subspace of V V a G L. Thus // is an L-subspace of V by property (8'). ■ Definition 8.1.14 Let \x be an L-subspace of V. Then fj, is said to be finite dimensional if [i is finitely generated, i.e., 3 x G T{\i) such that x* is finite and ii = (x); otherwise fi is infinite dimensional. Theorem 8.1.15 Suppose that L is a dense chain. (1) Suppose that property (7') holds. If V is finite dimensional, then every L-subspace fiofVis finite-valued. (2) Suppose that property (8') holds. If every L-subspace fi of' V is finitevalued, then V is finite dimensional. Proof. (1) Suppose that 3 fi which is infinite-valued. Then Im(/z) contains either a strictly increasing or a strictly decreasing infinite sequence. Hence V contains either a strictly descending infinite chain or a strictly increasing infinite chain of subspaces by property (7'). Hence V is infinite dimensional. (2) Suppose that V is infinite dimensional. Then V contains both a strictly ascending or a strictly descending chain of subspaces. Hence by either (1) or (2) of Theorem 8.1.13, V has an L-subspace which is infinite-valued. ■ Lemma 8.1.16 Suppose that property (7f) holds.

Let x € Lv.

Then V a

elm((x)),s(Xa)c(x)a. Proof. Since x Q (x), Xa Q (x)a- T h u s sUa) Q s((x) a )- Since (x) is an L-subspace of V, (x)a is a subspace of V. Thus s((x) a ) = (x)a • That is, s(Xa)Q(X)a■

320

Theorem 8.1.17 Suppose that L is a chain. Suppose that property (7') holds. Let /J, be an L-subspace ofV. Iffi is finite-valued, then ji has a basis. Proof. Let Imfju) = { a i , . . . , a r } , where a\ < ... < ar. Then V = /i a i D ... D AW• Let vT C fjLar be a basis for /x 0r . Suppose that vr has been extended to a basis Br U . . . U i?i+i of fiai+1 such that £& C AW for A; = z + 1 , . . . , r. Then £ fc C / / a ; t \ / V + i for fc = i + 1,...,r - 1. Extend B r U . . . U B»+i to a basis # r U . . . U Bi of fiai. By induction, we have a basis B = B r U . . . U B\ of V>ai = y such that B» C ^ a i \ ^ a i + 1 for i = 1 , . . . , r (^a r+1 = 0)- Let x G /x*. Then /x(x) = a^ for some i. Thus x G /x0i = s(Br,... ,Bi). Let 5 = {x a m | x e B, am = fi,(x), ra = 1 , . . . , r } if ai > 0 and 5 = {#a m | x G B , a m = n(x),m = 2 , . . . , r } if ai = 0. Since x G s(B r ,...,-Bi), a^ < ai+i < . . . < a r , and B r U . . . U Bi = x ( S ) 0 i , we have that x G s(x(5) a i ) Q (x(5)> a . , where the inclusion holds from Lemma 8.1.16. Thus xai C (x(«S)) = ( 5 ) . Hence /x(x) = ai < (S) (x) < /J>(x). Thus (S) = fi. By Theorem 8.1.3, S is free. ■ Corollary 8.1.18 Suppose that L is a chain. Suppose that property (7') holds. If V is finite dimensional, then /J, has a basis. Proof.

The result follows from Theorems 8.1.15 and 8.1.17.

■

( 9 ; ) V x € L v , i f Im(x) < oo, then Im((x)) < oo. Corollary 8.1.19 Suppose that L is a chain. Suppose that properties (7') and (9') hold. If /J, is finitely generated, then fi has a basis. Proof. There exists x £ ^(AO s u c n that A4 — (x) a n d X i s finite-valued. By property (9'), (x) is finite-valued, i.e., /x is finite-valued. ■ (2') Let S G 5(V), 5 ^ 0. V x a C (S), 3 <S' C S and y6 G 5 with b < a such that xa C ( 5 ' U {7/5}) and x a g (S 7 ). (5') Let S G FS(V). If x a C (5Ufo 6 }) and xa % (S) for b < a, then l/6C(5U{xa}). Lemma 8.1.20 Suppose that properties (2') and (5') /ioZd. Le£ S G «S(V). T/ien V xa G 5 , (<S\{za}) (x) =0 if and only if\f xa eS,xa£ (S\{xa}). Proof. Suppose that 3 xa G S such that xa C ( 5 \ { x a } ) . Then (5\{x a }) (x) > a > 0. Conversely, suppose that 3 xa G S such that (5\{x a }) (x) = b > 0. If a < 6, then xa C ( 5 \ { x a } ) , the desired result. Suppose that a > b. Then rr6 C ( 5 \ { x 0 } ) . By property (2'), 3 «S' C cS\{x a } such that x 6 C (<S') and 3 yc e S' such that x 6 g («S'\{2/c}), c < b. By property (5'), yc C ((57\{2/c}) U {x&}). Once again, we have that it is not the case that V yc G S, Vc % (S\{yc}). ■

321

Definition 8.1.21 Let S G FS(V). ThenS is a minimal of(S) i/V xaeS,xa £ {S\{xa}).

set of

generators

Theorem 8.1.22 Suppose that properties (2') and (5') hold. Let S G FS(V). Then S is free if and only if S is a minimal set of generators of (S). ■

8.2

L-Subspaces, L-Subgroups, and L-Subfields

In this section, we show that the results of Section 8.1 can be applied to the concepts of freeness for L-subspaces, relative p-independence and algebraic independence for L-field extensions, and p-independence and torsion-free in­ dependence for L-subgroups. We assume throughout this section that L is regular. If V is a vector space over a field F and xx is an L-subset of V, then xx is called an L-subspace of V if xx(a; — y) > /JL(X) A /JL(X) and xx(rcc) > /JL(X) VX,T/ G V and Vr G F. Let F be a field and let xx, v be L-subfields of F such that /x I) ^- If 7 is an L-subfield of F such that v C 7 C xx, recall that 7 is called the algebraic closure of v in xx if 7 is algebraic over v and V xa C xx, x a is algebraic over z/ implies a:a C 7. That 7 exists follows from Theorem 5.1.19. In the following, we let V denote a vector space over a field F, K a subfield of F, and G a commutative group. Let x € L G . In this section and the next, -< x ^~ denotes the intersection of all L-subgroups of G which contain x- We consider the following cases for s and () : (a) Let 5 G P(V) and x € Lv. Let s(5) = the intersection of all subspaces of V which contain S and (x) = the intersection of all L-subspaces of V which contain x- Let xx be an L-subspace of V such that /x* = V. Then () maps J-(fi) into itself since V x £ -^"(A0> (X) i s the intersection of all L-subspaces of V contained in /x and containing x; (b) Let 5 G P(F) and x £ £ F - Let M a n d ^ be L-subfields of F such that /x D 1/ and xx* = F Let F / i f have characteristic p > 0. Let s(5) = the intersection of all subfields of F which contain i/*(/x*p) U S and (x) = the intersection of all L-subfields which contain ^(/xp) and x- Then () maps ^"(/x) into itself since V x £ ^"(A0> (X) is the intersection of all L-subfields of F contained in xx and containing ^(xxp) and x; (c) Let S G F ( F ) and x £ £ F - Let /x and 1/ be L-subfields of F such that xx D z/ and xx* = F Let s(5) = the algebraic closure of v*(S) in F and (x) = the algebraic closure of v{x) in 1^- Then V x £ ^(AO? (X) H xx is the algebraic closure of z/(x) in /x; (d) Let 5 G P(G) and x ^ LG. Let /x be an L-subgroup of G such that /x* = G. Let s(S) = the intersection of all subgroups of G which contain pxx* U S and (x) = the intersection of all L-subgroups which contain pzx and x-

322

Then () maps J7^) into itself since V x € J r (^), (x) i s the intersection of all L-subgroups of G contained in /x and containing p/j, and x; (e) Let S G P(G) and x ^ LG. Let /x be an L-subgroup of G such that /i* = G. Let 5(5) = the torsion closure of -< S >- (the subgroup of G generated by S) in G and (x) = the torsion closure of -< x >- hi 1Q. Then V x £ ^(AO* (x) fl /i is the torsion closure of -< x >- in /i. Then from known results [261, p. 50, p. 97, p.129] and Section 2 of [164], 5 satisfies properties (1) — (5) for cases (a) — (e) and it is clear that () satisfies properties (1'), (3'), and (4') for cases (a) — (e). Lemma 8.2.1 Let V be a vector space over the field F. Let x £ Lv. £ G Lv as follows: \/x G V,

Define

n

£(x) = v{A7=iX(^i) I ^ O i ^ i = x,a,i e F,Xi e V,i = 1,... ,n;n e N}. Then (x) = £. Proof. Clearly, x Q £• We now show that £ is an L-subspace of V. Let a G F and a; G V. Then £(ax) = V{A^ = 1 x(^)| YJi=i aixi = axi ai € Fixi e V, i = l , . . . , n ; n 6 N} > V J A ^ x ^ i ) ! E?=i a a ^ = ax > a * G F ' x ^ G ^> i = l , . . . , n ; n G N} = V{Af =1 x(x;)| a ( E i L i a i x 0 = axi ai ^ F,x{ e V, i = l , . . . , n ; n G N} > V{Af=1x(a:i)i E 2 = i a ^ = x, a^ € F , ^ G V, z = l , . . . , n ; n G N} = £(x). That is, £(ax) > f(x) Va e F,x e V. Let x,y G V. Then £(x + y) = V{A?=1x(x*)| E I L i <W = i + y, o< e F . i i € V, i = 1 , . . . , n ; n 6 N} > V{A{x(u<) A x f a ) | i = 1 , . . . ,n} | £ " = 1 a ^ = x, £ " = i M i = y, a.i,bi e F,Ui,Vi eV, i = l,...,n; n eN} = (V{A^ = 1 x(^)| YA=I a ^ = x, a{ G F , ^ G V, i = l , . . . , n ; n G N}) A (V{A? = 1 x(^)| £ ? = i M i = 2/, bieF,ViSV, i = l , . . . , n ; n € N } ) = £(*)Af(y). ■ For any of the cases (a) - (e), let x a C (S), where 5 G PS(/JL). Then (5) (x) > a > 0. Hence by Lemma 8.2.1, Theorem 2.1.9, and Theorem 5.1.10, 3 a finite subset S' of S such that xa C (S'). Since 5 ' is finite, property (2') clearly holds. We now consider case (a). Let /i be an L-subspace of V. Let x G .T 7 ^)- Then (x)* = (n{7|7 is an L-subspace of V, x Q 7})* Q H{7*|7 is an L-subspace of V, X Q 7} = *(x*)- By Lemma 8.2.1, six*) Q (x)* . Thus property (6') holds. Property (9') also holds by Lemma 8.2.1. That properties (7') and (8') hold follows easily. Proposition 8.2.2 Consider case (a). Then property (5') holds for ().

323

Proof. Suppose that x a C ( 5 U {yb}) and xa <£ (S), where 0 < b < a and S G FS(V). Then by Lemma 8.2.1, x = £ 2 n = 1 kiXi + kyy where A*, A: G F, M ° ) a n d (si) a i € 5, 2 = 1 , . . . ,n, such that a < axA . . . A a n A 6. Hence 2/ = E^iC-fc" 1 )***, + A:-1^. Since K a ) 2 / 6 C ( 5 U {x a }). ■ Recall that V x G L v (or L F , or LG) (x) C ^ if and only if /z is an Lsubspace of V (or F, or (?) in the sense of Definition 8.1.1. For cases (a), (b), and (d), we have that V ^ G .F(/z), (x) Q fi. In order to apply the results of Section 8.1, we thus note that 7 is an L-subspace of V (or L-subfield of F containing v(np), or L-subgroup of G containing p/x) if and only if 7 is an L-subspace of V (or F, or (?) in terms of Definition 8.1.1 for case (a) (or case (b), or case (d)). If 7 is an L-subspace of F, G in terms of Definition 8.1.1, then 7 is an L-subfield of F, 7 is an L-subgroup of G for case (c), (e), respectively. Therefore, all the results of Section 8.1 hold for case (a), i.e., for L-subspaces of a vector space. Let fi be an L-subspace of V. Suppose that the map B is ^-admissible for /x [186, Definition 5.1, p. 368]. Then Ua>oB(a) is free in fi by Theorem 8.1.3. If B is a basis of zx according to [186, Definition 5.5, p.370], then Ua>oB(a) is a fuzzy basis of /x according to Definition 1.6 and conversely by [186, Theorem 5.4, p. 369] and Theorem 8.1.8. Proposition 8.2.3 Properties (6') and (97) hold for cases (b) — (e). Proof. (b) By Theorem 5.3.14 and Theorems 5.1.10 and 5.1.12, (x)* = v(ljP)(xY = ^*(/**P)(x*) = s(x*)- T ^ s property (6') holds. Property (9;) holds by Theorem 5.1.10. (c) By Theorem 5.1.12, v(x)* = ^*(x*)- Since (x) /v(x)1S algebraic, (%)* /v* (x*) is algebraic by Theorem 5.1.15. Thus s(x*) 3 (x)* • Let 6 = l8(x*)- Then (5* = s(x*) and so 8*/v(x)* 1S algebraic. Hence 6/v(x) is algebraic. Thus 6C(X). Hence s(X*) = $* Q <X>* • Thus s(x*) = (x>* • Hence (X) = l s(x *) and so Im((x)) = 2. (d) By Theorem 1.5.4, Proposition 2.4.16, and Corollary 2.1.11, (x)* = (p^ + -< X >-)* = (PAO* + -< X >-*= PV>* + -< X* >-= s(x*)- T n u s property (6;) holds. Property (9') holds by Theorem 2.1.9. (e) By Corollary 2.1.11, -< x >^*=^< X* >- • Since (x) is the torsion closure of -< x >~ m 1G'» (X)* / -< X* >- i s t n e torsion subgroup of G/ -< x* >- by Theorem 2.3.9. Thus s(x*) = (x)* • Hence l s ( x * } = (x) and so Im((x» = 2 -

■

Hence all the results of Section 1 up through Corollary 8.1.9. hold for cases (b) — (e). For cases (c) and (e), this follows since if x Q A*, X 1S (maximally) free in 1^* if and only if x is (maximally) free in fi. We now interpret the concept of freeness for cases (a) — (e). Definition 8.2.4 Let \i be an L-subspace ofV. Let x G ^(AO- Then x is called linearly independent m / i i / V x i , . . . , Xk G x*> V a, a i , . . . ,afc G L\{0}

324

with x(xi) > aii and V r» € F, ri(xi)ai + . . . + rk(xk)ak = Oa implies r a = . . . = rk = 0, where ri(xi)ai = (r{Xi)ai, i = 1 , . . . , k. x is called a basis for [i if x is linearly independent in fi and fj,=^x>- • Theorem 8.2.5 Let \i be an L-subspace of V. Let x € «F(/-0- Then x is free with respect to case (a) if and only if x is linearly independent in //. Proof. Suppose that x ls n ° t linearly independent. Then 3 Xi E x*> a> ai E L\{0} with x(xi) > a25 a n d ci E F, i = 1 , . . . , A;, such that c\(x{)ai + . . . + Ck(xk)ak = 0a and not all c» = 0, say c\ ^ 0. Then c\X\ + . . . +cfca;fc = 0, a = ai A . . . Aajfc, and #i E s(z 2 , ..., xk). Thus (xi) a C (x - £ i ) . Hence (x~ xi) (xi) > t > 0. Thus x is not free. Conversely, suppose that x is not free. Then 3 x E x* such that (x — x) (x) = a > 0. Thus a?a C (x — x). Hence (x — x) (x) > a > 0. By Lemma 8.2.1, 3 Xi E X*\{ x } and Q E F, i = 1 , . . . , fc, such that x = ci^i + . . . + ckxk. Hence (—x)a -f ci(#i) a + . . . + ck(xk)a = 0 a . Thus x is not linearly independent. ■ Definition 8.2.6 Let \i and v be L-subfields of F such that v C fi. Let x £ .F(^). Then x is called relatively p-independent in [i/v if E

(fc*i-*r)«*1...«r((«l)*1)«i • • • ( W ' ) « r = 0„,

w/iere fc^...^ E ^*(/i* p ), t^...*,., a, a i , . . . , a r E L\{°}> |/(Mp)(**i...*r) ^ w*i...ir> x( x j) > aj> 0 < h , • • • ,ir < P implies the kilmmmir = 0. x ^5 called a relative p-basis of \xjv if x is relatively p-independent in \xjv and fi = v(fip)(x)' Theorem 8.2.7 Let fi and v be L-subfields of F such that v C ^. Let x £ T{\i). Then x is free with respect to case (b) if and only if x is relatively p-independent in ji/u. Proof. Suppose that x is not relatively ^-independent in fi/v. Then there is a summation = 0 a as in Definition 8.2.6 with A;^...^ ^ 0. Hence ^ kilmmmir i (xi)11 ... (xir)lr = 0 and A{A{uilmmmir\ i\.. . i r } , a1? . . . , a r } = a. Thus for some Xj E { x i , . . . , xr}, say Xj = xi, x\ E i/*(/z* p )(x2,..., xr). Hence {x\)b C z/(/xp)((x2)a2, ...,(2; r ) a r ) for some 6 E L\{0}. Thus (xi) 6 C ( x - x i ) , i.e., {x~xi)(xi) > s > 0. Thus x is not free. Conversely, suppose that x is not free. Then 3 x E x* such that (x - #) (x) = a > 0. Thus xa C (x — x) = ^0* p ) ( * - * ) • Hence a; = E ^ . . ^ *ii...x r (^i) il • • • (xr)ir, where A^...ir E i/"0x*p), X i , . . . , X r E X*\{^}- Thus - ^ 6 + Eii...ir(fcii...v)fc((^l)2l)6"-((^)ir)6

= 0ft,

where 6 = A{A{z^(^)(A:il...^)| ix,..., z r }, x(z), x(xi), • • •, xO*v)}. Since - 1 ^ 0, x is not relatively p-independent in [ijv. ■

325

Definition 8.2.8 Let \x and v be L-subfields of F such that v C [i. Let x E Then x is called algebraically independent over v if

J~(IJL) .

E (kn-^K-ir(MlU ii...ir

■■■((Xr)i-)ar

=0„,

where A^...^ G v*, uilmmmir, a, a x , . . . , ar e L\{0}, K ^ i - v ) > ^t!...»r, x f o ) > aj implies the kilmmmir = 0. x is called a transcendence basis for \ijv if x is algebraically independent over v and fJ>/v(x) is algebraic. Theorem 8.2.9 Let ji and v be L-subfields of F such that v C /j,. Let x € J-'(n). Then x is free with respect to case (c) if and only if x is algebraically independent over v. Proof. Suppose that \ is n o t algebraically independent over v. Then there is a summation^ 0 a as in Definition 8.2.8 with fc^...^ ^ 0. Thus J2i ir ^i...i r (xiY1 ... (xr)tr = 0 and a = A{A{t4i1...ir|zi .. . i r } , a i , . . . , a r } . Hence for some Xj G {^I, . . . , xr}, say Xj = xi, x\ is in the algebraic closure of z/*(x2, • • •»#r). Thus (xi)^ is in the algebraic closure of ^((x 2 )a 2) • • •, (xr)ar) for some b G L\{0}. Hence (xi)t> C (x — x\). Thus (x — x\) (x{) > 0. Hence x is not free. Conversely, suppose that x is not free. Then 3 x G x* such that (x — #) (x) = a > 0. Thus x a C (x — x), the algebraic closure of i/(x — #)• By Theorem 5.1.15, x is in the algebraic closure of z/*(x*\{#})- Hence x* is not algebraically independent over v*. By Theorem 5.1.24, x is not algebraically independent over v. ■ Definition 8.2.10 Lei /i 6e an L-subgroup of G. Let x £ «^"(A0- Then x is called p-independent in /x i/ V £i, . . . , #& G x* anfll V a i , . . . , afc G L\{0} x a n with x( i) > *> iO*a)ai + . . . + nk(xk)ak G p/i, UiXi ^ 0, implies p\nu z = 1 , . . . , fc. x ^ ca//ed a p-basis of /i 2/ x ^ p-independent in /J, and ji = pfi

+ ^X^Theorem 8.2.11 Lei /J 6e an L-subgroup of G. Let x € ^"(AO- TAen x ^ free in fi with respect to case (d) if and only if x is p-independent in fi. Proof. Suppose that x is not p-independent in /x. Then 3 xi G x*> ni ^ Z, ai G L\{0} such that n^x; ^ 0, x(xi) > a;, z = 1 , . . . , A;, n i ( x i ) a i + . . . + nk(xk)ak Q PP"> a n d P lni f° r some i, say i = 1. Thus 1 = ;m + nii> for some integers u and v. Hence x\ = pux\ + n\vx\. Now p\i{n\X\ + . . . + nfc^fc) > a, where a = fliA...Aafc. Thus n ^ i + . . . + nkxk G (p/x)a- Hence n\X\ G (p/i) a + -< X2, • • • ,^fc >- • Since also xi G (p/x)ai + -< ™i^i ^S! (p^)o + ^ ^1^1 >^> ^ 1 ^ ( W ) a + -< x2, • • • , xk

y • T h u s Xi G (pj^)o + -< X ~ xl

- xi ^ a C (p/x + -< x - ffi >-)a, where u = a2A...Aak.

>-uQ (Pfj)a + ~< X

Hence (x - ^1) (^1) =

326

(pfi 4- -< x ~ xi y)(xi) > a > 0. Thus x is not free. Conversely, suppose that x is not free. Then 3 x G X* such that (x - x) (x) = a > 0. Thus xa Q (X ~ x) = pfj, + X x - a >~ • Hence x = pg + J2%=i nix^ w n e r e 0 € / A a* ^ X*\{^}> and m G Z, i = 1 , . . . , ft. Thus x 4 E i = i ( - ^ i ) ^ = M a n d s o *r + E i L i ( - n i ) ( x 0 r = P9r C p/z, where r = /z(#) A x(^i) A . . . A x(xk). Since p j l , X is not p-independent. ■ Definition 8.2.12 Let \i be an L-subgroup of G. Let x € ^(/-O &e 5 ^ c ^ *^a* V x G x* > ^ ^5 ° / infinite order. Then x is called torsion independent in [i ifM x i , . . . , x f c G x* a n ^ V a i , . . . , a f c G £\{0} w#A x ( ^ ) > a*> ^> ^i(^l)ax + . . . 4- nk{xk)ak = 0 a , n; G Z implies n^xi = 0, z = 1 , . . . , ft. x is called a torsion-free basis of /i if x is torsion independent and \i is the torsion closure of -< x^~ in \i. Theorem 8.2.13 Let [i be an L-subgroup of G. Let x € F(li)- Then x is free in fi with respect to case (e) if and only if x is torsion independent. Proof. Suppose that x is not torsion independent in \i. Suppose 3 x G x* which has finite order, say nx = 0 for some n. Then xa C (0G) C (x — x) V a G L. Hence x is not free. Suppose that every element of x* is of infinite order. Then 3 X; G x*> n i G Z, a^ G £\{0}, x(^i) > a*, & = 1 , . . . , ft, Wi(#i)ai -f . . . 4- nk(xk)ak = 0a and n^x; ^ 0 for some x;, say x^ = x\. Then n i x i 4. . . 4- nkxk = 0 , a = ai A...Aafc, and niXi G^ X2, . . . , xk >- . Thus n ^ i G -< X — xi ^a and so n i ( x i ) a C ^ x — x ^ • Hence (xi) a C (x — x\). Thus (x — x\) (xi) > a > 0. Hence x is not free. Conversely, suppose that x is not free. Then 3 x G x* such that (x — x) (x) = a > 0. Thus xa C (x - x). Hence 3 n G Z, n ^ 0, such that nxa C-< x — # >- and so -< x — # >- (nx) > a > 0. By Theorem 2.1.9, nx = nix1 + . . . 4- nkxk for some n; G Z, X* G X*\{^} such that (x-x)(xi) > a, i = 1 , . . . ,ft. Thus - n ( x ) a 4 n i ( x i ) a 4 . . . 4 nk(xk)a = 0a- H x is of finite order, then x is not torsion independent. If x is of infinite order, then n ^ 0 => nx ^ 0 and so x is not torsion independent. ■ In order to determine which results hold for cases (b) — (e), we must examine properties (7') and (8') for these cases. Example 8.2.14 Let L = [0,1]. We consider case (b). Let F = P(x), where P is a field of characteristic p > 0 and x is an indeterminate over P. Define the fuzzy subsets \x and v of F as follows: fi(z) = 1 if z G Fp, fi(z) = \ if z G F\FP; v(z) = lifzeP, v(z) = \ if z G F\P. Then \i and v are fuzzy subfields of F such that / I D I / . Clearly, \i = (/x). However, s(pi) = J/*(IX* P )(^I) = F D F p = /xi. TTia* is, fi is a fuzzy subspace of F, but fia is not a subspace of F fora = l. Thus property (T) does not hold. Since s(0) = ^*(/i*p)(0) = F, 0F

327

is maximally free in fi. However, OF is not a basis for \x even though 0 is a relative p-basis for fJ,*/v* since (OF) = V(/J,P)(OF) = K ^ p ) c M- Thus JJL does not have a basis even though fi is finite-valued, i.e., Theorem 8.1.17 does not hold. Note (Op) {%) = v(/J,p)(x) = \ =^ 0. Hence x\ is not free even though x\ Example 8.2.15 Let L = [0,1]. We consider case (d). Let G = Z(p°°), where p is a prime. Then G =-< #i, # 2 , . . . ,Xi, . . . >-, where X{ has order p% and pxi+i = X{ for i = 1,2, Define the fuzzy subset fj, of G by JJL(Z) = 1 if z G^ x\ >- and /i(z) = ^ifz€G\^xi)^. Then \x is a fuzzy subgroup of G. Now fi = (/i). However, s(/xi) = pfj,* + -< x\ y= G D-< £i >-= /zi. 77ms property (7') does not hold. Similarly, as in Example 8.2.14, [x does not have a basis even though /i is finite-valued. Proposition 8.2.16 Property (8') holds for cases (b) and (d). Proof. Consider case (b). Let x € LF. Suppose that s(xa) = Xa, i.e., i / V X x a ) = Xa V aeL. Then V a e L, Xa 3 ^*(M* P ) = K^ p )* 2 K^ p )a, where the equality holds by Theorems 5.1.12, 5.3.3, and 5.3.14. Thus x 2 i/(/xp). Since s(xa) = Xa, Xa is a subfield of F. Thus x is a n I^subfield of F. Hence x = K / ^ X x ) , i-e., X = (x) • Consider case (d). Let x € I/ G . Suppose that s(xa) = Xa, i.e., p/i* + Xa=XaV a € L. Then V a £ L, Xa ^ PV* = (PAO* 2 (p/^)a, where the equality holds by Proposition 2.4.17. Hence x 2 PM- Since s(xa) = Xa, Xa is a subgroup of G. Thus x is an L-subgroup of G. Hence X = .PM + X> i-e-> X — (x) • " Proposition 8.2.17 Property (77) Zio/ds /or cases fcj and (e). Proof. Suppose that x = (x) • Then as in the proof of Proposition 8.2.3 for cases (c) and (e), x — lx* since x* = (x)* = 5 (x*) by property (6'). Thus Xa = X* and so s(xa) = s(x*) = X* = Xa V ae L\{0}. ■ Thus Theorem 8.1.15(1), Theorem 8.1.17, and Corollaries 8.1.18, 8.1.19 hold for cases (c) and (e). These results are holding in a trivial way since if X is an L-subspace of F (or G), then (x) = lx* a n d so (x) is two-valued. However, we are mainly interested in fi/u for case (c) and [i for case (e). For X € ^F{fj), (x) H \x is the algebraic closure of v(x) m M f ° r c a s e ( c ) a n d the torsion closure of ^ x ^ m M f ° r c a s e ( e )- Thus properties of (x) concerning bases can be immediately applied to [ijv for case (c) and to fi for case (e). The next theorem also immediately yields desired results. Theorem 8.2.18 (1) Consider case (c). Let fi and v be L-subfields of F such that /iD v. Then fi has a basis with respect to Definition 8.1.6, i.e., fi/u has a transcendence basis.

328

(2) Consider case (e). Let fi be an L-subgroup of G. Then /x has a basis with respect to Definition 8.1.6, i.e., /x has a torsion-free basis. Proof. (1) The desired result follows from Theorem 8.2.9 and the fact that for x € -^"(AO* X is a transcendence basis of \ijv if and only if x* is a transcendence basis of p? jv*, Theorem 5.1.24. (2) The desired result follows from Theorem 8.2.13 and the fact that for X £ «^"(A0 > X 1S a torsion-free basis for p, if and only if x* is a torsion-free basis for//*. ■ Example 8.2.19 Let L = [0,1]. Consider case (c). Let F = P(x), w/iere P is a field and x is an indeterminate over P. Define the fuzzy subsets p and v of F by n(z) = 1 = v[z) if z G P and p(z) = \, v{z) = 0 if z G F\P. Then p and v are fuzzy subfields of F such that [i^v. Now pa = F for a G [0, \] and Ha = P for a G (5,1]. Thus s(pa) = pba^ CL e. [0,1] since P is algebraically closed in F. However, (p) = lj? D p. Thus property (8') does not hold. Example 8.2.20 Let L = [0,1]. Consider case (e). Let G =^< x y be an infinite cyclic group. Define the fuzzy subset p of G by /x(0) = 1 and p,(z) — \ if z G G\{0}. Then p, is a fuzzy subgroup of G. Now pa = G if a G [0, ^] and Va = {0} if a G (5,1]. Thus s(pa) = pa V a G [0,1]. However, (p) = 1G D p. Thus property (8') does not hold. Even though property (8') fails to hold for cases (c) and (e), we have the following version of Theorem 8.1.15(2). Theorem 8.2.21 (1) Consider case (c). Let v be an L-subfield of F. If every L-subfield pi of F such that p D v and 1M* = (/x) is finite-valued, then F/v* has a finite transcendence basis. (2) Consider case (e). If every L-subgroup pi of G such that 1^* = (p,) is finite-valued, then G has a torsion-free basis. Proof. (1) Suppose that F/v* has an infinite transcendence basis T. Let *i, t2, . . . e T and set T = T \{*i,* 2 , - . - } - Then s(V U {tu . . . , U}) C s(T' U { t i , . . . ,*i + i}) since ti+1 is not in s(T U {tl9 .. .,£;}), i = 1, 2 , . . . . Let ai G L be such that a i + 1 > a», i = 1,2,.... Define the L-subset /x of F by p(z) = 1 if z G s(T U{ti}), M W = <* if 2 G s(T' U {*i,... ,^+i}) \ s(V U fa,...,*<}), i = 1,2,.... Now F = s(T) = U £ l 5 ( r U { t i , . . .,*<}) and so /x is defined on all of F. Hence /x is an L-subfield of F and /i D v. Now F = ix*. Thus 1M* = (/i) and /x is infinite-valued. (2) Suppose that G has an infinite torsion-free basis T. Let ti, t2, . . . G T and set 2* = T \{t l f t 2 , . . . } . Then s(T U fa,.. .,*<}) c a(T' U fa,..., ti+i}) since t i + 1 is not in s(T' U fa,.. .,*<}), z = 1,2,.... Let a* G L be

329 such that a,i+i > di, i = 1,2, Define the L-subset \i of G by fi(z) = 1 if Z G S(T' U {*!», ^ ) = fli if 2 G s(T' U {«!, . . . ,*i+i}) \ s(T' U {ti, . . ., ti}), 2 = 1 , 2 , — Now G = s(T) = U g ^ T ' U {tu..., U}) and so ^ is defined on all of G. Hence p, is an L-subgroup of G. Now G = p*. Thus 1^* = (p) and p is infinite-valued. ■ Theorem 8.2.22 Property (5') /io/^5 for cases (b) — (e). Proof. Consider case (b). Let S G FS(F). Suppose that x a C ( 5 U {yb}) and xa % (S). Then xj = k0j + kxjy + . . . +fc P -i,j2/ p " 1 , where % G I/*(AX* P )(5*) and (a*) 0 = (A;0j)a +(A;li)a2/a + . . . + ( V u i ^ " ^ , where ( K M P ) ( S ) ) ( ^ ) > a for i = 0 , 1 , . . . ,p — 1; j = 1 , . . . ,p — 1. We can solve for y in terms of x, x 2 , . . . , x p _ 1 and rational functions of these same k^. Now v(pP)(S) applied to these rational functions of the kij yield values > a by the definition of addition and multiplication of fuzzy singletons. Hence ya C (S U {# a }) • Consider case (c). Let S G FS(F). Suppose that xa C (SU {yb}) and o:a % (S). Then x a is algebraic over v(S)(yb) and so x is algebraic over v*( S*)(y). Hence rn(y)xn + . . . + r\{y)x + 7*0(2/) = 0, where r;(y) G */*(<S*)(2/) for some n and (i/( 5))(/c) > a for every A: G i/*(«S*) which appears as a coefficient in any ri(y). By rearranging terms in the expression rn(y)xn -f- . . . -f ri(y)x + r o(y) = 0 we obtain qm(x)ym + . . . + qi(x)y + go0*0 = 0, where qi{x) G v*(S*)(x) and the coefficients of the qi(x) in z/*(5*) are rational expressions of the same k appearing in the r;(y). Hence ya C (5 U {x a }). Consider case (d). Let S G FS(G). Suppose that xa C (5 U j ^ } ) and xa£(S) . Then x e pp* + ^ S* U {y} >- and x £ p/x*+ -< 5* >- . Hence x = pg + Y^inixi + n2/> where g € p*, X{ E S*, rii, n are integers with n and p relatively prime and xa = p# a + ]Ci n *( x i)a + nVa, where (xi) a i G <S, di > a and ^(y) > a. Now 3 integers u, v such that 1 = up + vn. Hence y = upy + vra/. Thus ?/ = p(uy - vg) + J2i(~vni)xi + v x a n d 2/a = p{uya v

9a)

+ J2i(-V7li)(Xi)a

+

v

*a Q (5U{xa}> .

Consider case (e). Let S G FS(G). Suppose that x a C ( 5 U {yb}) and z a g (S). Then nx = Ylinixi + .72/ f° r integers n, n*, j and where (xi)ai G 5 implies a; > a. Thus - j y a = ^2i rii(xi)a - nxa. Hence ya Q (S U {xa}) • ■ By Theorem 8.2.22 and comments preceding Proposition 8.2.2, Theorem 8.1.22 holds for cases (b) - (e).

8.3

Nonexistence of Bases

Lemma 8.3.1 Let x, y eV

be such that x £ s(y). Then s(x) fl s(y) = s(0).

330

Proof. Suppose there exists z G s(x) H s(y), z £ 5(0). Then z G s(x) and z € s{y) and so x G s(z) and y e s(z) by condition (5). Thus s(x), s(y) C s(z) C s(ar), 5(2/) by conditions (1) and (4). Hence s(x) = s(z) and s(y) = s(z) and so s(x) — s(y), a contradiction. Thus s(x) D s(y) = 5(0). ■ Let V = { x i , . . . , xn}. Let 5 be the identity map on P(V). Then V is a basis for V and V = \Jl=1s(xi). This example explains the need for the assumption in the following lemma. Lemma 8.3.2 Let V be an n-dimensional space, n > 2. Let { # i , . . . , x n } be a basis for V. If V ^ UjL^Xi), then there exist n -f- 1 subspaces Ui of V, i = 1 , . . . , n + 1, such that Ui n £/j = 5(0) /or z, j = 1 , . . . , n, i ^ j . Proof. Let £7* = 5(x;), z = 1,... ,n. Then for i ^ j , Ui D Uj = 5(0) by Lemma 8.3.1. Let y eV\ Uf=1s(xi). Set [7 n+ i = s(y). Then Un+i nUi = 5(0) by Lemma 8.3.1, i = 1 , . . . , n. ■ We make use of the following conditions on a space V. (8") V regular L, V has the property that V L-subsets /x of V, /xa is a subspace of V V a € L implies /i is an L-subspace of V. (10") V regular L, V has the following property: Suppose that VX-subspaces /x of V and Va: G V, if x is a basis for *x, then /x(x) = A{fi(s)\s G 5 ' } , where Sf is the smallest subset of x* such that a; G ( 5 ' ) . Theorem 8.3.3 Suppose that conditions (8") and (10") /ioZd /or a space V with a finite basis {xi,..., x n } , n > 2, 5ixc/i £/ia£ V 7^ yj^=1s{xi). There exists a regular complete Heyting algebra L and a finite-valued L-subspace /J>ofV such that 11 does not have a basis. Proof. Let L D {0, &o>&i»«- • >&n+i, 1}, where biAbj = 60, i, j = 1, . . . , n + l , i T^ j . By Lemma 8.3.2, there exists subspaces £7i,..., E/n+i of V such that Z7» D Uj = 5(0), z, j = 1,2,..., n + 1; i ^ j . Define the L-subset \x of V as follows: /x(z) = 1 if x G 5(0), ^(x) = b{ if x G t/;\ 5 (0) for i = 1 , . . . ,n + 1, p(x) = 60 otherwise. Clearly, /xa is a subspace of V V a G L. Thus /x is an L-subspace of V by condition (8"). If /x has a basis x> then x* is a basis for xx* by Theorem 8.1.7. Hence the cardinality of x* is n. Let z G /x*. Let S" C x* be the unique smallest subset of x* such that z G 5(5'). Then by condition (10"), jx(z) = A{fi(x)\x G 5 ' } . Thus Im(xx) < n +2 since | S ' | < n, but by the definition of /x, Im(/x) = n + 3, a contradiction. Thus \x does not have a

basis. ■ The cases (a), (b), and (d) referred to in the sequel are defined in Section 8.2. We review them briefly. For case (a), V is a vector space over the field F and 5(5) is the intersection of all subspaces of V which contain S, where S CV. For case (b), F is a field of characteristic p > 0, v is an L-subfield of

331

F, and s(S) = v*(Fp)(S) V S C F. For case (d) G is a commutative group and 5(5) = pG + -< 5 >-, where p is a prime, 5 C G, and -< 5 >- is the subgroup of G generated by 5. It is clear that in these cases, the assumption in Lemma 8.3.2 holds. Theorem 8.3.4 Property (8") holds in case (a) V L-subspaces 7 of a vector space V, in case (b) for those intermediate L-subfields 7 of a given L-field extension ji/v such that 7 D y(/z p ), where /x* = F, and in case (d) for those L-subgroups 7 of G such that p/j, C 7 C /i /or a pzven L-subgroup ^, where li* = G. Proof. Let L be regular. Let 7 be an L-subset of V. Suppose that 7 a is a subspace of V V a G I/. (a) Let x, y e V. Let 7(2;) = u and 7(2/) = v. Let a = uAv. Then x, y G 7a- Hence x — y e ^a since 7 a is a subspace by hypothesis. Thus 7(2; 2/) >a = uAv = j(x) A7(2/). Let k e F and rr G V. Let j(x) = a. Then x G 7 a and so kx e j a . Thus 7(far) > a = 7(x). Hence 7 is an L-subspace of V. Thus V satisfies condition (8"). (b) An argument entirely similar to that in (a) shows that 7 is an L-subfield of F. Since 7 a is a "subspace" of F in the sense of case (b), 7 a 2 ^*(^* p ) V aeL. Thus 7 D i/(^ p ). (d) As in case (a), it follows that 7 is an L-subgroup of G. Since 7 a 15 PM* V a G L, 7 D p/x. ■ Let // and 1/ be L-subgroups of G such that JJ, D v. v is said to be strongly compatible in /i if A{V(z)| 2; G 1/*} > V{^(z)| 2: ^ 1/*}. Theorem 8.3.5 Property (10") /io/ds m case fa,) V L-subspaces /z, m case (b) for those L-field extensions /i/z/ with ii/v(np) strongly compatible, and in case (d) for those L-subgroups fj, such that pfi is strongly compatible in fi. Proof. Let L be regular. Let // be an L-subspace of V. Suppose that x is a basis for \x. Let x eV. Let S' = { # i , . . . , xn} be the smallest finite subset of x* such that x G s(S'). Let a; = /i(xi), i = 1, ..,n. Then x(xi) = a*, z = 1 , . . . ,n. (a) /x(x) = ((xi) a i ,...,(a: n )a n >(a:) = V{(E?=i(fci)i(^)aJ(x)| A* G F, /x(xi) = a», z = 1, . . . , n, n G N} = CCiLi( fe i)i (**)<*<) (a) = ai A . . . A a„ since # has a unique expression of the form Yl7=i ^ixi(b) p(x) = V{(E(*i 1 ...i„k 1 ...* n ( O ^ k ) * 1 • • • ((*»)«■>)(*) \kilm..in G I / * ^ * ) , p fc K^ )( *i"...in) = wii...i»> 0 < z i , . . . , i n < P, M ^ ) = a*, i = 1 , . . . , n, n € N} = (£(ki...i» W . . * n (OOai)' 1 . •. ( W O S (*) = A{^,..i n A ax A . . . A an\ 0 < 2 i , . . . , in < p} = ai A . . . A an since fj,/v(fip) is strongly compatible and x has a unique expression of the form ^ ^ . . . ^ ( ^ I ) 2 1 . . . (xn)ln.

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(d) n(x) = V{(pgu + Td=iki(xi)*i)(x)\ 9 £ G, Pfi(pg) = u, k{ e Z, Ufa) = a.i, i = 1 , . . . , ra, n G N} = (V{w | x = pg + ^ = 1 fc;X;, -p < h < p, i = 1 , . . . , n, /x(^) = ix}) A ai A . . . A an = a\ A . . . A an since p\x is strongly compatible in [i and in the expression pg + Yl7=i ^ x * f° r x ' ^ ^s U I U C l u e if ~P < ki < p, i = 1 , . . . ,n. ■

8.4

Existence of Bases

We assume in this section that L is a dense chain and [i is an L-subspace of V. We also consider the following properties, where x € «^"(A0(I)VXGX*,XW = ^W; (2) x* is free in V; (3) If x G s(x*)i then s € s(SXfX), where S ^ = {rG X*IMr) > M ( X ) } We make use of the following condition. (10') Let x be an L-subset of V. Suppose that x G s(x*)- Let S' be the smallest finite subset of x* such that x G s(S'). Then fj,(x) > A{/x(s')| s' G S'} V L-subspaces /i of V. If /i is an L-subgroup of a group (if, *) and ii x, y E H with /i(x) ^ /i(?/), then n(x*y) = fi(x)Afi(y). This type of result has been the key in the derivation of structure results for L-substructures of algebraic structures. Our axiomatic approach has allowed us to put this important result in a general setting, as the next theorem shows. Theorem 8.4.1 Suppose that V satisfies condition (10'). Let x, y, w G V. Suppose that w G s({x, y}), w £ s({x}), and w £ s({y}). 7//x(x) ^ fJ>(y), then fi(w)

=fi(x)Afj,(y).

Proof. Assume that fi(x) > fi(y). By condition (10'), /i(w) > /x(x) A /i(y). Suppose that /J>(w) > fj,(x)Ap,(y). Then fJ>(w) > fi(y). Hence jj,(y) < /J,(X)A/J,(W). By (5), the exchange property, y G s({x, w}). Hence by condition (10'), fi(y) > fi(x) A IJL(W), a contradiction. Thus fi(w) = fi(x) A n(y). H Theorem 8.4.2 Suppose that V satisfies condition (10'). Suppose that x £ F{p) satisfies properties (1), (2), and (3). If pi has the sup property and x is maximal with respect to satisfying properties (1), (2), and (3), then /z* = s(x*)Proof. Suppose that /x* ^ s(x*)- Then there exists z G /A* such that z £ s(x*)- Since \i has the sup property, there exists y G /x* such that /i(y) is maximal in {^{z) \ z G fi*\s(x*)}- Define the L-subset rj of V as follows: v(x) — x(x) if x £ X*> v(x) = M x ) ^ x — 2/J a n d T?0E) = 0 otherwise. Then 77(2/) > 0 and so x C r/. Clearly, 77 satisfies properties (1) and (2). Let x G s(x*) and w G s({x, 2/}), w £ s({x})- Then w ^ s(x*) and u> G 5(77*). Suppose

333

that w £ s(SWiTJ). Let S' be the smallest finite subset of x* such that such that x G s(Sf). Then S' U {y} is the smallest finite subset of 77* such that w G s(S'U{y}). Now if 5" = {xu . . . , x n } , then fj.(w) > n(xi) A.. .Af^(xn)Afi(y) (since w ^ s(£u,,y)) = /JL(X) A ^(3/) (since x satisfies property (3) and by (10'). Thus fi(y) = (JL(X) by Theorem 8.4.1. Hence fi(w) > fi(y) which contradicts the maximahty of fj.(y). (If S' = 0, then fi(w) > fi(y) similarly.) Thus w G s(SWiY)Hence rj satisfies property (3). However, this contradicts the maximahty of xThus/i* = s(x*). ■ Theorem 8.4.3 Let \ € F(v) satisfy properties (1), (2), and (3). If fi* = 5 (x*)> then x is maximal with respect to satisfying properties (1), (2), and (3). Proof. Since x* is a basis of /x*, x is maximal with respect to satisfying property (2) by Theorem 8.1.7, and hence maximal with respect to satisfying properties (1), (2), and (3). ■ Theorem 8.4.4 Suppose that V satisfies condition (10'). Suppose further that every L-subspace 7 of V with 7 C fi is such that 7 = ji on s(0). If fi has the sup property, then [i has a basis. Proof. Let S = {x| X € T{ii) and x satisfies (1), (2), and (3)} . Now 0y G S. Let C C S be a chain. Let Y = UxecX- For all x G C, x(x) = M(^) V x G x*- Thus {x*| X € C} is a chain. Also F * = (U x € C x)* = Ux€CX*It thus follows that 77 satisfies properties (1) and (2). Let x G 5(77*). Then £ £ 5 (x*) f° r some x £ C. Hence 77 satisfies property (3). Thus S has a maximal element by Zorn's Lemma, say x- By Theorem 8.4.2, fi* = s(x*)- Let z G fi*. Then z G s(x*) and so 2 G s(SZtx) by property (3). Hence ji(z) < fi(r) V r e SZfx- Let S' C SZ}x be the smallest finite set such that z G s(S'). Suppose that S" ^ 0. Then by condition (10'), Kz) > A M r ) l r € s'}- T h u s /i(z) = A{/i(r)| r G 5 ' } . Hence M (z) > (X) (z) > A{(x) (r)\ r G S'} (by condition (10')) = A{/x(r)| r G S"} (since r e x*) = Kz). Thus ^z(z) = (x) (z). Suppose that Sf = 0. Then ji(z) — (x) (z) by the hypothesis. Hence fi = (x). Thus x is a basis for )Lx. ■ Suppose that /x and v are L-subgroups of G such that \x^ v. Then z/ is said to be compatible in 11 if A{ii(z)\ 2 G / } > V{yLi(2:)| z ^ 1/*}. Theorem 8.4.5 Property (10') /10/c^s m case (a) V L-subspaces fi, in case (b) for those L-field extensions fijv with fi/iy(fip) compatible, and in case (d) for those L-subgroups /J, such that pfi is compatible in \i. Proof. Suppose that x G s(x*)- Let S' be the smallest finite subset of x* such that x G s(Sf). Let S' = { x i , . . . , xn}.

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(a) Here x = YJi=ikix^ where k{ e F, i = l , . . . , n . Hence /J,(X) > A{fi(kiXi)\ i = l , . . . , n } = A{v(xi)\i = l , . . . , n } . (b) Here x = Z ^ . i ^ i Y 1 • • • ( * n ) S ^ . . . ^ e */*(M* P ), 0 < i i , . . . , i n < p. Hence fi(x) > A{/i(fc; 1 ...ijA^(xi)A.. .A/x(xn) |0 < i i , . . . , i n < p } = M^ijA . . . A /x(xn) since /i/u(fxp) is compatible and K^ p )* = (^*)(M* P )(d) Now x = pg + J27=i hxii w n e r e g E G and /^ G Z, z = 1 , . . . ,n. Hence /x(a?) > fi(pg) A /x(fciXi)A . . . A fi(knxn) > fi(pg) A /z(xi) A . . . A /x(xn) = /i(xx) A . . . A fi(xn) since p^x is compatible in [i. ■ Remark 8.4.6 Every L-subspace v of V with v C /j, is such that fi = i/ on 5(0) m case fa,) V L-subspaces ji, in case (b) for those L-field extensions {ijv with fj,/v(fj,p) strongly compatible, and in case (d) for those L-subgroups /J, such that pfj, is strongly compatible in \x. Proof. The proof is easy since s(0) = {0} in case (a), s(0) = v{yP) in case (b), 5(0) = np in case (d). (In case (a), we use v(0) = 1 if v is an L-subspace

afV.) ■

8.5

L-Bases

In this section, L is a dense chain. The results in this section are from [127]. We no longer distinguish between the notation s and (). Definition 8.5.1 Let V be a vector space over F and /J, an L-subspace ofV. A subset X of V is said to be L-linearly independent over F with respect to ii if V finite subsets {xi,..., xn} of X, { x i , . . . , xn} is linearly independent over F and V r* G F, i = 1 , . . . ,n, /JL(Y%=I TiXi) = ^?=i^(rixi)• Example 8.5.2 Let V = R 2 . Define the L-subset /x of V by /i((0,0)) = 1, /x((0, R\{0})) = ±, and/x(F\{(0,0)},R) = \. Then x = (l\0)andy = (-1,1) are linearly independent, but are not L-linearly independent with respect to \x. Also fi(x) = fi(y) and fj,(x + y) > n(x). Proposition 8.5.3 Let V be a vector space over F and let ji be an L-subspace of V. Then any subset XofV such that 0 ^ X which has distinct /J, values is linearly independent over F and L-linearly independent with respect \x. Proof. It suffices to prove the result for X finite. The proof is by induc­ tion on \X\. If \X\ = 1, then the result is clearly true. Suppose that the result is true V subsets of V with cardinality n. Assume that \X\ = n + 1. Let {xi, # 2 , . . . , x n +i} be a set of vectors in V with distinct \i-values. By the induction hypothesis {x\, x
335

{ x i , x 2 , . . . ,xn+i} is not linearly independent and thus x n + i = J2iesTiXii where S C { 1 , 2 , . . . , n } , 5 ^ 0 and V i G 5, r» ^ 0. Then /x(x n +i) = and KiesKrixi) = hiesKxi) hence ^(x n +i) G {/x(a;i), jz(x 2 ),... , ^ ( x n ) } . This contradicts the fact that {^i,X2,... , x n +i} has distinct fi-values. Thus {#i, X2, • •., £ n + i } is linearly independent. Clearly, {xi, X2,..., £ n +i} is L-linearly independent. ■ Definition 8.5.4 Let V be a vector space over F and let fi be an L-subspace of V. An L-basis of V with respect \i is a basis for V which is L-linearly independent with respect to \i. If V is a vector space over F and \i is an L-subspace of V such that dim(V) = n, then | //(V)| < n + 1. It follows that if B is an L-basis for V with respect to fj,, then the L-subset x of V defined by x(6) = /x(6) V6 G L? and x(^) = 0 \fx ^ 5 is a basis for fi. Theorem 8.5.5 Let V be a vector space over F with a basis B = {^a}a£fiLet ao G L\{0} and {aa}aen C L\{0} be such that OQ > aa V a G fl. Le£ 0 ^ z £ V. Then z has a unique representation as z — Yl7=i rivoti> where ri =fi 0. Le£ [i € Lv be defined as follows: ^i(z) = A?=1/J,(vai) = A™=1aai and /u(0) = ao. Then B is an L-basis of V with respect to /J,. Proof. Let x,y G V\{0}. Then x and y can be uniquely expressed as x = T,ieCuDx xiv<*i, V = T,iecuDy Viv<*i s u c h t h a t C n ZX,. = 0, C7 n £>y = 0, DxDDy = 0, CL)DX and CuDy are finite and non-empty and V i G CUD X , a^ G F\{0} and V i G CUA,, ?/; G F\{0}. Suppose r,se F\{0} and rx+sy ^ 0. Let Z = {i G C | rxi + 57/i = 0} and N = C\Z. Suppose that C,Dx,Dy,Z and N are all nonempty. In case some of these sets are empty, the proof of the theorem is almost identical. Now /j,(rx + sy) = V>(52ieC(rxi + syi)vai +J2ieDx (rxi)v<*i + T,ieDy (syiWi) = KT,ieN(rxi + m)vai + E ^ , ^ ) ^ * + £; 6 i( 5 ^KJ. Since all coefficients in the above linear combination are non-zero, by definition of fj, we have fi(rx + sy) = (AieNfi(vai)) A (AieDxKv*i)) A (A» e D v /z(v a J) = ( A ; G j v a a J A ( A i e ^ a a i ) A ( A i G D y G a J > AieCUDxUDy^ai = ( A i G C U D ^ a a i ) A

(Ai£cuDyaai) Thus In the case, where /x(rx + sy) = fi(0) fi(0x + sy) = fi(sy)

if r, s ^ 0 and rx + sy ^ 0, then /x(rx + sy) > n(x) A /x(?/). rx + sy = 0, since /z(0) = ao > VfJ>(B) we must have > [i(x) A /J>(y). If either r or s is zero, say r = 0, then > fi(x) A fi(sy) > JJ,(X) A u(y). ■

Definition 8.5.6 A set B C L is said to be upper well ordered if for all non-empty subsets C C 5 , VC G C Definition 8.5.7 A set B C L is said £o Ziave an increasing monotonic limit x £ L if x is a limit of a monotonically increasing sequence in B.

336

Proposition 8.5.8 A set B C L is without any increasing monotonic if and only if it is upper well ordered. I

limits

Proposition 8.5.9 All upper well ordered subsets of L = [0,1] are countable. Proof. Suppose B C L is an uncountable upper well ordered set. Since B is upper well ordered, a predecessor function exists and is defined by predecessor (a) = V({6 G B\ b < a}) V a G B\{A(B)}. Let d(a) = a-predecessor(a) V a G B\{A(B)}. Then d(a) > 0 V a 6 B\{A(B)}. Since B\{A(B)} is uncountable > T,aeB\{A(B)\d(a) = °° b u t c l e a r l y EaeB\{A(B)} d ( a ) m u s t b e smaller than or equal to trie length of interval L. This is a contradiction and hence B must be countable. ■ Clearly, we can construct B C [0,1] which is upper well ordered with an infinite number of decreasing limit points. For example, consider B = {^ +

£ | n , m e {2,3,4,...}}. Lemma 8.5.10 Let V be a vector space over F, \i an L-subspace of V such that fJ>(V) is upper well ordered, and let U be a proper subspace of V. Then 3 w G V\U such that V u G U, fi(w + u) = fJ>(w) A fJ>(u). Proof. Since ^{V) is upper well ordered, 3 w G V\U such that /i(w) = V[/x(V\!7)]. Let u G U.li fi(u) ^ fi(w), then fi(w + u) = fi(w) A/J,(U). If jj,(u) = li(w), then fi(w+u) > /j,(w) A/JL(V). Since w-{-u G V\U and /i(w) = Vn[(V\U)], fi(w + u) < fi(w) = n{u) and hence /J,(W -f u) = JJL(W) A /J>(u). ■ Lemma 8.5.11 Let V be a vector space over F, \x an L-subspace ofV such that n(V) is upper well ordered, and let U be a proper subspace of V. Suppose Bf is an L-basis for U with respect to fi\u. Then 3w G V\U such that B+ = B'U{w} is an L-basis for W with respect to /i|w, where W = (B + ) and (B+) is the subspace generated by B + . Proof. Let w G V\U be such that fi(w) = V[/i(V\U)}. Then by Lemma 8.5.10, w is L-hnearly independent from B'. Let B+ = B' U {w}. Clearly, B+ is an L-basis for W with respect to ji\w, where W = (B+). ■ Theorem 8.5.12 V L-subspaces /J>ofV for which /J>(V) is well upper ordered, V has an L-basis with respect to fi. Proof. Let fi be any l-subspace of V for which /i(V) is upper well ordered. Let fi = {B C V\B is L-linearly independent}, fi is partially ordered set under set inclusion. Let C be a totally ordered subset of ^ and let A = UsecB. Clearly, A is an upper bound for C. Suppose x x , . . . ,xn G A. Then 3 B^),

337

-Ba(2)j • • •»Ba(n) £ C such that di G Ba(ny Since C is totally ordered 3 k such that # 1 , . . . , xn G Ba(ty. Since £a(fc) is L-linearly independent x i , . . . , x n are L-linearly independent. Thus A is upper bound of C in fi. Hence by Zorn's Lemma there exists a maximal element B' in Q. Suppose (Bf) = U is a proper subspace of V. Then by Lemma 8.5.11, 3 w G V\?7 such that £ ' U {u>} = J3+ is an L-basis for W with respect to (JL\W> where W = (B+). This is a contradiction since B' is a maximal element in fi. Hence (Bf) = V and B' is an L-basis for V. M Corollary 8.5.13 Let V be finite dimensional and let /x be an L-subspace of V. Then V has an L-basis with respect to JJL. Proof. Since V is finite dimensional, fi(V) is finite and hence upper well ordered. The result now follows by Theorem 8.5.12. ■ Let V be a vector space and \i\ and /x2 be L-subspaces of V. Recall that M x G V, (/xi +H2)(x) = V{/xi(y) A fi2(z)\ x = y + z, y,z G V} = V{/xi(y) A V2(x-y)\ ye V}. Proposition 8.5.14 Let V be a vector space over F and let [i\ and \i
< [fii(xi) A /x2(x - xx)\ A [fii(x2) A /x2(y - x2)\ (8.1)

but [/xi(xi) A \i2{x - xi)] A \\i\(x2) A fi2(y ~ x2)\ = Mi(^i) A Hi(x2) A /x2(x xi) A ji2(y - x2) < Mi(^i + x2) A /x2(x + y — xi — x2) Thus 3 x 3 = x x + x2 for which (8.1) is false. Hence (/xi + /x2)(:r + y) > (/xi + fJ>2)(x) A (/ii + /J,2)(y). (3) Clearly, (fun^V) Q ^(V)^2(V) and (/xi+/x 2 )(F) C ^(V)^2(V). Since ^i(V) and ii2{V) are upper well ordered, (/xi n ^ 2 ) ( V ) a n d (Mi + ^ 2 ) 0 0 are upper well ordered. Hence by Theorem 8.5.12, fi\ fl /i 2 and /xi + /x2 have an L-basis. ■

8.6

Dimension of L-Subspaces

Throughout this section L = [0,1].

338

Definition 8.6.1 Let V be a vector space over F and let [i be an L-subspace of V. The dimension of \i with respect to V is defined to be dim(/i) = Vx a basis of v

Clearly, dim is a function from the class of all L-subspaces to [0, oo] = [0, oo) U {oo}. An L-subspace ii is fuzzy finite dimensional if dim(/i) < oo. Proposition 8.6.2 Let V be a vector space and (j, an L-subspace ofV. Iffi is fuzzy finite dimensional, then V has an L-basis with respect to /J,. Proof. Suppose fi(V) C L has an increasing monotonic limit. Then 3 a sequence {xi\l < i < oo} C V such that {/j,(xi)\l < i < oo} is a strictly increasing sequence with limit a. We may suppose that fi(xi) = b > 0. By Proposition 8.5.3, {xi\l < i < oo} is linearly independent. Consider the fol­ lowing sequence of bases for V : Hn = the extension of the linearly independent set {xi\l < i < n} to a basis of V. Now clearly we have ^2xeH fJ>(x) > n&, which implies that dim(/z) = oo. This is a contradiction. Hence by Proposition 8.5.8, /i(V) is upper well ordered. Hence by Theorem 8.5.12, V has an L-basis with respect to /x. ■ L e t V - = /i- 1 ([o,l]). Proposition 8.6.3 Let \x be an L-subspace of a vector space V such that dim V = n < oo. Then if B is an L-basis for V with respect to fi and B' is any basis for V, then £ v G B , i*(v) < ^2veB fi(v). Proof. Since V is finite dimensional, | M ( ^ \ { 0 } ) | = k < dimV. Let p(V\{0}) = {ai\l < i < k} be such that a; > a;+i. Since B is an Lbasis for V with respect to fi, B D V"* is a basis for the vector subspace Vp and B' D V^ is an independent subset of V^. Thus |JB7 n VJf* | < | B fl V^ | Vi, 1 < i < k. Define recursively a set of injections {/i, / 2 , . •., fk} as follows: Let / i be any injection from B' D Vj?1 into B n V^1. Such an / i exists since l-B'nV^ 1 ! ^ |-BH V^ 1 ! and clearly fi(v) < fi(fi(v)) V v e B ' f l V * 1 . Given / n _ i to be an injection from Bf fl V£ n _ 1 into B D V£ n _ 1 such that fi(v) < /x(/ n _i(v)) V v € B' fl V^'1 let gn be any injec­ tion from B' H T£» into (B n y / f - ) \ / n _ i ( B / n V^ 71 ' 1 ). Such a # n exists since

\B'DT^\

- \B'nv^\

-1 B'nv^~l\

< \Bnv^\

- \B'nv£n-l\

= \BnV^\-\fn^{B'nV^)\ = K B n ^ A / n - i C S ' n l ^ - 1 ) ! .De­ f a fine fn : B H V^ " -* £ n V°» as foUows: If v e B' n V^*1"1, then let /n(^) = fn-i(v), otherwise let fn(v) = gn(v). Clearly, fn is an injection and since gn(B' n 3£») a ; « , n G { 2 , . . . , fc}, it foUows that /K(V) < fi(fn(v)) V

339

v G B* n V£n. Since V£k = V and \B'\ = \B\, fk is a bijection between B' and B. Now £ v 6 B M") = £ „ < * ' M A W ) > E . G B ^ /*(*)■ ■ Lemma 8.6.4 Le£ V be a finite dimensional vector space and \x an L-subspace ofV. Then V a G /i(V)\{0}, V* is finite dimensional. Proof. If V£ is infinite dimensional and B is an L-basis for V with re­ spect to //, then 5 fl V£ is infinite since £ n V£ is a basis for V* . Thus E ^ G B Kv) > TtveBnv? Kv) > TtveBnv? o = oo. Thus dim(^) = oo, which is a contradiction. Thus V!? is finite dimensional. ■ Theorem 8.6.5 Let V be a finite dimensional vector space and fi an L-subspace ofV. Then dim(/i) = ^ZveB A*(v), where B is any L-basis for V with respect to

wnere Proof. It is sufficient to show that YlveB' l^(v) - Y^veB^)^ B' is any basis for V. By Lemma 8.6.4 V a > 0, V£ is finite dimensional and B fl V* is an L-basis for \xyo.. Since B' fl V£ is an independent subset of V^, by Proposition 8.6.3, we have J2veB'nva M v ) < ^2veBnva viv)- This is true V a > 0 and hence £ V G J B , /x(v) < £ v € B M V ) - ■

Definition 8.6.6 Lei /x 6e an L-subset of a set S. The cardinality defined by card(/j,) = ^ S 6 5 M ( 5 ) -

of ji is

Let V be a finite dimensional vector space and \i an L-subspace of V with an L-basis B. By Theorem 8.6.5 dim(/x) = card(^|#). Let T£ = ^ _ 1 ( a ) and H; = tx-^(a,l}). Lemma 8.6.7 Let V be a finite dimensional vector space and fi an L-subspace of V. Then any L-basis B of V with respect to \i can be constructed in the following way: Let fi(V\{0}) = {ai,a2,... ,afc}. For all a;, i = 1,2, ...,fc, define Bai recursively (starting with Bai) such that Bai is any maximal linearly independent set of vectors inT** that extend the basis \Jj V{/z 2 (y\{0})} and /x2(0) > V{/xi(F\{0})}. Then 3 an L-basis B of V with respect to ui, /x2, ^ I H [12 and \i\ + ii 2 . In addition if, A\ = {x G V\/JLI(X) < / ^ O E ) } , A2 = V\A\, then V v G B H Ai, 0*1 n f^)(v)

= Vi(v),

(Mi + M2)(v) ='/x 2 (v)

andM v G B n A2, (/xi D fi2)(v) = /x2(v),

(Mi + A^2)(v) = t*i(?).

340

Proof. We prove the result by induction on dim V = n. Clearly, the result is true if dim V = 1. Suppose that the theorem is true for all L-subspaces with the dimension of the underlying vector space equal to n. Let /zi, // 2 be two L-subspaces of the vector space V such that dim V = n + 1 > 1. Let B\ = {v{\l < i < n + 1} be any L-basis for V with respect to \i\. We may assume that \i\{v\) < y>\{vi) Vz, 2 < i < n + 1 . Let H = ({vi\ 2 < i < n + 1}). Since n + 1 > 1, H ^ {0}. Clearly, dim H = n. Consider the following two L-subspaces, fii\H and I^2\H- By the induction hypothesis 3 an L-basis B' for LT with respect t o ^ i | # , /x 2 |ij, VIIH^^IH and /xi|fj+/i 2 |ij. Also Vf G B'flAi, (Milif n/x 2 |if)(v) = Mi|if(v), (/xi|jf+ / Z 2 | H X V ) = M 2 | H (^)and V v G F n A 2 ,

(Mijif n/x2|ff)(v)

= A*2|H(V),

fa\H + I^2\H)(V) =^i|i*(v).

We shall now show that B' can now be extended to B such that B is an L-basis for V with respect to /zi, /X2, //i fl ^2 and /ii -f /i 2 - Furthermore V

u e B r i A i , fa n /x2Xf) = ^i(v), fa + /x2)(v) = M20>) and V v e ^ n ^ , (^ifl/X2)(v) = ^2(^)5 (Mi+A^X^) — ^i{v)- First we have to show that V x G if, (/xi 4- /i2)|if W = (Milif + A^lffX^) (8.2) Let x G # \ { 0 } . Then fa + /x2)|if (x) = Vfafa) A /z2(x - Xi)|xi € V} = (V{/xi(xi)A/x 2 (x-xi)|xi G iJ})V(V{^i(x 2 )A/x 2 (^-a:2)|^2 G V\H}). Since x G ^ \ { 0 } wehave/zi(x)A/z 2 (x-x) = Mi(^)A/i 2 (0) < V{//i(xi) A / x ^ x - x i ) ! x\ G # } , ^i(O) A ^ 2 (x - 0) = /ii(0) A /x2(x) < V{/zi(xi) A /z2(x - xx)\ xx G H}. Since ^i(O) > V/x2(JJ\{0}) and fi2(0) > V^i(ff\{0}), m(x) A/^(O) = ^i(x) and /ii(0) A /i 2 (x) — M2(#)- Hence jzi(x) V W ( x ) < V{/xi(x!) A \±2{x - a?i)| xi G # } .

(8.3)

Suppose that V{/xi(xi) A ^ 2 ( z - a?i) |x x G i f } < V{/xi(x2) A /i 2 (x - x2)\ x2 G V V O - (8.4) Then 3 z G V \ i f such that V{//i(xi) Afi2(x — xi)| x1 G i f } < fJ>i(z) A/x2(x — z). By (8.3), /zi(x) V /z2(x) < /xi(z) A IIQ(X - z). (8.5) Since z G V \ i ? and fii(V\H) = pifa) < tiiiyi) V z, 2 < z < n + 1, we must have/ii(x) > /ii(^). Hence by (8.5), fJ>i(x)\/fj,2(x) < //i(x)A^ 2 (x —z). It can be easily checked that the last inequality never holds. Hence (8.4) is false. Thus V{jtxi(xi)A/x2(x —xi) \xi € H} > \Zfa(x2) A fi2(x - x2) \x2 G V\H}. Clearly, this is also true if x = 0. Hence V x G H fa + ^2)|i/(^) = V{/xi(xi) A ^ ( ^ — a?i)| x1EV} = (V{/xiOri)A/z 2 (z-Zi)| Xi G LT})V(V{/xi(x2)A/Lt2(a;-a;2)| x 2 G V\iif}). But since V{/xi(xi) A ^ ( ^ — xi)\ xi e H} > V{/xi(x2) A ^2(3: ^2)| ^2 G V\H}, we have (/xi + H2)\H (X) — Vfafa) A / ^ ( z - xi)| xx G H} = V{/xi|/f(xi) A /i2|if (z - ^ i ) | xi 6 i f } = (/XI|H + !12\H)(X). This shows that

341

(8.2) is true. Clearly, (8.2) implies that B' is L-linearly independent in /xi -f-/x2. Let v' G V\H be such that /z 2 (i/) = Vfi2(V\H). Clearly, such v' exists since /x2 is finite-valued. By Lemmas 8.5.10 and 8.5.11, v' is an extension of the L-basis B' with respect to JJ,2\H to B = B' U {*/} an L-basis with respect to /z2. Since /xi(V\iJ) = /zi(vi), we have that v' is also an extension of an L-basis B' with respect to \X\\H to an L-basis B with respect to ji\. Now we shall show that v' is an extension of the L-basis B' with respect to /xi|// D /J>2\H to B an L-basis with respect to \i\ n JU2. If v ; 6 4 i , then since /ii is constant on V\H,

(MI n/i2)(Ai n (y\#)) = ^(V) andV2GA2n (y\H), (^ n w)(^) < ^(V). Hence if i/ G A I , then (m n /x2)(v') = V(/ii n // 2 )(VV0- If t/ G A 2 , then ^2(^0 < Hi(v'). Since /x2(?/) = V>2(V\.ff) and /ii is constant on V \ i f we must have Ai n ( V V 0 = 0. Thus if*/ G A 2 , then (/zi n/x 2 )(t/) = V(/xi n / z 2 ) ( y \ # ) . By Lemma 8.5.11, it now follows that t/ extends the L-basis B' with respect to /ii| # H/i 2 |if to i? an L-basis with respect to ^1H// 2 . We now show that v' is also an extension of Bf an L-basis with respect to /xi|# + /x2|// to 5 an L-basis with respect to /xi + /i 2 . Suppose that 3 z G V \ i ? such that (/ii + /z2)(?/) < (/^i + /z 2 )(z). Now z can be written as z = a(v' + v), where a ^ 0 and v € H. Hence (/^+// 2 )(u') < (/xi + /x2)(z) = (/^i + /* 2 )(a(i/ + v)) = (/n + /x 2 )(t/ + v). Then 3 xi G V such that V it; /xi(iy) A /x2(*/ — w) < /xi(xi) A /x2(t/ + v — xi).

(8.6)

In particular, (8.6) is true if w — 0, i.e., //i(0)A/z 2 (i/) < /xi(xi)A/z 2 (i/-hi>—xi). But since /^(O) > V/z2(V\{0}) we have /z 2 (t/) < /xi(xi) A /x2(v7 + v - xi).

(8.7)

If #i G if, then since v e H we must have v — x\ G if. Thus by Lemma 8.5.10, /z2(v7 + v — xi) = /^(^O A /x2(v - xi) and so (8.7) becomes /z 2 (i/) < /ii(xi) A /x2(^') A \±2{v — xi), which is impossible. Thus xi G V\H. Let w — v' in (8.6). Since /x2(0) > V//i(y\{0}) we have /zi(i/) < /xi(xi) A /z 2 (t/ + v - xi).

(8.8)

Since /xi(V\ff) = Mi(vi), Mi(^) = Mi(^i). Hence (8.8) is false. Then V z G V\ff, (MI 4- V>2){yf) > (/xi + fJ>2)(z)- Thus by Lemma 8.5.11, i/ is an extension of the L-basis B' with respect to fii\n + A*2|tf to an L-basis 5 with respect to /ii + /x2- Now we shall show that if v' G Ai, then (/xi + /x 2 )(f') = /x2(?/) and if v' G A 2 , then (/xx + // 2 )(i/) = Mi(v7)- Now (m + /x2)(v7) = V{/ii(xi) A A^K - »i)|ari G V}. Let w be such that V{/xi(xi) A /i 2 (i/ - xi)| x1 e V} = /J,i(w) A// 2 (i/ — w).By substituting x x = 0 and then x x = v' and since //i(0) > V{/i 2 (y\{0})} and /x2(0) > V//i(F\{0}), /xiK) V/x2(*/) < fi1(w)Afi2(v/-w). Suppose that /*i(t/) V n2{v') < / x i H A W ( v 7 - w). (8.9)

342

If w e H, then by Lemma 8.5.10 (since B = B'\j{v'} is an L-basis with respect to /z2), (8.9) becomes / i i M V / ^ M < Pi(w) A/X2(v7) A/z2(w). This is not true and thus w G V\H. But now since /xi(i/) = fi\{w) the inequality (8.9) is false. Thus m(t/) V fi2{vf) = m(w) A / i 2 K - w) = (/ii + AX2)(t/)(8.10) The result now follows from (8.10).

■

Corollary 8.6.9 Let V be a finite dimensional vector space and /xi, \i2 two L-subspaces of V such that /^(O) > Vfi2(V\{0}) and /x2(0) > V^i(F\{0}). T/ien dim(fj,i + fi2) =dim(/j,i)+dim(/j,2) — dim (/xi fl/x 2 ). Proof.

Let 5 be the L-basis as in Theorem 8.6.8. Then

dim(/ii + /x2)

=

E V G B ( ^ I + ^)(v)

= =

EVGA1nJBM2(^) + E V GA 2 nB^l( V ) E ^ ^ n B H*(v) + E v G A 2 nB ^ 1 W + E v € A 2 n S W( v ) + E^GAxHS /*lW - E V€ A 2 HB W(v) - E t ^ n B Ml(v) E V G B Mi(v) + J2veB W(v) - E v eA 2 nB w ( v ) -Ev€A!nBMi(v) dim(^i) + dim(^ 2 ) - E V G B ( M I n w)(v) dim(/xi) + dim(^ 2 ) - dim(/xi fl ^ 2 ) . ■

= = =

Example 8.6.10 LetL = [0,1]. SupposeV = R 2 . Define Hi and \x2 as follows: /ii((0,0)) = §, W ((0,R\{0})) = i , w ( n ( 0 , R \ { 0 } ) ) = i ^ ( ( 0 , 0 ) ) = l, // 2 ({(x,x) | x € R\{0}}) = ±, ^ ( ^ { ( x , ^ ) ! x G R}) = i . 77ien m and \x2 are L-subspaces of V and ^ ( ( 0 , 0 ) ) > V{/i 2 (F\{(0,0)})} and/z 2 (0,0) > V{Mi(n{(0,0)})}. ATo^( / x 1 n M2 )((0,0)) = f, ( / x 1 n M 2 )({(x,x)| x e R\{0}}) = ±, ( / ^ n ^ X ^ U M I x € R}) = i , (jii+/X2)((0,0)) = f, (/ii+M 2 )((0,R\{0») = ±, (/xi + /x 2 )(F\(0,R)) = ^ and B = {(0,1), (1,1)} w a /uzz?/ basis with respect to ^ 1 , \x2, pi D /x2 and \i\ + /i 2 . 27ms dim(fii + ^ 2 ) = ^ + 5 = J , * K f t n M2) - i + I = &, diro^i) = i + I = f, dim(^2) = | + 5 = ftifence dira(/zi)+dzra(^ 2 )-dim(/zi n/z 2 ) = f + ^ - ^ = § =dim(fi1 + /x2). Theorem 8.6.11 Le£ /i 6e an L-subspace of a finite dimensional vector space V,U a vector space over F, and f : V -+ U a linear transformation. Then dim(fx\Kerf)

+ dim(f(/j,)\Imf)

= dim(fi).

Proof. Suppose that Ker/ ^ 0. If Ker/ = 0, then the proof is similar. Now let jB Ker / be an L-basis for Ker/ with respect to /x|Ker/, and Byx be an extension of BKerf to an L-basis with respect to \i (this is clearly possible by repeated application of Lemma 8.5.11). BVx U BKerf = B is an L-basis with

343

respect to \x and BVx fl i?Ker/ = 0- We first show that f(BVx) = Bimf is an L-basis for I m / with respect to /(/x)|i m /. Clearly, L?im/ is a basis for Im/. Let v\, V2,..., Vk G Bvx and r i , r 2 , . . . , rk G R not all zero. By definition, we have / ( M ) ( E t i ^ / K ) ) = V{M*)|s € / - H E t i n / C ^ ) ) } if f-KZlirifivi)) + 0 and / ( M ) ( £ L I ^ / ( « i ) ) = 0 if / ^ ( E L I r j f a ) ) = 0- Since £ * = i r * / ^ ) ) € Im/, we have / ( M X E L I ^ / K ) ) = V{M*)I * G r H E L i ^ / ^ ) ) } . By the hnearity of / and by the property of / _ 1 , we get / ( ^ ) ( X l i = i r ^ / ( ^ ) ) — r v z s u V{JU(X)|X = ^ + X)i=i * *> GKer/}. If z e Ker/ then z = 0 or z = YA=I ih wnere Ui G i?Ker/> not all Si are zero; so if x = z+ ^2i=1 r{Vi, z G Ker/, then either fi(x) = /z(0 + ^ i = 1 nvi) or /x(x) = / x ( ^ = 1 siui + 2 i = 1 r»Vj) and thus fi(x) = {/\^=il^{siUi)) A (Ai=1fi(riVi)), which is clearly smaller than or equal to Tnus K12i=irivi)' f(fJL)(^2i=irif(vi)) = tf=iKrivi)- B v t n e s a m e argument we get f(fj)(f(vi)) = fi(vi). Thus / ( / i J C t i n / W ) = A^i f(v)(riVi) and hence i?i m / is an L-basis for Im/. Now by definition of fuzzy dimension, we get dim(^) - EveBKerfuBvx ^v) = J2veBKerf ^v) + E v € Bv« M<>)- B u t b ^ the above, we have if z G (-Bva:), then f(ji)(f(z)) = /z(z) and thus dim(//) =

D t , € B K e r / M*>) + ^veBVx

/(M)(/M) = £vGBKer/ MM + £™GBIm/ / ( ^ ) H

dim(/x| K er/)+dim(/(^)|i m / ).

=

■

Corollary 8.6.12 The result of Theorem 8.6.11 extends to finite dimensional L-vector spaces. M

8.7

Existence and Nonexistence of Bases

In this section, L = [0,1]. Theorem 8.7.1 Let /J, be an L-subspace of a vector space V. If the dimension of V is countable, then fi has a basis. Proof. Consider a chain of subspaces V\ C Vz C . . . C Vk . . . of V, where the dimension of Vk is k and Ujg^Vjb = V. Suppose 3 x l 5 . . . ,#/- G V such that Vi = (xi,...,Xi) and if v = Yl%j=i T3xh w n e r e rj G F, j = 1 , . . . , i, then jji(y) = A* =1 ^(xj), for i = l,...,fc, the induction hypothesis. Let Xk+i G Vfc+i\Vfc be such that n(xk+i) = V{/x(v)|v G Vfc+i\Vfc}. Such an xk+i ex­ ists since /x is finite-valued on the finite dimensional subspace Vfc+i. Then { x i , . . . , Xfc+i} is a basis of Vfc+i- Let x = E ^ 1 r^x*, rfc+i ^ 0. Suppose that n(Yli=irixi) ¥" ^(rfc+i^fc+i)- Then by Theorem 8.4.1 and the induction hy­ A pothesis, we have that /u,(x) = /x(IZi=i r*x*) A Mrfc+i^fc+i) = ^i=iKrixi) r x = Tnen ^(r fc+1 x fc +i) = /\i=iH(riXi). Suppose that / i ( £ ; = i i i) M^fc+i^fc+i)-

344

tix) > KHi=irixi) hn(rk+1xk+1) = ti(Hi=irixi) = Krk+iXk+i). By the choice of xk+i, /x(x fc+ i) > fi(x). Thus /x(x) = fi(^2i=slriXi) A /z(r fc+ ix fc+ i) = Aj=1/x(riXi)A/x(rA.+ia:fc+i) = A ^ V f o s i ) . Since V = U g l ^ , { x i , x 2 , . . . ,ar fc ,...} is a basis for F. The desired result now holds since fx = (x), where x is the L-subset of V defined by x(xk) — Kxk) f° r & = 1,2,... and x(^) = 0 if x £ {ari,X2,...,x f c ,...}. ■ E x a m p l e 8.7.2 Ze£L = [0,1]. Let V be a set of sequences x = (#i, . . . , # & , . . . ) , where xk G F. Define the fuzzy subset \xofVas follows: H(x)

f 1 \ 1 - £pr

z/a: = 0 if x1 = x2 = ... = Xi-i =0,Xiy£ 0.

Le£ a^ = 1 — -^Y, i = 1,2, We s/io^ £/ia£ /J, is a fuzzy subspace of V which does not have a basis. Since the level sets of /J, are subspaces of V, JJ, is a fuzzy subspace ofV. Suppose that V has a fuzzy basis with respect to //, say B. Suppose that x,y G B D /x _1 (ai), where x ^ y. Then x = ( 0 , . . . , 0, #;, ...) and y = ( 0 , . . . , 0,2/i,...), where xi ^ yi and \i{xiy — yix) > a;. Since x and y are fuzzy linearly independent, ji(xiy—yix) = fi(xiy)Afi(yix) = a;. However, this is impossible. Thus \B n /i~1(ai) | < 1. Hence B = {z{ \ i = 1, 2 , . . . } , where Z{ G ji~1{ai). We can assume without loss of generality that zi = ( 0 , . . . , 0,1, * , . . . ) , where 1 appears in the i-th component of Zi, i = 1, 2 , . . . . Then v = Yl^i zi ^s well-defined and v G V. Clearly, v is not a finite linear combination of the Z{. Hence V does not have an L-basis with respect to /i. We now consider which L-subspaces have a basis. Let (ia be an L-subspace of Va Ma G A, A a nonempty index set. Consider a vector space V = 0 a e A Va and a function fi : V —» L defined by the equality //(#) = A ^ L ^ a - ) for an element x = J2i=ixi> xi € K^, where the oti are distinct. Clearly, /i is an L-subspace of V which we call the direct sum of the ^ a , denoted 0 a € i 4 / i a . Then \x = naGA^<*. Let fj, be an L-subspace of V. Let H* = /x _1 (( a > 1]) and VJf = M _1 ([«, 1]), where a G L. For all elements a G /x(V), consider the quotient space V"/H" We will assume H° = {0} if a = /z(0). Define a function /i a : V^/fl? -> [0,1], as foUows: Vx G V^/fl^

M a W

" \

a

ifaf^O.

We shall show that fia is an L-subspace of V^/H^. Indeed, the inequality fJ>a(rx + 5t/) > fia(x)

A ^a(|/)

(8.11)

345

is true if aa(x) = a or na(y) = «• Let iia{x) = na{y) = A*(0). Then x = y = 0, fi a (rx + sy) = /i(0) and the inequality (8.11) is true too. Denote V = ®a<EimM V^/jffJ and \J = n a G / x ( y ) /xa. We say that L-subspaces u\ of V\ and /x2 of Vz are isomorphic if there exists an isomorphism / : V\ —> V2 with the property ^i(x) = /Z2 (/(#)) for all are Vi. Theorem 8.7.3 ^4n L-subspace [lofV

has a basis if and only if / / = /x.

Proof. The L-subspace y! has a basis because it is sufficient to choose the union of bases of the spaces V£/H£ with respect to fif. An isomorphism \i' = fi leads to existence of a basis of \x. Conversely, suppose ji has a basis /?. Set £ = 0* = {a}. If fi(ei) = a, then we define e» = e* + # £ G V ^ / i ^ . Consider the linear transformation / : V —> V7 defined by /(e;) = e; on the basis and extended by linearity. We shall show that Ker/ = 0. Indeed, let x = rxeix + . . . + r t e i t + wie^ + . . . + wsejs G Ker / , fifej = ... = /i(eit) = a, n / 0 , . . . , r t ^ 0, /x(eji) > a,...,/x(e i a ) > a. Then r{eix + „ . + r t e i t + w{ej1 + . . . + wseja = 0. Thus we obtain n e ^ + . . . + rteit = 0 since V' is the direct sum of the spaces V£/H£. Hence r\e\x + . . . + rteit G iJ£. This is a contradiction since B is an L- oasis of V with respect to ^. Now we shall show that f(V) = V. Let x G V° /H° and x is its representative in V£. Then x = n e ^ + . . . + rteit + wie^ + . . . + waeja, M e *i) = • • • = Ken) = a, ri ¥" 0, . . . rt ^ 0, ^(^1) > a, . . . , A*(ejJ > a> a n ( i w e n a v e # = f{rieh + • • • + rt^h)Consequently, the map / : V —> V7 is a isomorphism of vector spaces. Clearly, the property ji{x) = / / ( / ( # ) ) is true. ■ Now we shall prove another equivalent formulation of this theorem. Theorem 8.7.4 Let \i be an L-subspace of V. Let {e?}, i G Ia, be a basis of the space V£/H°, e% be a representative of~e% in V*. Then (1) {ef} are linearly independent in V and fi(^2aie^) = A//(e?); (2) ji has a basis if and only ifV= ({ef\a G /z(V),2 G Ia}) • Proof. (1) Suppose r\e"x + . . . -f rteft + y = 0, where n ^ 0 and y is a linear combination of elements e\,b > a. Then we have r^e^ + . . . + rte%t = 0 in V£/H£, but this is impossible. (2) If V = (e?), then the map / : V —> V, /(e?) = e? gives the isomorphism /x' = /i. Then by Theorem 8.7.3, \x has a basis. Conversely, let \i have a basis. Then a = y!. Let 73 denote (e?) .JThen we have V' = Df. But {e?} is a basis for 5 . Therefore, D' =5. Hence V = D. ■ Definition 8.7.5 Let /x 6e an L-subspace of V. A set of vectors B is called a maximal L-linearly independent system if B is an L-linearly independent system and for all x G V, x £ B, the set Bl){x} is not L-linearly independent.

346

Theorem 8.7.6 Let fi be an L-subspace of V. Then fi has a basis if and only ifV = (B) for any maximal L-linearly independent system B. Proof. The sufficiency is obvious. We shall prove the necessity. Let B = {x^ be a maximal L-linearly independent system. Denote x~i = xi + H£ G V"/Hp, where fi(xi) = a. If the #;'s do not generate V', then there exists ele­ ment y G V^/H^ which do not belong to (x^ . Let y be some representative of y in V*. Then B U {y} is an L-linearly independent system. But this is impos­ sible. Hence (x{) — V1 and we have (xi) = V by Theorem 8.7.4. Consequently, (B) = V since (xi) = (x~i). ■ R e m a r k 8.7.7 If the condition of isomorphism V = (B) is replaced by the condition of equality V = (B), then Theorem 8.7.6 is not true. Indeed, consider the vector space V of Example 8.6.10. Let v be a vector with all coordinates being equal to 1, and ei be vectors with i-th coordinate being equal to 1, others being equal to 0. Let V = (v) + ({e»| i G N}). Then B = {e*| i G N} is a maximal L-linearly independent system in V and (B) ^ V. But on the other hand fi has an L-basis since V has countable dimension. E x a m p l e 8.7.8 Example 8.7.2 follows from Theorem 8.7.3. Indeed, in this case dim V?i/H?ii = 1 for any i and V' has countable dimension. Therefore, V' ^ V and \i has no basis. Theorem 8.7.9 Let U be a sub space in V having finite codimension. Let ji be an L-subspace ofV. If jj\u has a basis in U, then \i has a basis. Proof. It is sufficient to prove the theorem for the case dim V/U = 1. Let V = U + ( / ) . Let x G Lu be a basis for /i\u. Set x* = B = {e*}. Extend x to V by defining it to map all elements V\U to 0. Call this extension x a l s o Now B is a basis for U. If {e;} U {/} is an L-linearly independent set in V, then x U fai is a basis for /x, where ji(f) = a x . Suppose {e^} U {/} is not an L-linearly independent set and e^, ej 2 , . . . are all vectors from B which have Rvalue equal to a\. Then we have / = r{ei1 + . . . + rteit in the V"1/!!"1 .We can assume a\ ^ 0. Hence eix -wfw2ei2- ... -wteit = 0 for some linear combination. Thus g = eilL -wf - w2ei2 - ... - wteit G H^2 for some a 2 > ai, a2 G fi(V). Now in the basis B, the vector eil is replaced by the vector / , and the resulting set is denoted B\. Let ( e ^ , . . . ) be all vectors from Bi which have /x-value equal to a 2 . If elements 7/, e^, ejl . . . are linearly independent in V£2/H°2, then g is added to Bi, otherwise for g we do the process which is similar to one for / , and the resulting set is denoted by B2. Thus we obtain a strictly increasing sequence {a^}, a\ < a2 <

347

If the sequence {fl^} is infinite then corresponding sequence of sets B\, B2,... has the following property: the set Bi is obtained from Bi-i by replace­ ment of a vector from ^~l{ai). We define the set B' as follows: an element x with property fi(x) < a* belongs to B' if and only if x G Bi. If the sequence {a^} is finite with last element a m , then the set B' is ob­ tained by addition of one element to £? m _i (according to the above-mentioned process). We will show that the obtained set B' is a basis for V. Indeed, the vector / lies in the (B'). Further the vector a belongs to B' or it was replaced in some step. Consider the first replaced vector e^. Since / , e^2, . . . eit G B' and g = eix — wf - w2ei2 — . . . — wteit G B', we have that eix hes in (B'). The other vectors are considered similarly. Finally, the set B' is an L-independent system by construction. H

8.8

Examples

The results of this section are taken from [189]. In this section, L will always be either [0,1] or the power set algebra P(S) = (P(S), U, fl, 0, S) of some set S. We note that the lattice P(S) is not regular. V will always denote a vector space over a field F. Let B be a linearly independent subset of V. Let x G V. A canonical representation of x by elements of B is a linear combination x — X^i=i ^i^i (ki G F, kiXi 7^ 0 and all X{ G B distinct). Likewise a canonical linear combi­ nation Y^i=\ ki^i of elements of B is one that is a canonical representation for some nonzero vector. The characteristic of the field F will be denoted by char F, and whenever V and U are vector spaces over the same field F, L(V, U) will denote the set of all linear transformations from V into U. Let X be a set. If fi is an L-subset of X, then X is called the carrier of /J,. Example 8.8.1 Let V = E n be the n-dimensional real vector space. Define fjbn : V -* P ( { l , . . . , n } ) by Vx = (a?i,... , x n ) , fin(x) = {k \ xk = 0}. 7t is easily shown that fin is an L-sub space of V. fin measures each x G V by the number of vectors in the standard basis to which x is orthogonal: the more basis vectors x is orthogonal to, the greater the degree of membership of x in fjin. It is easily shown that jin has a basis; in fact, the standard basis of V is the carrier for a basis of fin. The above example presents a family {/zn} of L-subspaces with a uniform definition, which is such that every member of the family has a basis. In gen­ eral, when a family of L-subspaces is given, determining which, if any, members of the family have a basis can be difficult. In some cases, combinatorial argu-

348

ments can be used to demonstrate the impossibility of the existence of a basis for certain members of the family. Example 8.8.2 Let F he an infinite field, let V = F[x] be the vector space of all polynomials over F, and let ii : F[x] —► P(F) he the L-suhset defined by\fp £ V, fi(p) = {k £ F\ p(k) = 0}. Then LL is an L-suhspace ofV with no basis: The proof that \x is an L-subspace of V is straightforward; we simply note that ji{p) = Fifp is the zero polynomial, and that LL(P) = 0 for all nonzero polynomials of degree 0. We will now show by contradiction that [L cannot have a basis. Let B = {pi,P2> • • •} be the carrier of a supposed basis. Fix any n > 1, and consider the subspace S = s({l, x, x 2 , . . . , x n }), the span o/{l, x, x 2 , . . . , xn} of all polynomials of degree at most n. Since each x-7 is in the span of finitely many Pi, there exists a finite m > 1 such that S Qspan{pi,... , p m } . By assumption, B is the carrier of a basis for LL, so the above inclusion implies that for a nonzero polynomial q of degree at most n, there exists a subset { g i , . . . ,qj} C { p i , . . . , p m } such that q = YH=i kiQi ^s a canonical representation. This in turn implies that fi(p) = A^=1/x(gi). Since each nonempty subset of F with cardinality at most n is the root set of some polynomial of degree at most n, we conclude from the above equation that each of infinitely many such subsets can he expressed as an intersection of the finitely many sets, namely the root sets of the polynomials p i , . . . , p m - However, this is impossible. Example 8.8.3 Let F be a field and V and U be vector spaces over F. Then L(V,U), the set of all linear transformations of V into U, is also a vector space over F. Let S be any subspace of {/, and define fx : L(V, U) —* P(V) by V/ € L(V,U), fi(f) = {x e V\f{x) e S}. That is, LL assigns to each linear transformation f : V —» U its preimage of the subspace S. Then LL is easily seen to be an L-subspace of L(V,U). We thus have a family of L-subspaces indexed by the set of all subspaces of U. In the special case in which S = {0}, we get the L-subspace /J, in which the membership grade of f is just the kernel of f. In this case, we regard the degree of membership of a linear transformation f as being measured by the extent of its kernel: the bigger the kernel, the greater the degree of membership. It is clear that the above L-subspace ii always has a basis if the subspace S of U coincides with U. We will show below that if S is a "large" proper subspace of C7, then the resulting L-subspace fails to have a basis. (By a proper subspace S of U, we mean that S ^ £/, but the possibility S = {0} is allowed.) The precise nature of the "largeness" alluded to above, depends on the characteristic of the scalar field F. We will need the following result, which seems to be of interest in its own right.

349 Theorem 8.8.4 If V is a nontrivial vector space over afield F, then V is not the union of less than (char F) — 1 of its proper subspaces if char F ^ 0; and V is not the union of any finite number of its proper subspaces if char F = 0. Proof. We proceed by contradiction. Let n > 2 be given with the provison that n < (char F) - 1 whenever char F ^ 0, and suppose that there exist proper subspaces Si,..., Sn whose union is V. We may assume without loss of generality that no Si is totally contained in the union of all other 5 / s for otherwise such an Si is redundant and may be discarded. For each i = 1 , . . . , n, choose Xi G Si\ Ui^jSj (1 < j < n). Consider now the sequence of n + 1 vectors xi + JX2 (1 < j < n + 1). We claim that all terms of the above sequence are distinct. Suppose by way of contradiction that there exist j and j ' with 1 < j < j ' < n + 1, such that zi + jx2 = xi +j'x2. Then ( ( / - j)l)x2 = 0, and since x2 ^ 0, (j 7 — j)l = 0. This is clearly impossible if char F — 0. Suppose that char F ^ 0. Then 0 < j ' — j U which send each x G V to a point in S, and let {xu \u G Iy] be a basis for V. Select a linearly independent subset {l/ij • • • >2/n} Q U\S such that the equivalence classes [2/1],..., [yn] constitute a basis for the quotient vector space U/S, and consider the family T — {fl I u G i y and 1 < i < n} of hnear transformations V —► 17 defined by Vxv G V, JuK^v)

^0

otherwise

at each member xv of the basis By. Clearly, T is a hnearly independent subset of HY, U) of cardinality | T\ = (dim V)n > 2n > n. It also clear that each ft

350

£ L(V, S), and that whenever f%u ^ /£, we have fu —fi £ L(V, 5), so that the equivalence classes [fu] and [f£] are distinct elements of the quotient vector space L(V,U)/L(X,S). Since the set {[fl] \u G Iy and 1 < i < n} is linearly independent in L(V, U)/L(V, S), we infer that the codimension of L(V,S) in L(V, U) is at least \F\ > n. Suppose now contrary to the conclusion of the theorem that /J, has a basis, and let B = {fu | u G 1} be the carrier of this supposed basis. (Here / denotes some index set, and not the closed unit interval.) Again letting [fu] denote the equivalence class of fu in L(V, U)/L(V, 5), we verify easily that the set {[fu] I [fu] 7^0 and a G / } = {[/„] | UV)%S

and a G / }

spans the vector space L(V, U)/L(V, S) (which has dimension at least n + 1) and therefore must contain some linearly independent subset { [ / J ] , . . . , [A*,+i]} of cardinality n + 1. Now for each i = 1 , . . . , n + 1, the hnear transformation /* may be assumed to belong to the carrier B of our supposed basis for //, and /* {V) 2 S for i < n + 1. Hence /x(/* ) = {x G V\ /*(x) G S} is a proper subspace of L(V, U), and by Theorem 8.8.4, we may select a vector x* G V \ U&1

M(/*).

By the choice of x*, each /*(£*) ^ 5, so that each of the n + 1 vectors [/*(#)] is nonzero in the n-dimensional vector space U/S. Choose scalars fci,..., fcn+i, at least two of them being nonzero, such that Yl7=i MA* 0*0] = 0. This gives us a hnear transformation /* = Y17=i ^A* s u c n that /*( x *) — S S L ^ A * ( X ) G 5. By dropping every A;* that is 0 and reindexing if necessary, we obtain a canonical representation / * = YlT=i ^A* (2 < m < n + 1) which nevertheless still satisfies the equation /*(#*) G 5, that is to say, 771

2=1

This implies that / / ( S i l i **A*) T^ n i = i M / * )> contradicting the assump­ tion that B is the carrier of a basis for ji. ■ As a corollary of the preceding theorem, we get the following result. Theorem 8.8.6 Let F be afield of characteristic 0, and let V and U be vector spaces over F with U being finite dimensional. Then for every proper subspace S of C7, the L-subspace fi of V, where \i : L(V, U) —► P(V) is defined by V/ G

L(y,u), M/) = {* e v\f(x) G 5} fails to have a basis. ■ Example 8.8.7 Let V and U be vector spaces over the field F and let S be a subspace of U. Define /x ; V -> P(L(V,U)) by Vz G V, \i{x) = {/ G L(V,tf)

351

| f(x) e S}. It is clear that ji is an L-subspace ofV. If we again consider the special case in which S is the trivial subspace of U, then we get the L-subspace in V that assigns to each x eV the subspace x1- of L(V, U) that annihilates x. Theorem 8.8.8 Let V and U be vector spaces over the field F with the dimen­ sion of V being at least 2 and that of U being at least 1, and let S be a proper subspace of U. Then the L-subspace fi of' V given by /J,(X) = {/ G L(V, U) \ f{x) G S} Vx G V, has no basis. Proof. Assume to the contrary that B = {x\,x2} U {xu\ u G A} is the carrier of a supposed basis for /z, and let y\ G U\S be any vector. Clearly, Xi — X2 is a canonical linear combination of elements of B. Define / * G L(V, U) by \fxv G B,

f*r 1

\- J °

&">-{

yi

if v e A

ift/e{l,2}.

Then /* G /x(xi — X2)\(fi(xi) fl ^(x 2 )). This contradicts the assumption that B is the carrier of a basis. ■ Example 8.8.9 Let V be a vector space over the field F. Define fi : V —> P(L(V,V)) by Mx G V, fi(x) = {/ G L(V,V)\f(x) = x}. Then p is an Lsubspace ofV. Theorem 8.8.10 The L-subspace fi ofV, where /i(x) = {/ G L(V,V)\f(x) x} Vx G V, does not have a basis if the dimension ofV exceeds 1.

=

Proof. Suppose that ji does have a basis with carrier B — {#i, x2} U {xu\ u € A}, where we allow the possibility that the index set A may be empty. Define a linear transformation /* : V —> V as follows: f*(x\) = x2, /*(#2) = Xi, and f*(xu) = xu Vu G A. Then / * fixes x\ + x2 while faihng to either fix X\ or x2. This shows that ^i{x\ +x2) ^ A^(xi) H 7^(0:2), contradicting the claim that B is a carrier for a basis of /z. ■ As a generalization of the above, we can fix a nonzero scalar k G F, and consider the L-subset /x : V -> P(L(V, V)) defined by Vx G V, /i(x) = {fe

L(V) | / ( x ) = kx}.

This L-subspace also fails to have a basis whenever the dimension of V is at least 2 : to prove this mimic the proof of the above theorem, except define /*(xi) = x2) f*{x2) = xi, and f*(xu) =xuMue A. Example 8.8.11 Let V be a vector space over the field F and let S be a subset ofV. Let L = P(V). The L-subset fi ofV defined by \/x G V, fi(x) = {y G V \ x G {S,y)} is an L-subspace ofV. Clearly, /i has a basis if the span of S is the

352

entire vector space V. We will show that at the opposite end of the spectrum, when the span of S is the trivial subspace of V, no basis can exist. When V has finite dimension n, this leaves unresolved the question of the existence of a basis when the dimension d of the span of S is in the range 1 < d < n — 1. Theorem 8.8.12 If V is a vector space of dimension n > 2, then the Lsubspace /i defined by /i(x) = {y e V\x G (y)} Vx G V lacks a basis, where L =

P(V).

Proof.

Observe that

_ / y if x = o * j " I (x) \{0} Hx^O

n M

If xi and X2 are two distinct vectors in the carrier of a supposed basis for fi, the fact that neither is a scalar multiple of the other implies that /i(xi + x 2 ) = (xi -f X2) \{0} 7^ 0 = (xi) fl (#2) • This is a contradiction. Therefore, no basis can exist. ■ The preceding results would seem to indicate that when L is not linearly ordered, L-subspaces are more likely to not have a basis than to have one. In the case that L is linearly ordered, we have seen that L-subspaces have a basis if they have the sup property. We conclude with an example of a fuzzy vector space with a linearly ordered lattice of membership values, which, despite the fact that it does not satisfy the sup property, nevertheless has a basis. Example 8.8.13 Let L = [0,1]. Let F be thefieldZ2 and let V = {/ : N -► Z 2 I f(n) — 0 for all but finitely many n}. Define a fuzzy subset fxofVbyVfe V, fi(f) = V{1 - 2~{ I f(i) = 1 and f(j) = 0 for all 0<j< i}. Then \i is a fuzzy subspace of V, with the operation of vector addition being defined pointwise using the addition operation in Z 2 . The set of vectors B = {fj | j G N} defined by [ 1 if i zzz i /i(i) = \ 0 ifj^i shows that ji lacks the sup property, but B is also the carrier of a basis for \i. The following example is the authors'. Example 8.8.14 Let L = [0,1]. Let U = {/ | / : R -► E}. Then U is a vector space over R with the usual definitions of addition of functions and scalar multiplication. Vr G [0,1], define fr:R->Rby f(x) = 0 if x G [-r,r] and f(x) = 1 otherwise. Suppose V x G l that 0 = ( c i / r i + . . . + Cfc/ r J(x), where a G R, i = 1 , . . . , k.

(8.12)

353

Now r i , . . . , r*fc are distinct. We assume without loss of generality that r\ < . . . < rfc. Let xi+i G (r^, r i + i ] , i = 1 , . . . , k — 1. ^4/so let y G R, y ^ [—1,1]- T/ien substituting y, Xk, • •., X2 /or x m equation (8.12), we obtain c\ + . . . + c^ = 0, ci + . . . + Cfc_i = 0 , . . . , c\ — 0. Thus Ci = 0, 2 = 1 , . . . , k. Thus B = {fr \ r G [0,1]} is linearly independent over R. Let V be the subspace of U spanned by B. Then B is a basis of V. Let f G V. Then there exists fr. G B and Ci G R\{0}, i = 1 , . . . ,fc, such that f = c\fri + .. . + Ckfrk- Letr — r\ A . . . I\r^. Then r = V{s G [0, l]|/(x) = 0 Vx G [—5,5]} since r±, . . . , r^ are distinct. (Note that if x G (r*i,r2] and r\ < . . . < r^, #ien /2(x) = . . . = fk(x) = 0 ^ ^ /i(x) ^ 0 and 50 it is not the case that /(x) = 0 Vx G [—r, r] /or some r > ri.) Define the fuzzy subset fiofVbyVfe V, fi(f) = r, where r = V{s G [0, l]|/(x) = 0 Vx G [-5,5]}. Let f,geV. Then f = c1fri + . . . + cfc/rfc and 0 = di<7Sl + . . . + dTYlgSm, where gj G B and dj G R\{0}, j = 1 , . . . ,ra. 77ms f+ g = e\htl -f . . . + euhtu, where htv G 5 and e v G R\{0}, v = 1 , . . . ,n, and t/ie expression e\htl + . . . + eu/i£u is obtained from c\fri + . . . + Ckfrk + di#si + • •. + dTngSTJl by combining like terms and simplifying. Now { £ 1 , . . . , tu} }. Hence n(f+g) > /x(/) A/x(o). Clearly, /x(/) = /x(~/) and /x(c/) > /x(/), where c G R. Hence [i is a fuzzy subspace of V. Now let f £ V. Then fi(f) — /z(/i) A . . . A /x(/fc). Hence by Theorem 8.5.12, //, /ias a basis. However, fi does not satisfy the sup property since [J>(V) is not wellordered under > . That is, the sup property is not a necessary condition for the existence of a basis for a fuzzy subspace.

8.9

EXERCISES

Let V be a vector space over the field F. Let fi G Lv and K G LF'. Then K is an L-subfield of F if K is an L-subgroup of the additive group of F and an L-subgroup of the multiplicative group of F. fi is an L-subspace of V over K if /J, is an L-subgroup of V and Vk e F and Vx G V, fi(kx) > K,(k) A /x(x). Let £(V, *;) denote the set of all L-subspaces of V over K. Let fi G £(V, K) and X G L v be such that x ^ ^- Define the L-subset (x) K of V as follows: (x) K (x) = A{i/(x) I 1/ G £(V,/c), x £ *4- Suppose that «(1) = 1. Let FK = {k e F \ K(k) = 1}. 1. (163) Prove that (x) K is the smallest member of £(V, K) which contains X2. (163) Suppose that L is a chain. Let fi G £(V, /c) and let \ ^ Lv be such that x ^ M- Define the L-subset f of V as follows: £(x) = 1 if x = 0, otherwise €(*)

=

V

( ( E ? = i ( f c i k (**)«,)(*) IfciG F,Xi G V, «(fci) = 6i, X(xi) = ai, i = 1 , . . . , n; n G N}.

354

Prove that £ = (x) K . Let \x G £(V,ft) and let x £ £ V be such that x Q V- Then (a) x is said to be free over K if \fx G x*> (x ~~ X ) K ( x ) ~ 0(b) x is said to be a basis of fi over K if x is free and (x)K = ^. (c) x is said to be weakly free over K if Vx G x*> (x — #) K ( x ) < (x) K (x). (d) x is said to be a weak basis of ji over K if x is weakly free over K ^d (x) K = M3. (163) Prove that if /J, G £(V, K;), then ^ is an L-subspace of V over FK. 4. (163) Suppose that L is a chain. Let [i G L(V,«). Suppose that A{fi(x) | x G V} > V{K,(k) | A; G K ; * \ F K } . Suppose that /x has the sup property. Prove that \x has a weak basis over K. Prove that if x is a basis for \i over FK, then x is a weak basis for /x over K. 5. (163) Suppose that L is a chain. Let [i be an L-subspace of V. Suppose that A{«(fe) | k G F } > V{^(x) | x G V^jO}}. Prove that if fi has the sup property, then \x has a basis over K. Prove that if x is a basis of fi over /c*, then x is a basis of fi over K.

355

Chapter 9

GALOIS THEORY AND GROUP L-SUB ALGEBRAS Throughout this chapter, L is a chain. We make a connection between Lsubgroups and L-subfields in two ways. First, we make the connection by means of Galois theory and then by means of group algebras. The results of this chapter are taken from [168,172,173,177]. We remove the restriction that if a is an L-subfield of F, then <J(0) = 1 = cr(l).

9.1

Galois Theory

In this section, we introduce a Galois theory which corresponds to the classical theory. Thus all field extensions are assumed to be separable in this section. The theory developed here ties together some of the known results about Lsubgroups and L-subfields and opens the door for bringing together many more. We lay down the definitions and preliminary results needed for the de­ velopment of a Galois theory. We give the Galois correspondence between L-subgroups of the group of automorphisms of a normal field extension and its L-intermediate fields. We then apply our results to give a necessary and suffi­ cient condition for the solvabihty by radicals of a fuzzy polynomial. We give a comparison between the fuzzy dimension of an L-intermediate field of a normal field extension and the fuzzy index of its corresponding Galois L-subgroup. Let A be a group. If if is a normal subgroup of A, we write H < A. Let F be a field. We also let 0 and 1 denote the additive and multiplicative identities

356

of F, respectively. Let a be an L-subfield of F. We let cr# denote {x G F \ o{x) > 3 z G F such that (7(2:) > 6 and f(z) = x O f(cr)(x) >bOxe f(cr)b. ■ Suppose that K is a subfield of F We let L(F/K) = {a\ a e L(F) and cr# 2 i f } and AK denote the set of all automorphisms of F which are the identity on K. Suppose that F/K is normal, [67, p. 43]. Then an intermediate field J of F/K is normal over K if and only if V / G AK, f(J) C J, [67, Theorem 10, p. 42] or [23, p. 185]. (Since F/K is separable by assumption, F/K is normal if and only if F/K is Galois, [67, p. 27], [67, Theorem 9, p. 40].) Definition 9.1.2 Let a G L(F/K), where F/K be normal over KifVfe AK, f(o~) C a.

is normal. Then a is said to

We note that if F is a finite field and K its prime subfield, then f(a) = a V / G AK and V a G L(F/K), Theorem 5.4.10. This is in accord with the situation in the crisp case. P r o p o s i t i o n 9.1.3 Let a G L{F/K), where F/K is normal. Then a is normal over K if and only if ab/K is normal V 6 G L such that b < cr(l). Proof. Suppose that a is normal over K. Let / G AK. Then f(cr) C a. Hence f(a)b C a 6 V 6 G L. By Proposition 9.1.1, /(cr 6 ) C a 6 V b G L such that 6 < cr(l). Thus o&/if is normal V b G L such that 6 < a(l). Conversely, suppose that crb/K is normal V b G L such that 6 < cr(l). Then /(crfe) C <7& V 6 G L such that 6 < a ( l ) , where / G AK. Hence /(
357

Definition 9.1.4 Let \±, v G L(A) be such that /J, C v. Then fi is said to be normal in is if\f L-singletons fa of A such that fa C v, fao \i = // o fa. If ji is normal in v, we write \i < v. If fi < 1^, we say that \i is normal in A. (See Definition 1.4.1.) Proposition 9.1.5 Suppose that ji, v G L(A). Then the following assertions hold. (1) fi <3 v if and only if fj,a < va V a G L such that a < //(e). (2) If fi< v, then /x* < v*. (3) fi< A if and only if /z(/ o g) = ^(g o / ) V / , g G A. Proof. The result is a consequence of Theorems 1.4.3, 1.4.4, and 1.3.1. ■ The reader may wish to refer to Theorems 1.3.12 and 1.4.11 and comments preceding Theorem 1.4.10 before reading the next result. Proposition 9.1.6 Suppose that /z, v G L(A) are such that /x < v. Define the L-subset v j [i of v*jr/x* as follows: V / o /x* G ^*/M*> (v/n)(f o /j,*) = V{v(g)\g°/J>* = /o/x*}. Then vj\i is an L-subgroup ofv*/fj,*. v/fi is called the quotient of v relative to /x. Proof. (v/fi)(go[i*o f oJJ,*) — (^/^(gofo/j,*) = V{v(h) \ ho/i* = gofo^i*} > V{v(h\ o/i 2 ) | h\ o/x* = go {j,*, h2o fj,* == /o/x*} > V{v(h\) /\v{h2) | h\ o/x* = #o/x*, h2o{i* = /o/x*} = (v{^(/ii) I hi o/x* = goii*})A(W{iy(h2) \ h2ofi* = /o/x*}) = (v/vL)(gof) A(v/iM)(fof). Clearly, (i// M )(/) = (v/Mf1). ■ Proposition 9.1.7 Suppose that /x G L(A). Let f G A. Define the L-subset ffi of A as follows: V ^ G i , f^g)

=

^gof-1).

Set A/11 = {ffi I / G A}. Call \A/fi\ the index of fi in A and write [A : /x]. Define o on A/fi as follows: V //z, gfi G A/ft, (//-O ° (9t*) = ( / ° 9)V~ Suppose that fi < A. Then A/fi is a group and A/fj, ~ A//x#. Proof. The proof follows from Theorem 1.3.11 and comments preceding Theorem 1.3.9. ■ Let /x G L(A). We recall from Definition 1.4.14 that /x is Abelian if /z* is Abelian and that /z is solvable if 3 /x; G L(A), z = 1,..., fc, such that fa < /Zi_i and fa-i/fii is Abelian, i = 1, . . . , fc, where /z0 = ^ and /zfc = e^e) (Definition

358

1.4.14). Also by Theorem 1.4.15, if /J, is solvable, then so is /x* and for v G L(A) if \x C v and v is solvable, then fi is solvable. Let G be a subgroup of A. Let F G = {a G F | / ( a ) = a V / G G}. Then G F is a subfield of F. We say that FG is Galois in F. We also say that F is Galois over FG or F / F G is Galois. Let K be a subfield of F. Let A K = {/ G A| /(fc) = ^ V f c G i i : } . Then AK is a subgroup of A. We call A x the Galois group of F/K. The following properties are known to hold, [67, p. 27]. (1) V subgroups H, G of A, H C G => FH D F G ; (2) V subfields if, J of F, K C J => A * D A J ; (3) V subfields if of F, i ^ 4 * ) D if; (4) V subgroups G of A, J4( F °> D G; (5) V subgroups G of A, F ( A ( F G ) ) = FG; (6) V subfields if of F, ^ ( F ( A K ) ) = A K .

Let A denote the set of all finite groups of automorphisms of F and X be the collection of all subfields of F which are Galois and of finite co-dimension in F. The main results on finite groups of automorphisms of fields are as follows, [67, Theorem 5, p. 29]: (1) If if G J and AK is the Galois group of if, then AK G A and FA* = K. If G G A, then FG G J and AF° = G. (2) If Ge A, then \G\ = [F : FG}. (3) If if G J and J is an intermediate field of F/K, then J E l , A3 is normal in A K if and only if J/K is Galois. The Galois group of J/K is isomorphic to AK/AJ. These results yield a bijection between the collection of subfields of F which contain a fixed field if in J and the collection of subgroups of the Galois group of F/K. Let if G 1. Let A denote the Galois group of F/K, i.e., A = AK. Since A is finite, every L-subgroup of A is finite-valued. Let C(F/K) denote the set of all finite-valued L-subfields a of F such that at, D if, where b is the maximal element in Im(cr). Let c be a strictly decreasing function of L into L which is convolutive, i.e., V a G L, c(c(a)) = a. The approach here has the advantage that c may be chosen to fit the application. We emphasize that A° in the following theorem depends on the choice of c. Theorem 9.1.8 Let a G C(F/K). Suppose that Im(a) = { 6 1 ? . . . , 6 m }, where bi < ... < bm and m > 2. Define the L-subset Aa of A as follows: ( c(bm) A°(f) = { c(bi) ( c(6i)

iffeAK\A^m iffeA w\A°>i,i iffeA°>2. ab

=

2,...,m-l

359 Suppose that Im(a) = {6}, where 6 > 0. Define the L-subset Aa of A as follows: Aa(f) =liffe{e} and A°(f) = c(b) if f e A\{e}. Then A* G L(A). Proof. Clearly, (A") c ( 6 i ) = AGb^ , i = 1 , . . . ,m - 1, and (A°)c[bm) = AK K (even if A = Aab™). Thus (Aa)c{bi) is a subgroup of A, i = 1 , . . . ,ra. That is, the level subsets of A° are subgroups of A. Hence Aa is an L-subgroup of A. Thus A° G L(A). The case, where Im(cr) = {b} is immediate. ■ We note that if abrn = K, then lm(Aa) = { c ( 6 m _ i ) , . . . , c(&i)}. This is in agreement with the crisp case. Theorem 9.1.9 Let \x G L(A). Suppose that Jra(/z) = { a i , . . . , a n } , w/iere ai < . . . < a n and n > 2. Define the L-subset F^ of F as follows: ( c(an) F^(x) = I c(aj) ( c(ai)

ifxeF\F^ ifxe F ^ + i \ F ^ , j = 2 , . . . , n - 1 i / x G F^ a 2.

Suppose that Im(jj) = {a}, where a > 0. Define the L-subset F ^ o / F as follows: F*(x) =lifxeKand F*(x) = c(a) if x € F\K. Then F» G C(F/K). Proof. Clearly, (F") c ( a .) = F ^ + i , j = 1 , . . . ,n - 1, and (F»)c{an) = F Thus (F^) c ( aj .) is a subfield of F, j = 1 , . . . , n. That is, the level subsets of F ^ are subfields of F Hence F ^ is an L-subfield of F Now ( F ^ ) c ( a i ) = F ^ 2 D K. Hence F ^ G C(F/K). The case, where Im(/z) = {a} is immediate. ■ We note that if fian = {e}, then Im(F^) = { c ( a n _ i ) , . . . , c(ai)}. This is in agreement with the crisp case. Example 9.1.10 Let K be a Galois subfield of F (1) Let a G C(F/K) be such that Im(a) = {6i, 62, 63}, w/iere 61 < 62 < 63 and o~S3 = if. TTien f c(63) A a ( / ) = { c(62) I c(bi)

i / / E A Y ^ 3 =0 * / / e i4ff*3\A^a iffeA°>2.

Hence Im(Aa) = {c(62), c(&i)}. 77ms 6y Theorem 9.1.9,

c(c(6i)) P ^ M -/ W " \ cfcfa))

*/*e*v (A " )c(bi) 1/ x G F ^ ^ i ) .

A£T

77ms F ^ cr. However, if we were to "remember" that Aa(f) 0, then we could define ( c(c(6i)) F A
I c(c(63))

i/xGF\F^^i) z/xGF^)c(bi)\F^)^2)

tfxGF^^).

= c(bs) if f G

360

Since (AG)c{bl) = AGb2 and (Aa)c{b2) = A°b^, it would follow that FA° = a. (2) Let a G C(F/K) be such that Im(o~) = {&i, 62}, wAene 61 < 62 and a62 = K. Then A°(f) = c(b2) if f e AK\A°b2 = 0 and Aa(f) = c(6i) if / G A°b2. Hence Im(Aa) = {c(&i)}- In the strict sense, we must apply the definition of FA. Then we would have FA = a. Now let b\ = 0 and 62 = 1 so that we have the crisp case. Then a may be identified with K and Aa identified with AK = A. Now FA = K by crisp arguments alone. That is, in the crisp case "remembering" also takes place. We note that K is a proper subset of F and so K is two-valued in a sense while AK = A and so A is one-valued in a sense. In the following result, we assume that "remembering" takes place. Theorem 9.1.11 Suppose that F/K is normal. (1) Define V from C{F/K) into L(AK) and $ from L(AK) into C(F/K) as follows: V a G C{F/K), T{cr) = A°, V \i G L(AK), $(/x) = F*. Then V is a one-to-one inclusion reversing correspondence of C(F/K) onto L(AK) and $ is a one-to-one inclusion reversing correspondence of L(AK) onto C(F/K). Moreover $ o T is the identity map of C(F/K) and T o $ is the identity map

ofL(AK).

(2) If ii G L(AK), then [AK : fi] = [F>* : K], where a = V(Im(fi)). If a G C(F/K), then [AK : A*] = [cr3 : K] when Im(a) = {6}, b > 0 and [AK : Aa) = [aS2 : K) when Im{a) = {&i,..., 6 m } with bx < ... < 6 m , m > 2. (3) Let a G C(F/K). Suppose that Im(cr) = { 6 1 , . . . , bm} with 61 < . . . < 6 m , m > 2. Then a is normal over K if and only if A° is normal in AK. If a is normal over K, then the Galois group of crb2/K is isomorphic to AK jA°. Suppose that Im(a) = {6}, b > 0. Then a is normal over K, Aa is normal in AK', and the Galois group of ab/K is isomorphic to AK/A°. Proof. (1) Clearly, Y and $ are well-defined. Let Im(cr) = {61,... , 6 m } , where &i < . . . < 6 m and let Im(/x) = { a 1 ? . . . , a n } , where a x < . . . < an. By Theorems 9.1.8 and 9.1.9, we have that T maps C(F/K) into L(AK) and $ maps L(AK) into C(F/K). Now abi D K and so F/abi is Galois, i = 1 , . . . , m. Also naj is a Galois group for j = 1 , . . . ,n, since all subgroups of AK are Galois groups. Since also abi D crbi+1, i = 1 , . . . ,m - 1, and /i a . D fiaj+1, j = 1 , . . . , n - 1, we have that A°bi C Aabi+±, i = 1 , . . . , m - 1, and F^aj C A
361

one-to-one and inclusion reversing and that $ o T and r o <3> are the identity maps of C(F/K) and L{AK), respectively. (2) Suppose that fi G L(AK). Then [AK : JJ] = [AK : fia] = [F^ : K] by Proposition 9.1.7 and the classical Galois theory, where a = \Z(Im(fi)). Suppose that a G C(F/K). Suppose that Im(cr) = {&i,... , 6 m }, where b\ < . . . < 6 m ,m > 2. Then [AK : A°] = [AK : (A') c ( 6 l ) ] = [AK : A<"*] = K : K}. Suppose that Im(a) = {&}, b > 0. Then [AK : Aa] = [AK : {e}] = [F : K] = [ab:K]. (3) By Propositions 9.1.3 and 9.1.5, a is normal over K ^ crb/K is normal V b G L such that b < a(l) <5 o~bi/K is normal, i = 1 , . . . , m <5> A°bi < AK, i = 1 , . . . ,m ^ (A a ) c ( 6i _ 1 ) o AK, i = 2 , . . . ,m + 1 <£> Aa <\ AK. Suppose that a is normal over K. Then by Proposition 9.1.7, ( A a ) # < AK and A K / A a ~ AK/(Aa)# and the latter group is isomorphic to the Galois group of (Jb2/K since {Aa)# = (Aa)c^1) = A a6 2. The case, where Im(cr) = {&} is immediate.

■

We let K[x] denote the ring of all polynomials in an indeterminate x over K. We let 1K[X] denote the set of all f(x) = ^2^=0(ki)Uix\ where f(x) = X^r=i k%x% G K[x] and Ui G L, i = 0 , 1 , . . . , n. We say that f(x) corresponds to f(x). For such an f(x) and corresponding / ( # ) , we agree that f(x) = f(x)ui where u = A{ui \ i = 0 , 1 , . . . , n } . Let ]j(x) = ]CJ=O(^J) V J X : 7 ^ ^K[X] and g{x) correspond to #(x) with v = A{VJ \ j = 0 , 1 , . . . ,ra}. We define addition and multiplication on 1K[%] &S follows: f(x)+g(x)

=

(f(x)+g(x))w

and 7(x)g(x) = (/(aO^x))™, where w = uAv, respectively. Then l#-[#] is a completely regular commutative semigroup with identity under addition, [210, Definition 1.1, p. 104], and a commutative semigroup with identity under multiplication such that the distributive laws hold. Hence 1K[X] is a semiring. Consider the equation f(x) = 0V. The largest value v can have for the equa­ tion to have an L-singleton for a solution for x is v — u. Let F = K(r\,..., rn) be a splitting field of f{x) over K, where 7*1, . . . , r n are the roots of f{x). Clearly, in a sense, % ( ( r i ) u , . . . , (rn)u) is the smallest L-subfield of F over which f(x) factors as (x — J r i ) u ) . . . (x - (rn)u). We call uK((ri)u,..., (rn)u) the splitting L-subfield of f(x) over IKDefinition 9.1.12 Let J/K be a finite dimensional field extension of charac­ teristic 0 and r G C(J/K). Then r/K is called an L-extension by radicals if there exists m, n^ G N and L-singletons (a;)^, (Mfc* C r such that ((bi)bi)ni =

362

(a>i)bi,bi > 0, a,i G K(bi,...,bi-i),i = l , . . . , m , r = M( b i)&i> • • • > ( b m k J ) > where b = V{6; | i = 1 , . . . , m}, and r* = J. Definition 9.1.13 £e£ /(x) G IK[X]. Then f(x) is said to be solvable by radicals if the splitting L-sub field of f{x) over K is contained in an Lextension by radicals. Theorem 9.1.14 Let f{x) G K[x). Then f(x) only if f(x) is solvable by radicals.

is solvable by radicals if and

Proof. Suppose that f(x) is solvable by radicals. Then there exists a field J and an L-subfield r of J as described in Definition 9.1.12. Now the splitting field UK{{r\)u, • • •, (rn)u) of f(x) is contained in r. Hence F = K(ri,..., rn) = (ux(ri)u,..., (rn)uy C r* = J. Since J is an extension by radicals, f{x) is solvable by radicals. Conversely, suppose that f(x) is solvable by radicals, say J = Jm 2 • • • 3 Jo = K, where Ji = Ji_i(6;), bi is a root of fi(x) = xni — a^ where ai G Ji-\, i = 1 , . . . , m. Define the L-subset r of J by r{x) = wV x G J, where UK((ri)u, • • •, (^n)u) is an L-splitting field for f(x) over K. Then clearly r is an L-extension by radicals and r 5 ^x((^i)u, • • •, (^n)u)- (We consider W K ( ( n ) u , . . . , (r n ) n ) as an L-subfield of J by assuming it equals 0 on J\F.) ■ Corollary 9.1.15 Let f(x) G K[x\. Then f(x) only if the Galois group of f{x) is solvable. ■

is solvable by radicals if and

Theorem 1.4.15 is immediately applicable here.

9.2

Dimension and Index

In this section, we let L = [0,1]. We compare the fuzzy dimension, [a : K], of an L-intermediate field a of F over K and the fuzzy index, [AK : Aa], of the L-subgroup Aa of AK. Definition 9.2.1 Let a G C(F/K). < bm. Define the fuzzy dimension

Let Im(cr) = {bu . . . , 6 m }, where bx < . . . of a over K, written [a : K], to be

m

[v.K}=Y,([°bi:K}-{abi+1:K})bi, 2=1

where [crbm+1 : K] = 0. Define the fuzzy dimension [F:a], to be [F:a] = [F:K]/[a:K].

of F over <7, written

363

The following result gives the motivation for Definition 9.2.1. It, together with Theorem 9.2.4, also brings into our Galois correspondence, results from the theory of fuzzy subspaces of linear spaces. Theorem 9.2.2 Let V be a finite-dimensional vector space over afield K. Let a be a fuzzy subspace of V, where Im(a) = {&i,..., 6 m } and b\ < . . . < 6 m . Let B — BiU . . . U Bra be a basis ofV, where B{U . . . U-Bm is a basis of a^ over K and Bi C &t)i\(Tbi+1,fc= 1, 2 , . . . , m, where cr^m+1 = 0. Define the fuzzy subset XofVas follows: x(x) = b{ if x E B{, i = 1,... ,m; x{x) = 0 otherwise. Then (1) x is a basis of a over K in the sense of Definition 8.1.6 and condition (10") of Chapter 8 holds; (2) (B, x) is a fuzzy basis of(V,cr) in the sense of Definition 8.5.4; (3) Y^!iL\ \Bi\bi is the fuzzy dimension of a over K in the sense of Definition 8.6.1. Proof. (1) That x 1S a basis of a over K in the sense of Definition 8.1.6 follows from the proof of Theorem 8.1.17. Let x G V, x^O. Then there exists a Bi such that either x £ (B{ U . . . Bm), x £ (I?i+i,..., Bm), i = 1 , . . . , m — 1, or x E (Bm). Thus a(x) = b{ = 6^ A . . . Ab m . Hence x £ {b £ B \ x(b) > x(x)}Thus x is a basis of a with condition (10") by Theorem 8.3.5. (2) The preceding argument together with Definition 8.5.1, Definition 8.5.4, and Corollary 8.5.13 yield the desired result. (3) Now £ 6 € B (7(6) = E 6 e S X(b) = TZx \Bi\bi = £ ^ i ( K : K\ -\abi+1 : K])bi. Hence the desired result follows from (2) and Definition 8.6.1 and The­ orem 8.6.5. I Definition 9.2.3 Let \x e L(A). Let Im(u) = { a i , . . . , a n } , where ai < . . . < an. Define the fuzzy order of u, written \u\, to be \u\ = 5^^ =1 (|/i aj I _ a |MOJ+I l) i» where \uan+1\ = 0- Define the fuzzy index of JA in A, written [A':p],tobe[A:iJL] = \A\/\n\. Suppose that F/K is normal. Let J be an intermediate field of F/K. Then [J:K} = [AK : AJ). Let a e C(F/K). We now consider when [cr: K] = [A K : ^4a] for some choice of c. Theorem 9.2.4 Suppose that F/K is normal. Let a G C(F/K) and Im(o~) = {fci,... , 6 m } , where b\ < ... < 6 m . TTien t/iere exists c such that [a : K] = [AK : Aa] if and only if the equation m

[F : K]/[a : K] = £(|(A*)c(fcm_i+l)| - Un^^Mb^^) has a solution for c(&i),..., c(6 m ).

(9.1)

364

Proof. By Definitions 9.2.1 and 9.2.3, we have that [F : a] [a : K} = [F : K] = \AK\ = [AK : A'WA'l Thus [a : K] = [AK : A"} if and only if [F : a] = \Aa\. Hence by Theorem 9.1.8, [a : K] = [AK : A"} if and only if [F:K) = [a:

K\ E7=l(l(^)c(6m_i+1)l - KA'U^lWbrn-i+l).

M

Consider a fixed a. Then Aa varies with the choice of c. However, the level sets of the Aa remain unchanged. Hence E ^ i ( l ( ^ ) c ( 6 m _ J + 1 ) l ~ l(^a)c(fem_,)l) remains constant for different c. Let r denote this constant. Corollary 9.2.5 Suppose thatF/K is normal. Let a e C(F/K) and Im(a) = {&i,..., 6 m }, where b\ < ... < 6 m . Then a solution to equation (9.1) exists if and only if [F : K]/[a : K] < r. Proof. Suppose that a solution to equation (9.1) exists. Then [F : K]/[o~ : K] < r since c(&i) < 1, c(bi) < 1 for z = 2, . . . ,ra. Conversely, suppose that[F : K]/[o~ : K] < r. The c(bi) can be chosen arbitrarily close to 1 and so the right-hand side of equation (9.1) can be made arbitrarily close to r. Also the c(b{) can be chosen arbitrarily close to 0 and so the right-hand side of equation (9.1) can be made arbitrarily close to 0. That is, the c(bi) can be chosen so that the right-hand side of equation (9.1) can equal any number in the interval (0,r). ■

9.3

Infinite Fuzzy Galois Theory

We give a generalization of the finite fuzzy Galois theory developed in the previous two sections to certain infinite dimensional algebraic field extensions F/K. The finite dimensional case in Sections 9.1 and 9.2 is a special case of the theory developed here. We assume that F is a splitting field over the field K of a set Q of separable polynomials, [67, p. 147]. As is well known, problems arise in the crisp infinite Galois theory, e.g., more than one subgroup may have the same subfield of invariants, [67, p. 148] and [153, p. 63]. This difficulty was overcome by Krull who restricted the correspondence to subgroups of the Galois group which are closed in the topology defined in [67, p. 149] or [66, p. 248] or [153, p. 60]. The full intermediate subfield - subgroup correspondence which holds in the finite dimensional case fails to hold if F/K is infinite dimensional, [67, Theorem 4, p. 150]. Not all the subgroups of the Galois group A of F/K appear in the correspondence. Let A denote the Galois group of F over K. The group operation on A is composition, o, of functions. We let e denote the identity of A. Let J be an intermediate field of F/K. Let AJ denote the closed subgroup of A in the Krull topology which leaves the elements of J invariant. A3 is called the Galois group of F/J. Clearly, AK = A. Let H be a subgroup of A. Let FH denote the subfield of invariants of H. Then FH D K and F<e> = F. We let L(F/K)

365

denote the set of all L-subfields a of F such that cr# D K. If a G L(F/K), then a(0) = 1 and |Im(^)| > 1 since the cases |Im(cr)| = 1 and |Im(/z)| = 1 are dealt with in Sections 9.1 and 9.2. Theorem 9.3.1 Let a G L{F/K).

Aa(f)

[ c(W{b G Im(a)\f $ Aab}) c(b*) = { c(A{b\b G Im(a)})

Then Aa G

Define the L-subset Aa of AK as follows: if 3b G Im(a) such that f <£ A°b, if 3 smallest b* G Im(a) and f G A°b Vb G Jra(cr) such that b > &*, if jB smallest 6* G Im(a) and f G A°b Vbelm(a).

L(AK).

Proof. Clearly, AG(f) = Aa{f~1) V / G AK. Let / , g G (A") r , where r G 7 Im^* ). Suppose that 3 bu b2 G Im(cr) such that / £ A a6 i and ^ ^ A ab 2. Let it! = v{6 G Im(cr) | / £ A*5} and w2 = V{6 G Im(cr) | # £ A a b } . Now either ui > tx2 or i^2 > ^ i , say i^i > 1x2- Suppose that f o g £ A°b for some 6 G Im(cr). Let ix = W{b G Im((j)| / o g £ AGh}. Suppose that u G Im(cr). Then clearly / o g £ jsf-u. a n c i s o u > Ul [s impossible else / , g G Aau and so / o g G A a ". Suppose that ix ^ Im(cr). Suppose that u > ui. Then 3b e Im(cr) such that u > b > u\. Hence / , g G A a6 and so / o g G A a 6 , a contradiction. Thus ixi > u. Hence A a ( / o p ) = c(u) > c(t*i) > r. Thus / o g G ( A a ) r . Hence ( A a ) r is a subgroup of AK. Suppose that 3 smallest 6* G Im(cr). Suppose that / o g e A°bV be Im(tr) such that 6 > 6*. Then A a ( / o 5 ) = c(b*) > r. Thus / o # G (A*7),.. Suppose that JB smaUest 6* G Im(l a )r. Hence ( A a ) r is a subgroup of AK. Now suppose that g eAab\/be Im(cr). Then / o # £ A ab if and only if / £ A°b. Thus Aa(f og) = a A (f) > r. Hence fog G (^4a)r. Thus ( A a ) r is a subgroup of AK'. That is, all level sets of A a are subgroups of A K and so A° is an L-subgroup of AK. Thus Aa G L(AK). m

366

We note that in Theorem 9.3.1, it is actually the case that Aa G L(AK). This will follow from Proposition 9.3.7. Theorem 9.3.1 contains the case, where |Im(cr)| < oo since 3 smallest b* G Im(cr). Theorem 9.3.2 Let fj, G L{AK).

Define the L-subset F ^ of F as follows:

[ c(\j{a G Im(fi)\x (£ F ^ } ) c(a*) F^(x) = I c(A{a\a G Im(/i})

Then F» G

if 3a G Im(/i) such that x (£ F»a, if 3 smallest a* G Im(/d) and x G F M a Va e Im(fi) such that a> a*, if fl smallest a* G Im(fi) and x G F ^ a Va G Im(fi).

L(F/K).

Proof. Clearly, F»(x) = F ^ ( - x ) = F ^ x " 1 ) V x G F, x ^ 0. Let x, y G ( F ^ ) r , where r G Im(F^). Suppose that 3 ai, a2 G Im()[x) such that x £ F^ a i and y ^ F^. Let ui = V{a G Im(^) | x ^ F ^ } and i;2 = V{a G Im(/x) | y ^ F ^ 0 } . Then either t;i > V2 or V2 > vi, say i>i > i>2- Suppose that x + 2/ ^ F ^ a for some a G lm(fi). Let v = V{a G Im(^) | x + y fi F^a}. Suppose that v G lm(fi). Then clearly x + y ^ F^ and so v > v\ is impossible else x, 2/ G F ^ v and so x + y G FAXv. Suppose that v £ Im(^). Suppose that v > v\. Then 3 a G Im()u) such that v > a > v\. Hence x, y £ F M a and so x -f y G F ^ a , a contradiction. Thus ^i > v. Hence F^(x + y) = c(v) > c(v\) > r. Thus x + ?/ G ( F ^ ) r . Similarly, xy G ( F ^ ) r . Hence ( F ^ ) r is a subfield of F Suppose 3 smallest a* G Im()u). Suppose that x + y G F ^ a V a G Im(^) such that a > a*. Then F^(x -f 2/) = c(a*) > r. Thus x + y G ( F ^ ) r . Similarly, xy G ( F ^ ) r . Suppose that ^ smallest a* G Im(/x) and x + y G F ^ 0 V a G Im(/x). Then clearly x + y G ( F ^ ) r . Similarly, X2/ G ( F ^ ) r . Hence ( F ^ ) r is a subfield of F Now suppose that 2/ G F^° V o e Im(/x). Then x -f 2/ ^ F ^ a if and only if x ^ F ^ . Thus F^(x + y) = F^(x) > r. Hence x + 2/ G ( F ^ ) r . Similarly, xy G ( F ^ ) r . Thus ( F ^ ) r is a subfield of F That is, all level subsets of F ^ are subfields of F and so F ^ is an L-subfield of F Since /x G C(AK), fia C A K and so F ^ D F A * = K V a G Im(^). Thus F ^ G L(F/K). ■ Theorem 9.3.2 contains the case, where |Im(/x)| < 00 since 3 smallest a* G Im(/i). Example 9.3.3 Let L = [0,1]. Let p be a prime and set K — Z p . Let C denote the algebraic closure of K. Then for any positive integer n, C contains a subfield Kn, where \Kn\ — pn. If m is also a positive integer, then Km C Kn if and only if m divides n. Let q be a prime and let F = Ug^ifgi. Then F is a proper subfield of C. Since K is perfect, every polynomial with coefficients from K is separable over K. Also Kqi is the splitting field of the polynomial xp

i

—

367

x over K. Let Q = {x^q - x \ i = 1, 2 , . . . } . Then F is the splitting field of Q over K. Define the L-subset a of F as follows: a{k) = 1 if k G K and a(k) = iTT {fk e Kqi\Kqi-i, i = 1,2,.... Then a G L(F/K) and fl a smallest 6* G Im(a). Clearly, 0 = A{b \ b G Im(a)}. By Theorem 9.3.1, Aa(e) = c(0). £j/ T/ieorera 9.3.2, 0 ^ Im(FAa) since FA*(x) = c(c(0)) <* x £ F ^ k o ) = F<e> = F, w/iere (Aa)c{0) = {e} since f G (A ff ) c(0) < ^ V 6 e ira(cr), / G Aa* (7^ Theorem 9.3.1J <^> V 6 G Jm( V x G F, / ( x ) = x 4=> / = eLet a G C{F/K). Condition G1 on a: Let b G L, where 6 < cr(0). If 3 / G A K such that b = W{b G Im(cr) | / ^ Aab}, then H 6> -A ab D A ^ . Proposition 9.3.4 Let a G C{F/K) and% G L, wherel) < j<76 if and only if b £ Im(a);

(2)nb>~A°»=AU^ab; (3) Ifnb>~Aab D A^, then c(b) G Im(Aa). (4) Suppose that condition G\ holds. If c(b) G Im(Aa), then either there does not exist a smallest 6* elm(cr) and b — b', where b' — l\\b \ b G Im(o~)} or PI, -A* b D A"Z. Proof. (1) Both statements are equivalent to the statement that V x G cr>, cr(x) > S. (2) g G nb>~Aab <^ V 6 > 6, V x G cr6, #(x) = x <£> V x G ^b>^b, g(x) = x O g e A t»b . (3) nh>^A0b D A^^3f

Gn6>^ab, / ^ ^ ^ 3 / e A

K

such that b =

b

V{6 G Im(cr) | / £ A° ] => c(b) G Im(i4"). (4) By (1) and (2), Aab* C P\b>b*Aab if 3 smallest 6* G Im(cr). By condition G\, the first implication in the proof of (3) becomes an equivalence. Now c(6) G lm{ACJ) implies either fl smallest 6* G Im(cr) and b = b' or 3 / G ^4K such that 6 = V{6 G Im(cr) | / ^ ACTb}. This yields the desired result by making the second implication in the proof of (3) an equivalence. ■ That is, under the condition G\ and the assumption that b ^b' when fl smallest 6* G Im(cr) and b' — A{b \ b G Im(cr)}, we have that (3) of Proposition 9.3.4 is an equivalence. Proposition 9.3.5 Let a G L(F/K). Suppose that b is not the largest element in Im(a). Ifbe Im(a), then c(b) G Im{Aa). Assume that condition G\ holds. Suppose also that if jB smallest 6* G ira(cr), then b ^ &', where b' = A{b \ b G Im(a)}. Ifc(b) G Im(Aa), thenb elm(a).

368 Proof. U

b

A b>^

b G Im(o-) <& o~ D ^b>^b

(by (1) of Proposition 9.3.4) & A b c

(since o^ and ab are Galois) <^ A^

C f)b>^Aab

(by (2) of Proposition

a

9.3.4) <& c(b) G lm(A ) (by (3) and (4) of Proposition 9.3.4). ■ Hence under condition G\ and the assumptions t h a t b^b' when jB smallest 6* G Im(cr) and b' = A{b \ b G Im(cr)} a n d b is not the largest element of Im(cr), we have t h a t b G Im(cr) if a n d only if c(b) G Im(^4 a ). P r o p o s i t i o n 9.3.6 Let a G L(F/K). Suppose that b is the largest element Jra(cr). Then c{b) G Im(Aa) if and only if a^ ^ K. cr-r- = K <^ A°^ = AK

Proof.

K

Galois) <3> V / G A ,

(since all intermediate fields of F/K

in

are

a

/ G ^ 4 ^ 44> c(6) ^ I m ( A ) , where the last equivalence

holds as follows: If c(b) G lm(Aa),

t h e n 3 / £ A ^ by t h e definition of A".

Suppose t h a t 3 / ^ A T T h e n f e AK

\ A^

and so either 6 = V{6 G Im(cr)

| / ^ A a 6 } or 3 6 > 6, 6 G Im(cr), / ^ A a b , a contradiction of the maximality of 6. Hence b = \/{b e Im(cr) | / £ A a b } and so c(b) G I m ( ^ ) . ■ P r o p o s i t i o n 9 . 3 . 7 Let a G L(F/K). (1) Suppose that b = W{b G Im(cr) \ f £ Aab} <<x(0). r / > e n ( ^ ) c ~ = n 6 > ^ .

for some f G AK,

w/iere 6

(%) Suppose that 3 smallest 6* G Zra(cr). T/ien (A a ) c ( 6 *) = n&>6*A a \ (3) Suppose that there does not exist smallest 6* G Im(o~). Let bf = A{b\b G Im{o~)}. Then (Aa)c{b/) = C\hAGb. Proof.

(1) g G (A*)^

^

A°(g)

> c(b) & g G A°b V b >b.

CT

(2) c(6*) (3) 0 G (A") c ( 6 ,) ^

&geA«>Vb>b*.

A"(<;) > c(fc') ^ p G ^ V 6 .

We see in Proposition 9.3.7 t h a t (Aa)c^

■

is an intersection of closed sets a

and so is closed, where r G {6,&*, b'}. Hence A

G

~L(AK).

Let /i G £ ( A K ) . C o n d i t i o n G 2 o n fi: Let a G L , where a < /x(e). If 3 x G F such t h a t fl = V { a 6 Im(/i) | x £ i ^ a } , then n ^i™** D i ^ T . P r o p o s i t i o n 9.3.8 £e£ /z G £ ( A K ) and a e L, where a < /z(e). T/ien following assertions hold. W K (2)C\ v y

=

u

a > a ^ ° # a n c ? o n ^ if a £ ~F** =FUa>^tMa;

a>a

(3) Ifna>2F^

Im(fi)]

'

D F^,

then c(a) e / m ( f ) .

tte

369

(4) Suppose that condition G2 holds. If c(a) G Im(F^), then either there does not exist a smallest a* G Im{ji) and a — a', where a' = A {a | a G Im(n)} or fl -F^ D F ^ . Proof.

(2) v

(1) Both statements are equivalent to the statement that V / € /*~, e n

a>2Ftla ^> V 0 > a, V / e A*a, f(y) =y^yfe

U

y <3> y E r

a

^ a

u o> ~ M a, /(*/) =

a

>

(3) H a > ^F^ a D ^ < ^ 3 x G n a >a F M a > ^ ^ ^ ^ € F such that a = V{a e°Im(/i) | x £ F^} => c(a) G fm(F^). (4) By (1) and (2), F**m C n a > a * F ^ a if 3 smallest a* G Im(/x). By condition G2, the first implication in the proof of (3) becomes an equivalence. Now c(a) G Im(F^) implies either jB smallest a* G Im(/x) and a = a' or 3 / G AK such that a = V{a G Im(/i) | / ^ F M t } . This yields the desired result by making the second implication in the proof of (3) an equivalence. ■ That is, under condition G2 and the assumption that a ^ a! when there does not exist a smallest a* G Im(/x) and a' = A{a \ a G Im(/z)}, we have that (3) of Proposition 9.3.8 is an equivalence. Proposition 9.3.9 Let /1 G L(AK). Suppose that a is not the largest element in Im(/i). IfaE Im(/i), then c(a) G Im(F^). Assume that condition G2 holds. Suppose also that if ]B smallest a* G 7ra(/x), then a ^ a', where a' = A{a | a G Im(ii)}. Ifc(a) G Im(FfM), thenb G Im(a). Proof. a G Im(^) <^> /x- D U a > ~^ a (by (1) Proposition 9.3.8) <^> F ^ C ir a>a^a (since /i£ and /i a are Galois) <=> F ^ C H ^ F ^ 0 (by (2) of Proposition 9.3.8) <^> c(a) G Im(F^) (by (3) and (4) of Proposition 9.3.8). ■ Hence under condition G2 and the assumptions that a ^ a' when jB smallest a* G Im(^) and a7 = A{a | a G Im(/i)} and a is not the largest element of Im(^), we have that a G Im(/z) if and only if c(a) G Im(F M ). Proposition 9.3.10 Let /1 G L(A K ). Suppose that a is the largest element in Im(fi). Then c(a) G Im{F^) if and only if fjr-^{e}. Proof. fir- — {e} <& F ^ = F (since all closed subgroups are Galois) <=> V x G F^ — F, x G F ^ <(4> c(a) ^ Im(F M ), where the last equivalence holds as follows: If c(a) G Im(F^), then 3 x £ F^ by the definition of F ^ . Suppose that 3 x £ F^. Then x G F\F^ and so either a = V{a G Im(/x) | x ^ F^} or 3 a > a, a G Im(|x), a: ^ F ^ a , a contradiction of the maximality of a. Hence a = V{a G Im(/x) | x £ F^a} and so c(a) G Im(F^). ■

370

Proposition 9.3.11 Let p G L(AK). (1) Suppose that a = V{a G Jra(/z) | x £ F^a} for some x G F, where a < /x(c). Then(F»)cQ=na>~F»-. (2) Suppose that 3 smallest a* G Im(fi). Then (i^) c ( a *) = n a > a * i ^ a . (3) Suppose that there does not exist a smallest a* G 7m(/x). £e£ a' = A{a | a G /m(/x)}. Then (F") c ( a /) = f \ F ^ . Proof. (1) 2/ G ( i ^ ) ^ <» F»(y) > c(a) & y G F»* V a > a. (2) y G (i^) c ( a * } ^ F"(j/) > c(a*) ^ y 6 F ' V a > a * . (3) 2/ G (F") c (a') ^ F*(y) > c(a') & y G F»* V a. ■

9.4

The Galois Correspondence

We restate Definition 9.1.2 and Proposition 9.1.3 since the assumptions con­ cerning F/K have changed. Definition 9.4.1 Let a G L(F/K). feA, f{a) C a.

Then a is said to be Galois over K ifM

Proposition 9.4.2 Let a G L(F/K). Then a is Galois over K if o~b/K is Galois V b G L such that b < o~(l). ■ Example 9.4.3 Let K = Q, the field of rational numbers, and let F = Q(\/2)Then F/K is normal. Let cr = 1^. Now AK = {e, / } , where e is the identity map on F and f is the automorphism of F which is invariant on K and maps y/2 to -V2. Now e G A°b V b G Im{a) = {0,1}. Hence Aa(e) = c(0) since 0 = 6* of Theorem 9.3.1. Now f G A°^ = AK, f (£ Aa° = AF = {e}. Thus Aa(f) = c(0) 62/ Theorem 9.3.1. Fence c(l) £ 7m(A a ). 77MS is in accord with Proposition 9.3.6. Thus 1 £ Im(FA> = K, where b is the largest element of Im{o~), then we define FA°\x) — b V x G K. We note that this is in accord with the crisp situation since K corresponds to IK which is two-valued, AK corresponds to \^K which is one-valued, and FA = K corresponds to IK which is two-valued. In the remainder of the paper, we agree that if o~~ = K, where b is the largest element of Im(cr), then we define FA° {x) = bVxeK. Similarly, if a F is the largest element in Im(/i) and //- = {e}, then we define A * (e) = a.

371

Lemma 9.4.4 (1) Let cr G L(F/K). Suppose that conditions d and G2 hold for a and A°', respectively. Then FAb*ab. Hence by Proposition 9.3.4(2), nb>b*Aab D A***. Hence 3 f G nb>b*A°b such that / £ A°b*. Thus nb>b*Aab^{e}. Hence by Proposition 9.3.7(2), (Aa)c{b^{e}. Thus F = F*e> D F ^ W ) and so 3 x G F, x $ F^U**). Thus 6* = c(c(b*)) e Im(FAa). (Hence the use of our "agreement" is not necessary at this stage.) Clearly, o~b* = F = (FA<J)b*. Suppose that b is neither the largest nor the smallest element in Im(cr) Suppose further that b ^ b' if jB smallest 6* G Im(tr), where b' = A{b \ b G Im(cr)}. Then by Propositions 9.3.5 and 9.3.9, it follows that b G Im(cr) <& c(b) G lm(Aa) ob = c(c(b)) G Im(F Ac(6) FiA"Ub) = no{b)>c&Fnb'>bA'b' ( ^ Proposition 9.3.7) = n c ( 6 ) > c £ , A b >b F (by Proposition 9.3.4) = n (b) ,<\ Ub'>bcrb> (since F is Galois over all the intermediate fields of F/K) = cr^, where the last equality holds from the following argument: x G H (b) ^\ Ub'>bcrb/ <=> V b < 6, x G Ub'>bab' <=> V b < b, 3 b' > b such that x G ob> <& V b < b, 3 b' > b such that a(x) > b' <^> cr(x) > b (since x is fixed) <=> x G o>>. Thus (FA )~ = cr^. Suppose fi smallest 6* G Im(cr). Let b' = A{b \ b G Im(cr)}. Then c(fc') G Im(A a ) since e G i a 6 V 6 G Im(cr). Now f e Aab \f b e Im( = F = Ubeim{a)crb = ab,. Also (FA = F. Thus ( F ^ ) 6 . = (jft/. Note b' = c{c{bf)) £ Im(F Aa*Ma- Hence by Proposition 9.3.8(2), n a > a * F ^ D F^m. Hence 3 x G n a > a * F ^ a such that x £ F^*. Thus n a > a * F ^ D X. Hence by Proposi­ tion 9.3.11(2), (i^) c ( a *) D K. Thus A K D ^ F " W ) and so 3 / G AK, f £ ^4(^)c(a*)i Thus a* = c(c(a*)) G Im(A F ^). (Hence the use of our "agreement"

372

is not necessary at this stage.) Clearly, /itt* = AK = (AF^)a*. Suppose that a is neither the largest nor the smallest element in Im(/i). Suppose further that a ^ ol if jB smallest a* G Im(/i), where a' = A{a\a G Im(/i)}. Then by Propositions 9.3.9 and 9.3.5, it follows that a G Im(/i) <3> c(a) G Im(F^) O a = c(c(a)) G Im(AFtl), where we note that I m ( i ^ ) has a smallest element. Suppose that a is the largest element in Im(/x). Then by our, "agreement", if necessary, a G Im(AF ). Suppose that a G Im(/x) is neither the largest nor the smallest element in Im( M ). Then (AF")~ = (AF") c{c{2)) = nb>c(^A^» (by Proposition 9.3.7) = nc{a)>c®fF"Ua) = n c W > e ( ^ n " > " f V ( b y Proposition 9.3.11) = n c ( a ) > c ( ~ } AF a'>a a' (by Proposition 9.3.8(2)) = ^cfa\>cQ\ Ua'>a/^a' (since all closed subgroups of AK are Galois) = /i^, where the last equality holds from the following argument: / G n c ( a ) > c ^ U a / > a /i a / OV a a /i a > ^ V a < a, 3 a' > a such that / G /i a ' ^ V a < a, ] a ' > a such that /i(f) > a' <^ M / ) > a (since / is fixed) ^ / e /i-. Thus 0 4 ^ ) ^ = A^;Suppose there does not exist smallest a* G Im(/i). Let a' = A{a | a G Im(/i)}. Then c(o!) G I m ( i ^ ) since K C F ^ V a E Im(/i). Now x G F a V a G Im(/x) <=> V a G Im(/z), V / G ii a , /(x) = x <^> x G if. Thus (F^) c ( a ,) = if by Proposition 9.3.11(3). Hence ^ ( ^ W ) = AK = U aGlm(Ax) /i a = /xa/. Also (A F ") a , = A K . Thus (A F / i ) a / = Ma- Note a' = c(c(a')) £ lm(AF") by Proposition 9.3.6. Hence Im(^) = lm(FA<J) and /xa = (FA<7)a V a e Im(/i)\{a}, where a is the largest element in ImQx). Since a is also the largest element in Im(AF^) and since [ia = (AF*)a V a < 2, ^ = (A**1)^. Thus n = FA". ■ _ Let L*(F/K) = {a G L ( F / i f ) | cr is finite-valued} and I * ( A K ) = {^ G I ( A K ) | fi is finite-valued} . Then a G L*(F/K) => Aa G I * ( A K ) and /x G L*(A K ) => i ^ e L(F/K). Also, a G L*(F/K) => cr satisfies condition d and /x G Z*(A K ) =* // satisfies condition G 2 . If £ is finite, then L*(F/K) = L(F/K) and L (^4K) = Z/(A K ). These statements are applicable to the following result. T h e o r e m 9.4.5 (1) Define T from L*(F/K) into L*(F/K) as follows: VaeL*(F/K), Vti,eV(AK),

intoT(AK)

and®

fromV\AK)

T((j)=A°, *(/x) = F ^ .

Then T is a one-to-one inclusion reversing correspondence of L*(F/K) onto L (AK) and $ is a one-to-one inclusion reversing correspondence of L (AK) onto L*(F/K). Moreover, <3> o T is the identity map of L(F/K) and T o <1> is the identity map of L (AK).

373

(2) Let \x G L*(AK). Then \i < AK if and only if F^ is Galois over K. If (i < AK, then the Galois group of F11* /K is isomorphic to AK/'/j,. Proof. (1) By Theorems 9.3.1 and 9.3.2 and comments following Propo­ sition 9.3.7, r and $ are well-defined and map L*(F/K) into Z*(AK) and T(AK) into L*{F/K), respectively. By Lemma 9.4.4, FA<J = a and AF* = /z. Thus $ o T is the identity map of L*(F/K) and r o $ is the identity map of L*(AK). Hence T maps L*(F/K) one-to-one onto ~L*(AK) and $ maps ~L(AK) one-to-one onto L*(F/K). Now a C a' => ab C a'h V b G [0, cr(0)] => ACTfo D A a ' b V 6 G [0, A a D A a ' (by Theorem 9.3.1). Similarly, fi C p' => F" 2 -FM • Hence T and $ are inclusion reversing. (2) By Propositions 9.1.5, 9.4.2, and the crisp case, \i < AK <& \ia < AK V a G Im(yu) <£> F ^ / i f is Galois V a G Im(/x) ^> (F^)b/K is Galois V 6 £ Im(F^) 4=> F ^ is Galois over J^. Suppose that fi < AK. Then ^ # < ^4K and so by the crisp case, the Galois group of F^* over K is isomorphic to AK//J,#. Now AK/fi# is isomorphic to AK / [i by Theorem 1.3.11. ■ Let n E C (A) and cr G C*{F/K). The relationship between the index of fJ>(Aa) in A K and the dimension of F^(a) over if is examined in Section 9.2

9.5

Group L-Subalgebras

We introduce the concept of a group L-subalgebra of a group algebra. We bring focus to part of fuzzy algebra by showing how group L-subalgebras can be used to make a connection between L-subgroups and L-field extensions. This con­ nection allows ideas and results to be transferred between L-subgroups and L-subfields. This idea was originated for the crisp case in [238] and extended in [40] and [157]. The concept of a modular L-field extension plays a funda­ mental role. L-field extensions whose supports are simple purely inseparable field extensions are the building blocks for modular purely inseparable field extensions, [182, 231]. In particular, it is shown in [182] and [231] that if F/K is purely inseparable of bounded exponent, then F/K is modular if and only if F/K is a tensor product of simple extensions. Throughout this section F denotes a field of characteristic p > 0. We let L(F) denote the set of all L-subfields of F. We let L(A) = {K G L(F)\ K C A } . We let L(X/K)

= {0 G F(F)

\ K C C A}. We let S(f)

= {xa\x

G f,

f (a?)

a}. Let K be a field and G a (multiplicative) commutative group. Let K[G] de­ note the group algebra of G over K, [209]. Then G is linearly independent over K. We assume throughout that when we write Yl7=i ki9i, the gi are distinct and ki =£ 0, where gi G G and k{ G K, i = 1,..., n.

=

374

Definition 9.5.1 Let \x be an L-subset of K[G). Then \i is called a group L-subalgebra of K[G] over /X|K if V\K is an L-subfield of K, H\G is an Lsubgroup of G, and V £ 2 n = 1 kigi G K[G], 0 ^ h G K, g{ G G, i = 1 , . . . ,n, n E N , M(Er=i **&) = AMfe) A /!( ju(fci) A/x(k2); 1 (2) Vgi,92 e G, M2122" ) > M01) A M02);

(3) Vfc eK,\/ge

G, M ( M = /i(fe) A Ms);

M Vx,7/ G K [ G ] , M ( * - y) > fi(x) A ^(2/); ^ Vx,y G if[G],/x(x3/) > /x(a:) A/xfo). Proof. (1) The result here holds since /J,\K is an L-subfield of K. (2) The result here holds since /X|G is an L-subgroup of G. (3) Immediate from Definition 9.5.1.

(4) Let x = Z7=i ki9iand

y = £?=i kWje

K

\°\-Then

*-y = H=i

M

for some k£ e K,g£ € G, u = 1,... ,q, i.e., 5^u=i ^ ^ i s a simplification of £IU kg* -E7=i *;4- T h u s K* -y) = A(MM') A M S H I i = 1,...,g} > A{A{/i(fci)A/x(0i) I i = 1 , . . . , n } , A{/x(fc<)A/x(p»') | i = 1 , . . . ,m}} = fi(x)An(y), where the inequality holds since each fc^ is either of the form fc;, or —A:'-, or fc* - k'j and {^1,...,g n , g[,...,p^} D {g'{,...,^'} and /x(fc) = fi(-k) Vfc G if. (5) Let x = ££=1 ki9i and 2/ = E ^ %$ e K[G\. Then n

m 771

xy -2t = = 1i Ji == li

= £« q

1u=l 4=1

for some fc^ G -K",^' G G, iz = l , . . . , g , i.e., Ylu=iku9u ls a simplification of xy. Thus /x(xi/) - A{KK) A M O I ^ = 1, • • • ,9> > A{A{/z(^) A ^ ( # ) I i = 1 , . . . , n } , A{/x(fc<) A /x(^)} I * = !»• • •» ™}} = M*) A /x(y), where the inequality holds since JJL has the properties that /x(fc + fc') > /x(fc) A/x(fc'), /i(fcfc') > /i(fc)A/u(fc/), a n d / x ( ^ ^ ) > fi(gi) A ^(g'A and eachfc£is of the form X] £ fcifci.

■

Corollary 9.5.3 Suppose that /i is an groizp L-subalgebra of K[G] over /x|^. Then u is an L-subalgebra (L-subspace and L-subring) of K[G] over H\K- H Corollary 9.5.4 Suppose that \x is a group L-subalgebra of K[G] over 1K. If \i has the sup property, then fi has a basis over IK when a is considered an L-subspace of the vector space K[G] over K.

375

Proof. The result here follows by the definition of a group L-subalgebra and Theorem 8.4.4. ■ Let E be a subfield of K. By a group subalgebra of K[G\ over £", we mean E[G% where G' is a subgroup of G. We note that if fi is a group L-subalgebra of K[G\, then (//| K)*[(/X|G)*] is a group subalgebra of K[G]. This follows since (II\K)*

is

a

subfield of K, (/X|G)* is a subgroup of G, and (//|G)* is linearly

independent over K and thus over (/JL\K)*Proposition 9.5.5 Suppose that /i is a group L-subalgebra of K[G] over H\KThen f = (HJC)1(HG)1Proof. E I L i **# G M* ^ M(E?=i fePi) > 0 ^ A{/i(^) A fi(9i) | i = 1 , 2 , . . . , n) > 0 <=> /x(fc») > 0 and fifa) > 0 , 2 = l J 2 , . . . , n ^ f c i e 0*1 #)* and ^ € (/*|G)*, i = 1,2,... ,n => £ ? = 1 **# G (MIK)*[(M|G)*]- NOW every element of (AI|K)*[(HG)*] h a s the form £ ? = 1 fc^, where k{ G (IAK)* and g{ G (/X|G)*, 2 = l , 2 , . . . , n . Thus the desired result follows easily. ■ Proposition 9.5.6 Suppose that fi is an L-subset of K[G]. Then \i is a group L-subalgebra of K[G] over \X\K if and only if \ia is a group subalgebra of K[G] over (^IA-)O V a elm(fi). Proof. Suppose that /i is a group L-subalgebra of K[G] over /i|#. Let a Elm(/x) and let G^ = (/i|G)a- Then G( a ) is a subgroup of G since /A\G is an L-subgroup of G. Clearly, \ia D (^|*r)a[G (a) ]. Let Y!i=iki9i € Ma- Then a < K Yli=iki9i) = A{/x|/c(fc») A / Z | G ( # ; ) I i = l , . . . , n } . Thus fi\K{h) > a and /Z|GG&) > a, z = 1 , . . . , n. Hence A* G (/i|«-)a and #; G ( H G ) ^ Thus E7=iki9i e MKMGW]. Thus f,a C (/x|*)a[G (a) ]. Hence M a = ( ^ ) a [ G ( a ) ] and so /xa is a group subalgebra of if [G]. Conversely, suppose that /j,a is a group subalgebra of K[G) over (fi\K)a V a Elm(/z). Let XlILi k*#» ^ fc[G]. Let a — / x (Sr=i ki9i)- Now ^ a = (/iji<:)a[Gr/] for some subgroup G' of G. Thus £ I U **# € (IAKUG'] and so £ ? = 1fctf*= YZi ki9i, where ^ G G' and fej £ (IAK)CL- Since G is hnearly independent over K, we have that n = m, ki = ty, and a and ^(^i) > a, z = 1 , . . . , n. Since /xa is a group subalgebra of K[G] over (//| #-)<*, fia is a subalgebra of K[G] over (MIKOO (V a Glm(/x)). Thus ^ is an L-subalgebra of ^[(3] over / i ^ . Hence a = M E I U fe*^) ^ A M ^ ) A M&) | i = 1,2,..., n} > a. Thus M X X i **#) = A{^(fcf) A fJ>(gi) | 2 = 1,2,..., n}. Hence /x is a group L-subalgebra of K[G\. M The following definition corresponds to Theorem 5.1.8. Definition 9.5.7 Let K be an L-subfield of k and 7 be an L-subgroup of G. Define the L-subset K[J] of K[G] as follows: Vx G K[G\, n

K[J](X) = V{(^(fei) Ui (^i) Vi )(a:)|«(fei) = u^gi) 2=1

= vui = 1 , . . . , n ; n G N}.

376

Proposition 9.5.8 Let K be an L-subfield of k and 7 be an L-subgroup of G. Then ^[7] is a group L-subalgebra of K[G] over K. Proof. Usual arguments show that K[J] is an L-subalgebra of K[G] over K. Let x G K[G\. Then x has a unique expression of the form Y^i=\ **£*• T n u s K[J](X) = A{n(ki) Aj(gi) | z = l , . . . , n } . ■ Proposition 9.5.9 Let K be an L-subfield of K and 7 be an L-subgroup of G. Then Va G L, a < 7(e), K,[y]a = K,a[ya\Proof. Now £ ? = 1 kigt G K[i\a <£> K[I}(Z?=I ki9i) > a & A{n(ki) A 7 ( # ) | i = 1,... ,n} > a <$> ki € Ka and gi G 7 tt , i = 1,...,n => ]£? = 1 **# G «a[7o]Now every element of Ka{ya] has the form Y%=i **0*> where A* G Ka and gi G7 a , i = 1 , . . . , n. Thus the desired result follows easily. ■

9.6

Construction of L-Field Extensions

Let D denote a field of prime characteristic p. Theorem 9.6.1 Let P and G be commutative groups and v be a homomorphism of P onto G. Let 7r be an L-subgroup of P and p be an L-subgroup of P such that p C 7r and p* = R. Define the L-subset 7r/p of P/R by (jr/p)(p * R) = V{7r(p * r) I r G R}, where R = Ker v. Then the following assertions hold. (1) n/p is an L-subgroup of G. (2) If IT is v-invariant, then 7r = v~l(ir/p). Proof. (1) The reader may wish to refer to Theorems 1.3.12 and 1.4.11, comments preceding Theorem 1.4.10, and Proposition 9.1.6 . (2) Let x eP. Then V-1(TT/P)(X) = ir/p(y(x)) = V{TT(P) | v(p) = v(x),p G P} = TT(X). M

Let P and G be commutative groups and 7 an L-subgroup of G. Let 1 -> R -^ P —» G -» 1 be a short exact sequence, where P is torsion-free, R is a subgroup of P, and i is the identity map. Then the group algebras D[P] and D[R] are integral domains, [152]. We let D{P) and D(R) denote their field of quotients, respectively. Let p and 7r be L-subgroups of P such that p C 7r, p* = i?, and 7r* = P. Suppose that p = v~1(7|{e}), where e is the identity of G and 7 = IT/p. Then we write 1—>p—>7r—>7—>1 and call 1 —► p —> ir —> 7 —> 1 a short exact L-sequence induced b y l — > . R A P A G - * 1 . Given G and 7, there always exists a short exact sequence 1 - ^ i j A p A G ^ l and an induced short exact L-sequence l - > / 9 - ^ 7 r - > 7 ^ 1 . This can be seen by taking P to be free and letting IT = v~x(j). Then TT is v-invariant. Also v~1(j)(z) =

377

y(v(z)) = 7(2 *R)Vz e P and (v'1 (l)Iv~l{l\e)){z * R) = V{v" 1 (7)(z * r) | r € R} = V{j(v(z*r)) \ r G R} = V{7(z*P) | r 6 P } = 7(2 *ii), where z * P is a left coset in P / i ? = G. Therefore, for every L-subgroup 7 of G, we can think of X/K as being constructed from some short exact L-sequence 1—> p —► 7r —► 7 —> 1, e.g., 7r = v _ 1 (7) and p = v~x{^\e). Proposition 9.6.2 Let 7 be an L-subgroup of G. Let N i J ^ P A C - ^ 1 be a short exact sequence and 1—> p —► 7r —> 7 —» 1 an induced short exact Lsequence. Let K[G] be the group algebra determined byl-+R-^*P^+G—>l. Let K be an L-subfield of K. Then V £ ? = 1 **& € #[G], n n

^ k / ^ K ^>^ i ) == AA{A{«(fci), V{7r(pi **r)|r r)|r G GP}} P}} || #& == Pip* **P,i R,i == 11,.. , . . ..,n} ,n}. ^/p}(J2 (A{«(fei), V{7r(pi ki9i>

Proof. Let x = Yl7=i ^i9i- Then K[TT/P](X) = A{K(A;;) A 7r/p(gi) \ i = l , . . . , n } = A{A{/-c(/ci), 7r/p(p;*P)} | # = p* *iJ, 2 = 1 , . . . ,n} = A{A{K(/C;), V{7r(pi * r) | r G P}} | & = Pi * P , i = 1 , . . . , n } . ■ We recall that from the exact sequence l - * f l A p A G - > l , w e construct the field extension F/K, where F = Z?(P) and # = £>(P). Let 5 be an Lsubfield of D. Let 1—> p —> 7r —► 7 —» 1 be a short exact L-sequence induced b y l — ► p A p A G — > 1 . Define the L-subset <5[p] of D[P] and the L-subset % ] of D[P] as in Definition 9.5.7. Then by Proposition 9.5.8, 6[p] and % ] are group L-subalgebras of -D[P] and D[P], respectively. Define the L-subsets 6(p) and 6(TT) of D(P) as follows: «(p)(s) 6(p)(x) === ^{(^=i^i)uM)vMZT=M)u'V V { ( E ? = 1 ( * ; k ( ^ ^ j)v^1(x)\xi,xj n,r'j e P; R; 6(xi) 8{x',j) = — ixj, u^pfc) = v*, Vi, ri, r^. G fi(xi) = uu*, p(n) = it 8{x r p(r^) = v'pi ^-; z = 1 , . . . ,,n;j n; j = 11,... , . . . ,,m;n,m m; n, m GeN} N} P( j) = = 1,...

ED;

if x G D(P) and S(p)(x) = 0 if x <£ D(R);

6{*)(x) 6{*){x) --== ^{(ZUi^uM^MZU^ip'MrH^l^xj V{(£?=1(^b;W

Pi,p'j €G P; P;6(xi) Ui,S(xj) = u'^{pi) iipirfa) = vu viy p.,pj fi(x.) = UiiSix'j) TT(PJ) = Vj; v'j; i = 11,, •. .•. •, , n; ; n,n,mm eGN} ir(Pj) n\jj == 11,,......,,mm; N}

€D;

if xGL>(P). Then <5(p) and <5(TT) are L-subfields of D(P) and 6(p) C 5(TT). We assume throughout that 6* = D. Recall that 7 = 7r/p, 7r* = P, p* = P, and 7* = G. Then («(TT))* = <5*(TT*) = L>(P) = P and (<5(p))* = <5*(P*) = 2)(jR) = X. Let A = S(n) and K = 6(p).

378

Theorem 9.6.3 Suppose that F/K is constructed from 1—> R ^ P ^> G -+ 1 and that X/K is constructed from the induced short exact L-sequence 1 —> p —> 7r —> 7 —► 1. TTien A/AC zs modular. Proof. Prom [238] or [40, Theorem 1.2, p.539], F/K is modular and so X*/K* is modular. Thus by Proposition 5.5.9, X/K is modular. ■ Suppose that F/K is constructed from G. For a subgroup G' of G, we say that an intermediate field F' corresponds to G' when F' is the fraction field of the group algebra K[v~1{Gf)]. Suppose that X/K is constructed from 7. For an L-subgroup V of G such that 7' C 7, we say that an //-intermediate field A' of X/K corresponds to 7' when A' = K(V_1 ("yf)). We also say that a subgroup S of P is compatible in P with respect to ir if and only if V{7r(p) | p G P\S} < A{ir(p) I p e S}. Lemma 9.6.4 Suppose that F/K is constructed from G and that X/K is constructed from 7. (1) Let 77 be an L-subgroup of G such that 7' C 7. Let X' be an Lintermediate field of X/K which corresponds to corresponds to 7'. Then A7* corresponds to 7'*. (2) Let G' be a subgroup of G. Let F' be an intermediate field of F/K which corresponds to G'. If v~l{G') is compatible in P with respect to 7r, then X' corresponds to 7', where X' = X\pf and 7' = 7 | c Proof. (1) We have that A' = Acf^" 1 ^ 7 )). Hence by Theorem 5.1.12, A'* = ^ ( ( v - ^ y ) ) * ) = Kdv-1^'))*). Now x e (v-HV))* <* v-\y)(x) > 0 <* i(v(x)) > 0 <4> v{x) e y * < ^ x € ^ ( V * ) - Thus ( V ^ y ) ) * = v - ^ Y * ) . Hence A7* = ^ ( v ^ y * ) ) , the fraction field of Klv'1^*)]. Thus A7* corresponds to

y*.

(2) We have that A7* = F' and y * = G7 since A* = F and 7* = G, respectively. Let x e F'. Then A(rr) A(s) ===

1 V{(ET v { ( n ==1 i ((a: ^ k<()«.,(Pik)(E^i(^)« P i k ) ( E ^ i ( ^ ) « j ( p5(pi-)t.j)i - ) t . j ) - 1 Wki,^ W k i , ^ GGl >A;

Pi,p'j P;«(a;i) = = Ui^ix'j) Ui^ix'j) = u'jtir(pi) ir(pi) = Vi,ir(j/j) Vi,v(j/j) = v'y, Pi.pJ- €G P\6{xi) v'y, ii = = l1,,.. .. .. ,,n; j = 1 , . . . ,m;n,m G N} n;j = l,...,m;nlmeN} i

= v{(Er=1(^)U4(Pi)„i)(Er=iK)^^)^)- (^)ki,^ G z?;

Pi,Pj Ge vt r-^^G K,PjG ' ) ; «5(xi) = tti, uu $(«£) Sty) = = u'j, u'j^ipi) Tt{pi) = = vvuu7r(pJ) 7r(pJ) -- t$; t$; i = 11,..., , . . . , n; , . . . , m; n; j = 11,..., m; n, m eG N} (by compatibility)

== V {(Er= ( ^ ) « i ( 4K(Pik)(E^i(^)«j(p;-)»'.)) v i ) ( E r = i K ) « ; ( P i ) - ' ) - 1 1(Wki,*; * , ^ eG 2?; V{(E2 D; =1i(a;<)« ft.pj G t r ^ G ' ) ; ^ ) = Ui^ix'j) = u'j, ( ^ ( T I G ' ) ) ^ ) == vvt,t, Pup'j G t r ^ G ' ) ; ^ ) = Ui^ix'j) = u'j, (y-^a^ipi) V(« 1 (7lG'))(Pi)='"j;« = l , - - --,,nn;;ii==::ll,,......, ,mm;;nn, ,TTnnGe N } , where the latter equality holds since n = t; _ 1 (7). ■

379 Let T be t h e torsion subgroup of G. Let q b e a prime and let Tq denote t h e ^-primary component of T. For each Tq, define t h e L-subset rq of G by V x e G, T x g( ) = 70*0 if ^ € T g and r g ( x ) = 0 otherwise. T h e n rq is an L-subgroup of G, r q * = Tq, and r g is t h e g-primary component of 7 by Proposition 2.3.3. There exists a unique maximal L-subgroup r of G such t h a t r C 7 and r is torsion, Theorem 2.3.8. Also, there exists a torsion-free basis x of 7, Definition 8.2.12, Theorem 8.2.18, and (x*) = (x)* , where (x*) is the subgroup of G generated by x* and (x) is t h e L-subgroup of G generated by x- For an L-field extension X/K, we let a denote t h e maximal separable algebraic closure of K in A, Theorem 5.2.6, a n d ip t h e maximal purely inseparable closure of K in A, Theorem 5.2.6. X/K is called r e g u l a r if and only if X/K is separable and a = K. T h e o r e m 9.6.5 Suppose that F/K is constructed from 1—> R -^> P -^> G —» 1 and that X/K is constructed from the induced short exact L-sequence 1 —► p —> 7r —» 7 —■> 1. T/ien £/ie following assertions hold. (1) Suppose that v~l(®qz£pTq) and v~1(Tp) are compatible in P with re­ spect to 71*. If K is neutrally closed in A, then cr corresponds to 0q^prq and tfj corresponds to rp. (2) Suppose that G is finitely generated and G = (X) a n d 7 i = y\{gi), i = l , . . . , n . Now K^dX))) cor­ responds t o (X) and K(v~1((gi))) corresponds t o (<&), i = 1 , . . . , n . T h u s 77 = A | / r ( v - i « x » ) corresponds t o x and A* = X\K^v-i^gi))) corresponds to 7;, i = 1 , . . . , n by Lemma 9.6.4. Now 77* = K^dX))) and A* = i ^ " 1 ((
380

= iv ®« Ai ®K • • • ®« Xn)* by Definition 5.3.5 and Theorem 5.3.3. Thus the desired result holds with v — rj ®K Ai &>K • • • ®K

Xn.

(3) Since 7 is torsion, 7* is torsion. Thus by [157, Corollary 1.4, p. 15], F/K is algebraic, i.e., X*/K* is algebraic. Hence X/K is algebraic, Theorem 5.1.2. (4) Since 7 is free, 7* is free. Thus by [157, Corollary 1.4, p.15], F/K is pure transcendental, i.e., A*/K* is pure transcendental. Hence X/K is pure transcendental Theorem 5.1.11 and Theorem 5.1.22. (5) Since 7 is torsion-free, 7* is torsion-free. Thus by [157], F/K is regular, i.e., A*/K* is regular. Hence K is algebraically closed in A. Thus X/K is regular by Theorem 5.3.18. (6) Since 7 has no elements of order p, 7* has no elements of order p. Thus by [157], F/K is separable, i.e., X*/K* is separable. Hence X/K is separable by Theorem 5.3.18. ■ Corollary 9.6.6 Suppose that F/K is constructed from l - > i j A p A ( ? - > 1 and that X/K is constructed from the induced short exact L-sequence 1 —> p —► 7r —► 7 —> 1. Then the following assertions hold. (1) If 7 is torsion and has no elements of order p1 then X/K is separable algebraic. (2) If 7 is torsion and p-primary, then X/K is purely inseparable. Proof. The desired results here follow from Theorem 9.6.5 and Theorem 5.2.3. ■ Lemma 9.6.7 it = A on P. Proof. Let z G P. Consider z = xy~x, where x,y G F, y ^ 0. Now x z x = YJi=i i i a n d V = lLjLix'jzji where Xi G D,zi G P,x£ G £>, Zj G P, i = 1 , . . . , n; j = 1 , . . . , m. Hence Yl7=i xizi — E j = i ^ j 2 ^ - T n u s n = m, xi = x\ and 2* = z
WJ^D^^W)))- 1 W I z=(EIU

*«*)(£« ^ r 1 } = V ^ ^ A T ^ A

TT(^) I i = 1 , . . . , n} I * = ( £ ? = ! z ^ ) ( E ? = i ^ i ^ ) " 1 } - Now TT(Z) = ^ ^ ( z j ) " 1 ) > 7r(z^) A 7T(Z<) and TT(^) > 7r(zz<) A TT(^), z = 1 , . . . ,n. Thus TT(Z) A TT(^) = 7 r ( ^ ) A TT(^), i = 1 , . . . ,n. Hence <5(?r)(z) < TT(Z). NOW <5(TT)(Z) > % ] ( z ) = 6[ir)(lz) = 6(1) ATT(Z) = TT(Z). Thus <5(TT)(Z) < ic(z). Hence A = 5(TT) = TT on

P. ■ Theorem 9.6.8 Suppose that F/K is constructed from G and X/K is con­ structed from 7. Then there exists an F-isomorphism u of K[P] ® F onto the group algebra F[G] such that V g G G, 3 z G P swc/i £/ia£ IA(Z ® z _ 1 ) = p and U(K[7T] ® A) > A[7], namely z G P swc/i £/m£ v(z) = #.

381

Proof. The existence of an F-isomorphism u of K[P] ® F onto F[G] such that V g eG,3z e P such that u(z®z~l) = # follows by [157, Theorem 1.2, p. 14]. Hence it remains to be shown that U(K[TT] ® A) > A[7]. Let K; G P[G]. Then w = YA=I ci9i, where Q G F and ^ G G, z = 1 , . . . , n. Let Zi G P be such that

u{zi®zr1) = gui = l r .. ,n. Then u Q X i a*®^ 1 *) w

z X

c

c

w

u^^z^zr1)^®

°i)) = E?=i (*i ® ; M1 ® 0 = Y?i=i9i i = - Also 7 (#) = (ir/p)(gi) = V{7r(^) I v(z^) = #;} = 7r(zi) since 7r is ^-invariant, i = l , . . . , n . Thus ® A)(w) = V{(/C[TT] 0 X)(z) I 1/(2) = w} > V{V{A{K[7r](xi) A A(^) | i

W(K[TT]

= 1 , . . . ,71} | 2 = E H i *»3/t, m G N} I u(z) = w} > A{K[TT](^) A Kz^a)

\

i = 1 , . . . ,n} = A{7r(z;) A A(z») A A(c») | i = 1 , . . . ,n} (by Lemma 9.6.7) = A{ir(zi) AA(ci) I z = l , . . . , n } = A{7(#) A A(Q) | i = l , . . . , n } = X[y](w). ■ Lemma 9.6.9 Suppose that tr and -K' are v-invariant. Then tr fl IT' and -K-K' are v-invariant. Furthermore, -K fl IT' = ^ _ 1 ( 7 n y ) and TTTT' = v~l{/y^t). Proof. It follows routinely that n D ir' is v-invariant. Let z,p G P be such that i;(z) = v(p). Then zk = p for some & G Ker i>. Suppose that z = xy, where x,y G P. Then 3r, s G P such that p = rs and v(x) = v(r) and i;(?/) = v(s), namely r — x and s = yk. Similarly, if p = rs, where r,s E P, then 3 x, y G P such that z = xy and v(r) = v(x) and 17(5) = V(T/) Thus Tnr'(z) = \/{7r(x)A7r'(y) I 2: = xy, x,y G P } = V{7r(r) A 7r'(s) | p = rs, r, 5 G P } = 7T7r'(p). Hence -K-K1 is v-invariant. It follows easily that -K fl 7r' = v - 1 ( 7 fl 7'). Now V~1(JJ,)(Z) = 77'OK^)) = V{7(x) A 7'(17) I v(z) = xy, x,y e G} = V{j(v(x)) A Y(v(2/)) | ^ = ^2/? x,yeP} = V{A{V{7r(x!) | ir(Xl) = TT(X)}, VJTT'O/I) | Tr'foi) = Tr'(y)}} \ z = xy, x,y G P } = V{7r(x) A 7r'(y) | z = xy, x,y G P } (by hypothesis) = -K-K'(Z). Hence 7T7T' = v~ 1 (77 / ). ■ Lemma 9.6.10 Suppose that F/K is constructed from G. Let 7 and 7' be L-subgroups of G. Then v _ 1 (7*) H ^"HY*) = fa""1 (7 nY))*Proof. a: G v _ 1 (7*) n V _ 1 (Y*) <^ x e v _ 1 (7*) «&<* x € V~1(/Y*) <& v(x) € 7* and v(x) G 7'* <^> j(v(x)) > 0 and Y M * ) ) > 0 <£> v" 1 (7)0*0 > ° a n d tr^YX*) > 0 ^ x G (V-^T))* n (^(V))* = (v-1(7) nv-HY))*- H e n c e

v 1 (7*)nv- 1 (Y*) = (^"1(7)n^"1(Y))*- Now Vx G P, (tr-^ntrHTOX*) = t r 1 (7)0*0 AV-HYX*) = 7(*>(a0) A Y W X ) ) = ( T H T O K ^ ) ) = v _ 1 (7nY)W-

Thus v - 1 ( 7 ) n i ; - : l ( Y ) = ^ ( T H V ) . ■ Suppose that P/iiT is constructed from G. Let Lf(F/K) denote the set of all L-subfields of F which are constructed from some L-subgroup of G. (Recall that TT is v-invariant.) Let A G Lf(F/K) and let L / ( A / K ) = {A /(n) | 7 A G Cf(F/K), K, C A' C A}, where the neutral closure is taken in A. Define • on L/(A//c) by 5(7r / ) (n) •5(7r , / ) ( n ) = 5(7rV / ) ( n ) . By Proposition 5.6.8, L / ( A / K )

382

is a lattice under • and n. We note that if X/K is algebraic, then • becomes the usual composition of fuzzy subfields. Let Lf(j) denote the lattice of all fuzzy subgroups 7' of G such that 7' C 7. Theorem 9.6.11 Suppose thatF/K is constructed from G. Let A, A' G Lf(F/K), where A is constructed from 7 and A' is constructed from 7'. Then the following assertions hold. (1) A and A' are linearly disjoint and (A D X')/8(v~l(i fl 7')) is neutral. (2) IfADA'', then X/X' is modular. (3) Lf(X/K) and Lf (7) are lattice isomorphic. Proof. Since A, A' G £f(F/K), 3 7, 7' G L/(G?) which A, A' correspond to respectively. (1) By Lemma 9.6.4, A* and A'* correspond to 7* and 7'*, respectively. By [157, Theorem 1.3, p. 15], A* and A7* are linearly disjoint over K0 = -D(ilo), where BQ = v " 1 ^ ) n V 1 ^ * ) - By Lemma 9.6.10, v" 1 (7*) fl v " 1 ^ * ) = (f~ 1 (7ri7 / ))*. Thus A* and A7* are linearly disjoint over 6*((f-1 (7 fl 7'))*) = (^(v - 1 (7ri7 / )))*, where the latter equality holds by Theorem 5.1.12. Thus A and A' are linearly disjoint over ADA' and (A fl X')/8{v~l(^ fl 7')) is neutral by Theorem 5.3.3. (2) By Lemma 9.6.4, A* corresponds to 7* and A'* corresponds to 7'*. Thus A*/A'* is modular by [40, Theorem 2.4, p. 542]. Hence A/A' is modular by Proposition 5.5.9. (3) Now Lf(X/K) ~ Uri) by V(5(7r/)(n) G L/(A/«), (5(7r')(n) -► 7', where v(Tr') = 7' : We have <5(7r')(n) -> V and ff(7r/7)(n) -> 7" =» <5(7r'7r")(n) = <5(7r')(n) • «(7r ,, ) (n) -► 7 V and <5(TT' H 7r")(n) = <S(7r')(n) n <5(7r")(n) - * y n 7" since by Lemma 9.6.9 V{-K'-K") = y'j" and V(TT' n TT") = 7' n 7". ■ Corollary 9.6.12 Suppose that F/K is constructed from G, where G is pprimary and that X is constructed from 7. If 7 is the direct product of Lsubgroups 7^ of G, i G / , whose supports are cyclic of order a power of p or are isomorphic to Z(p°°), then there exists an L-subfield A' of F such that K C A' C A, A/A' is neutral, and A' is a tensor product over K of' L-subfields A; of F such that K C A^ C A, A^*/K* is purely inseparable, and the intermediate fields o/Ai*//c* appear in a chain, i G JT. Proof. Since G is p-primary, F / i f is purely inseparable. Now A* is the direct product of the 7* since 7 is the direct product of the 7^ Theorem 1.5.9. Also A* corresponds to 7* by Lemma 9.6.4, and so A* is the tensor product over K of intermediate fields Li such that the intermediate fields of the Li/K appear in a chain, [40]. The desired result now follows by denning the fuzzy subsets Xi of F to equal A on Li and 0 on F\Li. ■

383

9.7

EXERCISES

[18] Let L — [0,1] and ji G L(F/K). Then \i is said to be fuzzy primitive if VIA G F, V roots r of the minimal polynomial of u over if, fi(u) = fi(r) if and only if K(u) = if(r). /i is called a fuzzy chain subfield if Mu,v G F, ^x(w) = /x(v) if and only if K(u) = K(v). fj, is called fuzzy simple if it has a minimal generating set S over K such that ua and vt> are in S with a > 6, then u G if (v). VIA G F, /i(u) = /Li(r) for all roots r in F of the minimal polynomial of u over if. If ^(F) is well-ordered under <, then \i is called fuzzy subnormal if Vii G /i a , MC1*) < M r ) f° r ^ roots r G F of the minimal polynomial of -u over / v Va G /i(F), where a! is the immediate successor of a with respect to < . /J, is called reduced if Va G Im(/i) and Vit, v G ^a\^a', Ma'0-0 = l^a'(v) implies if (tx) = if (v). /x is called fuzzy Galois if F/K is Galois and \i is fuzzy normal. \i is called fuzzy subGalois if F/K is Galois and \i is fuzzy normal. The following results and their role in the Galois correspondence appear in [18]. 1. Prove that all intermediate fields of F/K are chained if and only if F/K has a fuzzy chain subfield. 2. Prove that ji is fuzzy simple if and only if fia is a simple extension over if VaG/x(F). 3. Prove that the following statements are equivalent: (a) fi is a fuzzy chain subfield of F/K] (b) /V(„) = K(u) Vu G F ; (c) /x is fuzzy simple, reduced, and of maximal chain. 4. Let F/K be a normal extension field. Prove that /z is fuzzy subnormal if and only if fia/f^a' is normal Va G A*(F). 5. Prove that the following statements are equivalent. (a) \x is fuzzy subGalois; (b) F/K is Galois and fia/Ha' is Galois Va G /x(F); (c) F / i f is Galois and f(/S) C ^ V /ia> -automorphisms / of /xa Va G

tin

384

Bibliography 1. Abadi, H. H and Zahedi, M. M., A density theorem on fuzzy prime spectrum of a ring, preprint. 2. Abadi, H. H and Zahedi, M. M., Some results on fuzzy prime spectrum of a ring, FSS 77 (1996) 235 - 240. 3. Abdukhalikov, K. S., The dual of a fuzzy subspace, FSS 82 (1996) 375 381. 4. Abdukhalikov, K. S., Tulenbaev, M. S., and Umirbaev, U. U., On fuzzy bases of vector spaces, FSS 63 (1994) 201 - 206. 5. Abou-Zaid, Salah, On fuzzy ideals and fuzzy quotient rings of a ring, FSS 59 (1993) 205 - 210. 6. Ahsan, J., Khan, M. F., Shabir, M. and Zaman, N., Rings characterized by their fuzzy submodules, Inform. Sci. 74 (1993) 247-264. 7. Ajmal, N., Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups, FSS 61 (1994) 329 - 339. 8. Ajmal, N. and Thomas, K. V., The lattice of fuzzy subgroups and fuzzy normal subgroups, Inform. Sci. 76 (1994) 1 - 1 1 . 9. Ajmal, N., The lattice of fuzzy normal subgroups is modular, Inform Sci. 83 (1995) 199 - 209. 10. Ajmal, N., Fuzzy subgroups with sup property, Inform Sci. 93 (1996) 247 - 264. 11. Ajmal, N. and Thomas, K. V., The lattice of fuzzy ideals of a ring, FSS 74 (1995) 371-379. 12. Ajmal, N. and Thomas, K. V., A complete study of the lattices of fuzzy congruences and fuzzy normal subgroups, Inform. Sci., 82 (1995) 198 218.

385

13. Akgul, M., Some properties of fuzzy groups, J. Math. Anal. Appl, 133 (1988) 93-100. 14. Alkhamees, Y., Fuzzy cyclic subgroups and fuzzy cyclic p-groups, J. Fuzzy Math. 3 (1995) 911 - 919. 15. Alkhamees, Y. and Mordeson, J. N., Fuzzy principal ideals and fuzzy simple field extensions, FSS, 96 (1998) 247-253. 16. Alkhamees, Y. and Mordeson, J. N., Fuzzy localized subrings, Inform. Sci, 99 (1997) 183-193. 17. Alkhamees, Y. and Mordeson, J. N., Local examination of fuzzy inter­ section equations, FSS 98 (1998) 249-254. 18. Alkhamees Y. and Mordeson, J. N., Reduced fields, primitive fields and fuzzy Galois theory, preprint. 19. Asaad, M., Groups and fuzzy subgroups, FSS 39 (1991) 323-328. 20. Asaad M. and Abou-Zaid S., Fuzzy subgroups of nilpotent groups, FSS 60 (1993) 321 - 323. 21. Assad, M. and Abou-Zaid, S., A contribution to the theory of fuzzy subgroups, FSS (1996) 355 - 369. 22. Balbes, R. and Dwinger, P., Distributive Lattices, University of Missouri Press, 1974. 23. Barnes, W. E., Introduction to Abstract Algebra, D. C. Heath, Boston, 1963. 24. Bhakat, S. K. and Das, P., Fuzzy subrings and ideals redefined, FSS 81 (1996) 383-393. 25. Bhambri S. K., Kumar, R. and Kumar, P., On fuzzy primary submodules, Bull. Cal. Math. Soc. 86 (1994) 445-452. 26. Bhambri S. K., Kumar, R. and Kumar, P., Fuzzy prime submodules and radical of a fuzzy submodule, Bull. Cal. Math. Soc. 87 (1995) 163-168. 27. Birkhoff, G., Lattice Theory, American Mathematical Society, Provi­ dence, Rhode Island (1967). 28. Biswas, R., Fuzzy fields and fuzzy linear spaces redefined, FSS 33 (1989) 257 - 259.

386

29. Borzoee, R. A. and Zahedi, M. M., Fuzzy algebraic extension generated by a fuzzy subset and maximal fuzzy algebraic extension, J. Fuzzy Math. I (1993) 649 - 657. 30. Bourbaki, N. (transl. by P. M. Cohn and J. Howie), Elements of Math­ ematics, Algebra II, Chapters 4 - 7 , Springer - Verlag, Berlin, New York (1981). 31. Bourbaki, N., Elements of Mathematics, Commutative Algebra, SpringerVerlag, New York, 1989, Chaps. 1-7. 32. Chen De-Gang and Gu Wen-Xiang, Product structure of fuzzy factor groups, FSS 60 (1993) 2229 - 232. 33. Cox, D., Little, J., and O'Shea, D., Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer-Verlag, 1992. 34. Das, P., Fuzzy vector spaces under triangular norms, FSS 25 (1988) 73 -85. 35. Das, P. S., Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264 - 269. 36. Deveney, J. K., An intermediate theory for a purely inseparable Galois theory, Trans. Amer. Math. Soc. 198 (1974) 287 - 295. 37. Deveney, J. K. and Mordeson, J. N., Splitting and modularly perfect fields, Pacific J. Math. 83 (1979) 45-54. 38. Deveney, J. K. and Mordeson, J. N., Distinguished subfields of interme­ diate fields, Canad. J. Math. 33 (1981) 1085-1096. 39. Deveney, J. K. and Mordeson, J. N., Uniqueness of subfields, Canad. Bull. Math. 29 (1986) 191-196. 40. Deveney, J. K. and Mordeson, J. N., Inseparable extensions and primary abelian groups, Arch. Math. 33 (1979) 538 - 545. 41. Dib, K. A., On fuzzy subspaces and fuzzy group theory, Inform. Sci. 80 (1994) 253 - 282. 42. Dib, K. A., Galhum, N. and Hassam A. A. M., Fuzzy rings and fuzzy ideals, J. Fuzzy Math., 4 (1996) 245-261. 43. Divinsky, N. J., Rings and Radicals, University of Toronto Press (1965).

387

44. Dixit, V. N., Kumar, R., and Ajmal, N., Fuzzy ideals and fuzzy prime ideals of a ring, FSS 44 (1991) 127 - 138. 45. Dixit, V. N., Kumar, R., and Ajmal, N., On fuzzy rings, FSS 49 (1992) 205 - 213. 46. Eke, A. C. and Eke, B. I., Special fuzzy generators and intermediate fuzzy field extensions, FSS 64 (1994) 407 - 413. 47. Eslami, E. and Mordeson., J. N., Structure of fuzzy subrings, Inform. Sci. 76 (1994) 57 - 65. 48. Eslami, E. and Mordeson., J. N., Completions and fuzzy power series subrings, FSS 82 (1996) 97 - 102. 49. Fuchs, L., Infinite Abelian Groups I, Pure and Applied Math., Vol. 36, Academic Press New York 1070. 50. Garmendia, A., Molero, M. and Salvador, A., T-fuzzy prime ideals, FSS 80 (1996) 245-254. 51. Garzon, M. and Muganda, G. C., Free fuzzy groups and fuzzy group presentations, FSS 48 (1992) 249 - 255. 52. Goguen, J. A., L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145 - 174. 53. Golan, J. S., Norm, semirings, and power algebra, in: Ramanujan Con­ ference on Algebra and Its Applications, Madra, India (to appear). 54. Golan, J. S., Making modules fuzzy, FSS 32 (1989) 91-94. 55. Gu Wenxiang and Lu Tu, Fuzzy linear spaces, FSS 49 (1992) 377 - 380. 56. Gu Wenxiang and Lu Tu, Fuzzy algebras over fuzzy fields redefined, FSS 53 (1993) 105 - 107. 57. Gu Wenxiang and Lu Tu, Some properties of divisible and pure fuzzy subgroups, Fuzzy Syst & Math. 6 (1) (1992) 48 - 53. 58. Gu Wenxiang and Lu Tu, The properties of fuzzy divisible groups, FSS 56 (1993) 195 - 198. 59. Gupta, K. C. and Kantroo, M. K., The intrinsic product of fuzzy subsets of a ring, FSS 57 (1993) 103-110. 60. Gupta, K. C. and Kantroo, M. K., The nil radical of a fuzzy ideal, FSS 59 (1993) 87 - 93.

388

61. Head, T., A metatheorem for deriving fuzzy theorems from crisp versions, FSS 73 (1995) 349-358. 62. Head, T., Erratum to "A metatheorem for deriving fuzzy theorems from crisp versions," FSS 79 (1996) 277-278. 63. Head, T., Embedding lattices of fuzzy subgroups into lattices of crisp sub­ groups, Proceedings, Biennial Conference of the North American Fuzzy Information Processing Society - NAFIPS 1996. 64. Heerema, N and Tucker, D., Modular field extensions, Proc. Math. Soc. 53 (1975) 301 - 306.

Amer.

65. Jaballah, A. and Mordeson, J. N., Minimal generating systems for fuzzy ideals, Soochow J. Math., 21 (1995) 183 - 192. 66. Jacobson, N., Lectures in Abstract Algebra, Vol. II, Linear Algebra, D. Van Nostrand Company, Inc. Princeton, New Jersey (1953). 67. Jacobson, N., Lectures in Abstract Algebra, Vol. Ill, Theory of Fields and Galois Theory, D. Van Nostrand Company, Inc. Princeton, New Jersey (1964). 68. Jacobson, N., Basic Algebra I, W. H. Freeman and Company, San Fran­ cisco (1974). 69. Jacobson, N., Basic Algebra II, W. H. Freeman and Company, San Fran­ cisco (1980). 70. Jiang Huabiao, The direct sum of fuzzy rings, J. National Defense Sci. & Technol. Univ. No. 3 (1985) 105 - 110. 71. Jiang Huabiao, Fuzzy ideals and quotient rings, Sci. Exploration, No. 1 (1986) 17 - 24. 72. Jiang Huabiao, Fuzzy subfields on a finite extension field, J. National Defense Sci. & Technol. Univ., No. 1 (1988) 65 - 69. 73. Jianli, Z., Kaiquan, S., and Mingshan, Y., Fuzzy modules over fuzzy rings, J. Fuzzy Math. 1 (1993) 531-539. 74. Karpilovsky, G., Topics in Field Theory, North-Holland Mathematics Studies 155, Elsevier Science Publishers B. V., 1989. 75. Katsaras, A. K. and Liu, D. B., Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58 (1977) 135 - 146.

389

76. Kim, Jae-Gyeom, Orders of the fuzzy subgroups and fuzzy p-groups, FSS 61 (1994) 225 - 230. 77. Kim, Jae-Gyeom, Fuzzy subgroups having property (*), Inform. Sci. 80 (1994) 235 - 241. 78. Kim, Jae-Gyeom and Kim Han-Doo, A characterization of fuzzy sub­ groups of some abelian groups, Inform. Sci. 80 (1994) 243 - 252. 79. Kim Jae-Gyeom, Fuzzy orders relative to fuzzy subgroups, Inform. 80 (1994) 341 - 348.

Sci.

80. Kim, Jae-Gyeom, Fuzzy subgroups and minimal fuzzy p-subgroups, J. Fuzzy Math., 2 (1994) 913 - 921. 81. Kim, Jae-Gyeom, Commutative fuzzy sets and nilpotent fuzzy groups, Inform. Sci. 83 (1995) 161 - 174. 82. Kim, Jae-Gyeom, Lattices of fuzzy subgroupoids, fuzzy submonoids, and fuzzy subgroups, Inform. Sci., 91 (1996) 77 - 93. 83. Kim, Jae-Gyeom, Join-reducibility of fuzzy subgroups, Inform. Sci., 91 (1996) 211 - 225. 84. Kohli, J. K. and Kumar, R., Fuzzy prime spectrum of a ring II, FSS 59 (1993) 223 - 230. 85. Kreimer, H. and Heerema, N., Modularity vs. separability for field ex­ tensions, Canad. J. Math. 27 (1975) 1176 - 1182. 86. Kumar, R., Fuzzy semiprimary ideals of rings, FSS 42 (1991) 263 - 272. 87. Kumar, R., Fuzzy irreducibile ideals in rings, FSS 42 (1991) 360 - 379. 88. Kumar, R., Fuzzy nil radicals and fuzzy primary ideals, FSS 43 (1991) 81 - 91. 89. Kumar, R., Fuzzy vector spaces and fuzzy cosets, FSS 45 (1992) 109 116. 90. Kumar, R., Fuzzy prime spectrum of a ring, FSS 46 (1992) 147 - 154. 91. Kumar, R., Certain fuzzy ideals of rings redefined, FSS 46 (1992) 251 260. 92. Kumar, R., Fuzzy cosets and some fuzzy radicals, FSS 46 (1992) 261 265.

390

93. Kumar, R., Fuzzy subgroups, fuzzy ideals, and fuzzy cosets: Some prop­ erties, FSS 48 (1992) 267 - 274. 94. Kumar, R., On fuzzy irreducible ideals and homomorphisms (Short Com­ munication), FSS 50 (1992) 355 - 356. 95. Kumar, R., Fuzzy nil radicals and fuzzy semiprimary ideals, FSS 51 (1992) 345 - 350. 96. Kumar, R., Fuzzy ideals and fuzzy semiprime ideals, Inform. (1992) 43 - 52.

Sci. 66

97. Kumar, R., On the fuzzy radical of a fuzzy ideal (Short Communication), FSS 53 (1993) 233 - 235. 98. Kumar, R., On the dimension of a fuzzy subspace, FSS 54 (1993) 229 234. 99. Kumar, R. Bhambri, S. K. and Kumar P., Fuzzy submodules: some analogues and deviations, FSS 70 (1995) 125-130. 100. Kumbhojkar, H. V. and Bapat, M. S., Not-so-fuzzy fuzzy ideals, FSS 37 (1990) 237 - 243. 101. Kumbhojkar, H. V. and Bapat, M. S., Correspondence theorems for fuzzy ideals, FSS 41 (1991) 213 - 219. 102. Kumbhojkar, H. V. and Bapat, M. S., On semiprime fuzzy ideals, FSS 60 (1993) 219 - 223. 103. Kumbhojkar, H. V. and Bapat, M. S., On prime and primary fuzzy ideals and their radicals, FSS 53(1993) 203-216. 104. Kumbhojkar, H. V., Spectrum of prime fuzzy ideals, FSS 62 (1994) 101 -109. 105. Kumbhojkar, H. V., Some comments on spectrums of prime fuzzy ideals of a ring, FSS 85(1997) 109-114. 106. Kunz, E., Introduction to Commutative Algebra and Algebraic Geometry, Birkhouser Boston, 1985. 107. Kuraoka, T. and Kuroki, N., On fuzzy quotient rings induced by fuzzy ideals, FSS 47 (1992) 381 - 386. 108. Kuraoka, T. and Suzuki, Nobu-Yuki, A simple characterization of fuzzy subgroups, Inform. Sci. 73 (1993) 41 - 55.

391

109. Kuroki N., On fuzzy semigroups, Inform. Sci. 53(1991) 203-236. 110. Kuroki N., On fuzzy ideals and fuzzy biideals in semigroups, FSS 5 (1981) 203-215. 111. Lajos S., A note on semilattices of groups, Ada. 33(1972) 315-317.

Sci. Math.

(Szeged)

112. Lee, K. Y. and Mordeson, J. N., Factorization of fuzzy ideals in Dedekind domains, J. Fuzzy Math. 5(1997) 741-745. 113. Lee, K. Y. and Mordeson, J. N., Fractionary fuzzy ideals and fuzzy invertible fractionary fuzzy ideals, J. Fuzzy Math. 5(1997) 875-883. 114. Lee, K. Y. and Mordeson, J. N., Fractionary fuzzy ideals and Dedekind domains, preprint. 115. Li, S. Y., Almost trivial fuzzy vector spaces, J. Fuzzy Math., 3 (1995) 633-636. 116. Liu Wangjin, Fuzzy invariant subgroups and fuzzy ideals, FSS 8 (1982) 133 - 139. 117. Liu Wangjin, Operations on fuzzy ideals, FSS 11 (1983) 31 - 41. 118. Liu Wangjin, L-fuzzy generated ideals and co-prime ideals, J. Sichuan Teachers College (Nat. Sci. Ed.), No. 3 (1985) 37 - 42. 119. Liu Wangjin, L-fuzzy prime ideals and radical ideals, J. Sichuan Teachers College (Nat. Sci. Ed.), No. 3, (1985) 67 - 72. 120. Liu Wangjin, Fuzzy prime ideals and fuzzy radical ideals, Inform. 46 (1990) 1- 12.

Sci.

121. Liu Wangjin, On some systems of simultaneous equations in a completely distributive lattice, Inform. Sci. 50 (1990) 185 - 196. 122. Liu Xilong and Yu Yandong, The isomorphisms of fuzzy linear spaces (1), J. Xuzhou Teachers College (Nat. Sci. Ed.), No. 1 (1986) 10 - 15. 123. Liu Xilong and Yu Yandong, The isomorphisms of fuzzy linear spaces (2), J. Xuzhou Teachers College (Nat Sci. Ed.), No. 2 (1986) 14 - 19. 124. Lowen, R., Convex fuzzy sets, FSS 3 (1980) 291 - 310. 125. Lu Tu and Gu Wenxiang, Abelian fuzzy group and its properties, FSS 64 (1994) 415 - 420.

392 126. Lu Tu and Gu Wenxiang, Some properties of fuzzy vector spaces, J. Fuzzy Math. 3 (1995) 921-931. 127. Lubczonok, P., Fuzzy vector spaces, FSS 38 (1990) 329 - 343. 128. Ma Jiliang and Yu Chunhai, Fuzzy rings (I), J. Northeast Normal Univ. (Nat. Sci. Ed.), No. 1 (1982) 23 - 38. 129. Majumdar, S. and Sultana Q., The lattice of fuzzy ideals of a ring, FSS 81 (1996) 271-273. 130. Makamba, B. B., Direct products and isomophism of fuzzy subgroups, Inform. Sci. 65 (1992) 33 - 43. 131. Makamba, B. B., Studies in fuzzy groups, Ph. D. thesis, Rhodes Univ., 1992. 132. Makamba, B. B. and Murali, V., Normality and congruence in fuzzy subgroups, Inform. Sci. 59 (1992) 121 - 129. 133. Malik, D. S., Fuzzy ideals of Artinian rings, FSS 37 (1990) 111 - 115. 134. Malik, D. S. and Mordeson, J. N., Fuzzy prime ideals of rings, FSS 37 (1990) 93 - 98. 135. Malik, D.S. and Mordeson, J. N., Fuzzy subfields, FSS 37 (1990) 383 388. 136. Malik, D. S. and Mordeson, J. N., Fuzzy maximal, radical, and primary ideals of a ring, Inform. Sci. 53 (1991) 237 - 250. 137. Malik, D. S. and Mordeson, J. N., Fuzzy primary representations of fuzzy ideals, Inform. Sci. 55 (1991) 151 - 165. 138. Malik, D. S. and Mordeson, J. N., Fuzzy vector spaces, Inform. Sci. 55 (1991) 271 - 281. 139. Malik, D. S. and Mordeson, J. N., Fuzzy subgroups of abelian groups, Chinese J. Math. (Taipei) 19 (1992) 129 - 145. 140. Malik, D. S. and Mordeson, J. N., Fuzzy direct sums of fuzzy rings, FSS 45 (1992) 83 - 91. 141. Malik, D. S. and Mordeson, J. N., Extensions of fuzzy subrings and fuzzy ideals, FSS 45 (1992) 245 - 251, 142. Malik, D. S. and Mordeson, J. N., Fuzzy homomorphisms of rings, FSS 46 (1992) 139 - 146.

393 143. Malik, D. S. and Mordeson, J. N., Radicals of fuzzy ideals, Inform. Sci. 65 (1992) 239 - 252. 144. Malik, D. S. and Mordeson, J. N., 7£-primary representations of L-ideals, Inform, Sci. 88 (1996) 227 - 246. 145. Malik, D. S. and Mordeson, J. N., L-prime spectrum of a ring, NAFIPS 97(1997) 273-278. 146. Malik, D. S., Mordeson, J. N., and Nair, P. S., Fuzzy normal subgroups in fuzzy subgroups, J. Korean Math. Soc. 29 (1992) 1 - 8 . 147. Malik, D. S., Mordeson, J. N., and Nair, P. S., Fuzzy generators and fuzzy direct sums of abelian groups, FSS 50 (1992) 193 - 195. 148. Martinez, L., Fuzzy subgroups of fuzzy groups and fuzzy ideals of fuzzy rings, J. Fuzzy Math., 3 (1995) 833-849. 149. Martinez, L., Fuzzy modules over fuzzy rings in connection with fuzzy ideals of fuzzy rings, J. Fuzzy Math., 4 (1996) 843-857. 150. Mashinchi, M. and Mukaidono, M., Generalized fuzzy quotient sub­ groups, FSS 74 (1995) 245 - 257. 151. Mashinchi, M. and Zahedi, M. M., On L-fuzzy primary submodules, FSS 49 (1992) 231-236. 152. May, W., Commutative Group Algebras, Trans. Amer. Math. Soc. 136 (1969) 139 - 149. 153. McCarthy, P. J., Algebraic Extension of Fields, Blaisdell Publishing Com­ pany (1966). 154. McCoy, N. H., Rings and Ideals, The Carus Mathematical Monograms, Number 8, The Mathematical Association of America, 1956. 155. Mordeson, J. N., Remark on coefficient fields in complete local rings, J. Math. Kyoto Univ. 4 (1965) 637 - 639. 156. Mordeson, J. N., Splitting of field extensions, Archiv Math. 26 (1975) 606-610. 157. Mordeson, J. N., Modular extensions and abelian groups, Arch. Math. 36 (1981) 13 - 20. 158. Mordeson, J. N., Fuzzy algebraic field extensions, FSS 45 (1992) 359 365.

394

159. Mordeson, J. N., Fuzzy-field extensions, FSS 47 (1992) 253 - 264. 160. Mordeson, J. N., Fuzzy transcendental field extensions, Chinese J. Math. (Taipei) 20 (1992) 55 - 68. 161. Mordeson, J. N., Fuzzy subfields of finite fields, FSS 52 (1992) 93 - 96. 162. Mordeson, J. N., Distinguished subfields and splitting of fuzzy subfields, Bull. Cal. Math. Soc. 84 (1992) 121 - 126. 163. Mordeson, J. N., Bases of fuzzy vector spaces, Inform. Sci. 67 (1993) 87 -92. 164. Mordeson, J. N., Generating properties of fuzzy algebraic structures, FSS 55 (1993) 107 - 120. 165. Mordeson, J. N., Splitting of fuzzy field extensions, J. Nigerian Math. Soc. 11 (1992) 33 - 44. 166. Mordeson, J. N., Fuzzy algebraic varieties, Rocky Mountain J. Math. 23 (1993) 1361 - 1377. 167. Mordeson, J. N., Fuzzy algebraic varieties II, Advances in Fuzzy Theory and Technology Vol. I (1993) (Edited by Paul Wang) 9 - 21. 168. Mordeson, J. N., Fuzzy Galois theory, J. of Fuzzy Math. 1 (1993) 659 671. 169. Mordeson, J. N., Fuzzy coefficient fields of fuzzy subrings, FSS 58 (1993) 227 - 327. 170. Mordeson, J. N., Bases of fuzzy algebraic structures, FSS 62 (1994) 185 - 191. 171. Mordeson, J. N., Invariants of fuzzy subgroups, FSS 63 (1994) 81 - 85. 172. Mordeson, J. N., Infinite fuzzy Galois theory, Advances in Fuzzy Theory and Technology Vol. II (1994) (Edited by Paul Wang) 99 - 109. 173. Mordeson, J. N., Fuzzy group subalgebras, J. Fuzzy Math. 3 (1995) 69 81. 174. Mordeson, J. N., Relative p-bases of fuzzy field extensions, Chinese J. Math., 22 (1994) 297 - 308. 175. Mordeson, J. N., Fuzzy intersection equations and primary representa­ tions, FSS, 83 (1996) 93 - 98.

395

176. Mordeson, J. N., Commentary: Definition of prime and primary fuzzy ideals, J. Fuzzy Math. 3 (1995) 345 - 349. 177. Mordeson, J. N., Fuzzy group subalgebras II, J. Fuzzy Math.. 3 (1995) 885 - 897. 178. Mordeson, J. N. and Peng, Chang-Shyh, Fuzzy intersection equations, FSS 60 (1993) 77 - 81. 179. Mordeson, J. N. and Peng, Chang-Shyh, Operations on fuzzy graphs, Inform. Sci. (1994) 159 - 170. 180. Mordeson, J. N. and Sen, M. K., Basic fuzzy subgroups, Inform. 82 (1995) 167 - 179.

Sci.,

181. Mordeson, J. N. and Vinograde B., Extension of certain subfields to coefficient fields in commutative algebras, J. Math. Soc. Japan 17 (1965) 47-51 182. Mordeson, J. N. and Vinograde B., Generators and tensor factors of arbitrary purely inseparable fields, Math. Zeit., 107 (1968) 326-334. 183. Mordeson, J. N. and Vinograde, B., Arbitrary Purely Inseparable Exten­ sion Fields, Lecture Notes in Math. 173, Springer-Verlag, Berlin Heidel­ berg, New York 1970. 184. Morsi, N. N. and Yehia, S. E., Fuzzy-quotient groups, Inform. Sci, 81 (1994) 177 - 191. 185. Morsi, N. N., Note on normal fuzzy subgroups and fuzzy normal series of finite groups, FSS 72 (1995) 379-383. 186. Muganda, G. C , Fuzzy linear and affine spaces, FSS 38 (1990) 365 - 373. 187. Muganda, G. C , Free fuzzy modules and their bases, Inform. (1993) 65 - 82.

Sci.,72

188. Muganda, G. C , On the existence of bases for fuzzy vector spaces, preprint. 189. Muganda, G. C , Fuzzy algebras based on algebraic properties, preprint. 190. Muganda, G. C. and Garzon, M., On the structure of fuzzy groups, Advances in Fuzzy Theory and Technology Vol. I (1993) (Edited by Paul Wang) 23 - 42. 191. Mukherjee, N. P. and Bhattachaxya, P., Fuzzy normal subgroups and fuzzy cosets, Inform. Sci. 34 (1984) 225 - 239.

396 192. Mukherjee, T. K. and Sen, M. K., On fuzzy ideals of a ring I, FSS 21 (1987) 99 - 104. 193. Mukherjee, T. K., and Sen, M. K., Prime fuzzy ideals in rings, FSS 32 (1989) 337 - 341. 194. Mukherjee, T. K. and Sen, M. K., Rings with chain conditions, FSS 39 (1991) 117 - 123. 195. Mukherjee, T. K., Sen, M. K. and Roy D., On fuzzy submodules of their radicals, J. Fuzzy Math., 4 (1996) 549-558. 196. Murthy, N. V. E. S., How fuzzy are fuzzy ideals?, J. Fuzzy Math., 4 (1996) 15 - 23. 197. Nagata, M., Note on coefficient fields in complete local rings, Mem. Coll. Sci. Uni. Kyoto 32 (1959 - 60) 91 - 92. 198. Nagata, M., Local Rings, Interscience Publishers # 1 3 , John Wiley and Sons Inc., 1962. 199. Nanda, S., Fuzzy fields and linear spaces, FSS 19 (1986) 89 - 94. 200. Nanda, S. Fuzzy algebras over fuzzy fields, FSS 37 (1990) 99 - 103. 201. Nanda, S., Fuzzy linear spaces over valued fields, FSS 42 (1991) 351 354. 202. Negoita, C. V. and Ralescu, D. A., Applications of Fuzzy Sets to Systems Analysis, Wiley, New York (1975) 54 - 59. 203. Ovchinnikov S., Continuous fuzzy groups, FSS 76 (1995) 253 - 257. 204. Pan, F., Fuzzy finitely generated modules, FSS 21 (1987) 105-113. 205. Pan, F., Exact sequences of fuzzy linear maps, FSS 27 (1988) 317-325. 206. Pan, F., Fuzzy quotient modules, FSS 28 (1988) 85-90. 207. Pan, F., The various structures of fuzzy quotient modules, FSS 50 (1992) 187-192. 208. Peng, B.,The isomorphism of fuzzy groups and some problems of calcu­ lation, J. South China Normal Univ. (Nat. Sci. Ed. Math. Phys. Spec. Issue) (1990) 158-162. 209. Passman, D. S., The Algebraic Structure of Group Rings, A. Wiley In­ terscience Publication, John Wiley Sz Sons, Inc. 1977.

397

210. Petrich, M., Introduction to Semigroups, Charles E. Merrill Publishing Co., A Bell and Howell Co., Ohio, 1973. 211. Ray, S., Isomorphic fuzzy groups, FSS 50 (1992) 201 - 207. 212. Ray, S., The free product of fuzzy subgroups, FSS 50 (1992) 225-235. 213. Ray, S., Analysis of the level subgroups of a fuzzy group, FSS 51 (1992) 323-331. 214. Ray, S., Solvable fuzzy groups, Inform. Sci. 75 (1993) 47 - 61. 215. Ren Yongcai, Fuzzy ideals and quotient rings, Fuzzy Math., 5 (4) (1985) 19 - 26. 216. Rosenfeld, A., Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512 - 517. 217. Rygg, P. T., On mimimal sets of generators of purely inseparable field extensions, Proc. Amer. Math. Soc. 14 (1963) 742 - 745. 218. Salili, S. and Mashinchi, M., On fuzzy isomorphism, J. Fuzzy Math. 4 (1996) 39 - 49. 219. Sanchez, E., Resolution of composite fuzzy relation equations, Inform. and Control 30 (1976) 38-48. 220. Sebastian, S. and Sundar, S. B., Commutative L-fuzzy subgroups, FSS 68 (1994) 115 - 121. 221. Sessa S., On fuzzy subgroups and fuzzy ideals under triangular norms (Short communication), FSS 13 (1984) 95-100. 222. Sidky, F. I., Essential and superfuluous fuzzy submodules, J. Inst. Math. Comput. Sci. Math. Ser. 4 (1991) 103-106. 223. Sidky, F. I. and Khatab, S. A., Nil radical of fuzzy ideal, FSS 47 (1992) 117- 120. 224. Sidky, F. I. and Mishref, M. A. A., Divisible and pure fuzzy subgroups, FSS 34 (1990) 377 - 382. 225. Sidky, F. I. and Mishref, M. A. A., Fuzzy cosets and cyclic and abelian fuzzy subgroups, FSS 43 (1991) 243 - 250. 226. Sidky, F. L. and Mishref, M. A. A., Nil radical of fuzzy ideal, FSS 47 (1992) 117 - 120.

398

227. Suzuki, Y., On the construction of free fuzzy groups, J. Fuzzy Math., 2 (1994) 1 - 15. 228. Swamy, U. M. and Raju, D. V., Algebraic fuzzy systems, FSS 41 (1991) 187 - 194. 229. Swamy, U. M. and Raju, D. V., Irreduciblity in algebraic fuzzy systems, FSS 41 (1991) 233 - 241. 230. Swamy, U. M. and Swamy, K. L. N., Fuzzy prime ideals of rings, J. Math. Anal. Appl. 134 (1988) 94 - 103. 231. Sweedier, Structure of inseparable extensions, Annal. of Math. 87(1968) 401 - 410; correction, ibid (2) 89 (1969), 206 - 207. 232. Szasz, G. Introduction to Lattice Theory, Academic Press, New York and London, 1963. 233. Wang, G., Pointwise topology on a complete distributive lattices (I) (in Chinese), J. Shanxi Normal Univ., 1 (1985) 1-17. 234. Wang, Z. D., TL-submodules and TL-linear subspaces, FSS, 68 (1994) 211-225. 235. Wang, Z. D., Primary TL-submodules and P-primary TL-submodules, FSS 88(1997) 237-254. 236. Wang, Z. D. and Yu, Y. -D., TL-subrings and TL-ideals, Part 2: gener­ ated TL-ideals, FSS 87 (1997) 209-217. 237. Wang, Z. D. and Yu, Y. -D., TL-subrings and TL-ideals, Part 5: Prime TL-ideals and semiprime TL-ideals, FSS, submitted. 238. Waterhouse, W., The structure of inseparable field extesions, Trans. Amer. Math. Soc. 211 (1975), 39 - 56. 239. Weinberger, A., Embedding lattices of fuzzy subalgebras into lattices of crisp subalgebras, Inform. Sci, 108 (1998) 51-70. 240. Weng Wenhui, Fuzzy subspaces of a linear space, J. Xiamen Univ. (Nat. Sci. Ed.) 21 (1982) 371 - 376. 241. Weng Wenhui, The countable decomposition theorem of fuzzy subspaces, J. Xiamen Univ. (Nat. Sci. Ed.) 27 (1985) 20 - 25. 242. Weng Wenhui, The basis of fuzzy linear space, J. Xiamen Univ. (Nat. Sci. Ed.) 27 (1988) 385 - 386.

399

243. Weiss, M.D., Fixed points separation, and induced topologies for fuzzy sets, J. Math. Appl. 50(1975) 142-150. 244. Winter, D. J., The Structure of Fields, Graduate Texts in Mathematics, Springer-Verlag New York Heidelberg Berlin, 1970. 245. Wu, Wangming, Normal fuzzy subgroups, Fuzzy Math. 1 (1981) 21-30. 246. Yu Chunhai, Fuzzy maximal ideals of fuzzy rings, Fuzzy Math. (1986) 21 - 23.

6(1)

247. Yu Yandong, Fuzzy linear spaces redefined, Fuzzy Math. 4 (3) (1984) 59 -62. 248. Yu Yandong, Finitely generated fuzzy linear spaces, Fuzzy Math. 6 (4) (1986) 83 - 88. 249. Yu Yandong, Finitely generated T-fuzzy linear spaces, FSS 30 (1989) 69 -81. 250. Yu Yandong, Mordeson, J. N., and Cheng, Shih-Chuan, Elements of L-Algebra, Lecture Notes in Fuzzy Mathematics and Computer Science 1, Center for Research in Fuzzy Mathematics and Computer Science, Creighton University 1994. 251. Yu, Y. -D. and Wang, Z. D., TL-subrings and TL-ideals, Part 1: Basic concepts, FSS 68 (1994) 93-103. 252. Yu, Y. -D. and Wang, Z. D., TL-subrings and TL-ideals, Part 4: Radicals of TL-ideals, FSS 88 (1997) 227-236. 253. Yu, Y. -D. and Wang, Z. D., TL-subrings and TL-ideals, Part 6: Primary TL-ideals and semiprimary TL-ideals, FSS, submitted . 254. Zadeh, L. A., Fuzzy sets, Inform, and Contr. 8 (1965) 338 - 353. 255. Zahedi, M. M., L-fuzzy extended and contracted ideals, BUSEFAL 41 (1989) 104 - 110. 256. Zahedi, M. M., A characterization of L-fuzzy prime ideals, FSS 4A (1991) 147 - 160. 257. Zahedi, M. M., A note on L-fuzzy primary and semiprime ideals, FSS 51 (1992) 243 - 247. 258. Zahedi, M. M., On L-fuzzy residual quotient modules and 7r-primary submodules, FSS 51 (1992) 333 - 344.

400

259. Zahedi, M. M., Some results on L-fuzzy modules, FSS 55 (1993) 355 361. 260. Zahedi, M. M. and Ameri R., On fuzzy projective and injective modules, J. Fuzzy Math. 3 (1995) 181-190. 261. Zariski, O. and Samuel, P., Commutative Algebra Vol. /, D. Van Nostrand Company, Inc., Princeton, New Jersey (1958). 262. Zhang Yue, Prime L-fuzzy ideals and primary L-fuzzy ideals, Fuzzy Math. 7 (1) (1987) 45 - 48. 263. Zhang Yue, Prime L-fuzzy ideals and primary L-fuzzy ideals, FSS 27 (1988) 345 - 350. 264. Zhang Yue and Peng Xiantu, Maximal fuzzy ideals and primary fuzzy ideals of a ring, Fuzzy Math. 4 (1) (1984) 115 - 116. 265. Zhang Yunjie and Zou Kaiqi, Normal fuzzy subgroups and conjugate fuzzy subgroups, J. Fuzzy Math. 1 (1993) 571 - 585. 266. Zhu, N., Fuzzy subgroup chains of a group and fuzzy groups, Fuzzy Math. 6(3) (1986) 7-11. 267. Zou K., Fuzzy ideals of a ring, J. Dalian Marine College 8 (2) (1982) 66 -71. 268. Zou, K., Fuzzy modules and subspaces of fuzzy modules, J. Henan Teach­ ers University (Nat. Sci. Ed.) 2 (1983) 34-43.

401

Index A a-cut, 3 a-level set, 3 Abelian, 357 L-subgroup, 20 L-subset, 10 algebraic, 166 algebraic L-variety irreducible, 275 reducible, 275 algebraic closure, 166 algebraic L-variety, 267 algebraically independent, 167, 325 associated primes, 215

B B-linearly independent, 142 basic L-subgroup, 61 basis, 143, 236, 317, 354

c canonical representation, 347 carrier of an L-subset, 347 Cartesian cross-product, 94 Cauchy sequence, 240 characteristic function, 2 closed under projections, 34 coefficient field, 235 comaximal L-ideals, 83 compatible, 170, 192, 333 complete, 240 L-direct sum, 95 complete direct product, 4, 20-21 completion, 257 composite, 175 Condition G\ on cr, 367

Condition GT. on /x, 368 conjugate L-subgroup, 11 contracted ideal, 233 converge, 240 coprime L-ideals, 83 Correspondence Theorem, 86 coset, 12 left, 12 right, 12 crisp power set, 30

D direct product, 22, 26 complete, 20-21 weak, 22, 26 distinguished intermediate field, 192 divisible L-subgroup, 50

E exponent of inseparability, 198 extended ideal, 233 Extension Principle, 4

F /-invariant, 97 factor L-subgroup, 15, 17 factor group, 14 finite L-subset, 2 finitely generated, 123 fractionary fuzzy ideal, 263 extended integral, 265 integral, 264 invertible, 263 maximal, 264 prime, 264 free, 69, 316, 354

402 maximally, 317 free L-submodule, 140 fuzzy power set, 2 subset, 2 fuzzy C subset, 36 fuzzy chain subfield, 383 fuzzy dimension, 362 fuzzy Galois, 383 fuzzy graph, 301 fuzzy index, 363 fuzzy invertible ideal, 263 Fuzzy Luroth's Theorem, 204 fuzzy order, 363 fuzzy power set, 2 fuzzy prime spectrum, 262 fuzzy primitive, 383 fuzzy principal, 129 fuzzy relational equation, 303 fuzzy simple, 383 fuzzy subfield, 161 fuzzy subGalois, 383 fuzzy subgraph Cartesian product of, 302 fuzzy subgroup, 7 normal, 10 fuzzy submodule, 134 fuzzy subnormal, 383 fuzzy subset, 2

G Galois, 358, 370 group, 358 over a field, 358 generate, 9, 76, 121, 236 generating set, 42, 283 5-minimal, 288 minimal, 42, 57, 283 generic point, 226 group factor, 14 quotient, 14 group L-subalgebra, 374

H Heyting algebra, 1

homomorphic, 17, 89, 137 weakly, 89, 137 homomorphism, 17, 137 weak, 17, 89, 137

I image, 4 increasing monotonic limit, 335 index, 39, 357 inf property, 216 infinite L-subset, 2 inseparable, 171 inseparable /-algebraic, 172 intersection of L-subsets, 2 inverse image , 4 inverse of an L-subset, 6 irreducible L-ideal, 98 element, 98 irreducible topological space, 226 irredundant representation, 142 isolated component, 215 isolated primary L-submodule, 159 isomorphic, 17, 89, 137 weakly, 89, 137 isomorphism, 17, 89, 137 weak, 17, 89, 137

L /-algebraic, 168 L-basis, 335 L-coefficient field, 243 L-contraction, 233 L-coset left, 59 L-decomposition, 210 L-direct sum complete, 95 external weak, 95 weak, 90 L-extension by radicals, 361 L-field /-algebraic extension, 168 /-purely inseparable extension, 172 algebraic closure, 166 algebraic extension, 166

403 algebraically closed, 166 compatible, 170 distinguished //-intermediate ex­ tension, 192 extension, 161 inseparable /-algebraic extension, 172 inseparable algebraic extension, 171 modular extension, 188 neutral extension, 166 regular extension, 379 separable, 181 separable /-algebraic extension, 172 separable algebraic extension, 171 transcendental extension, 166 L-ideal, 72, 87 ^-primary, 205 £-prime, 205 ^-primary, 208 associated prime, 157, 208 comaximal, 83 contracted, 233 coprime, 83 equivalent, 284 finitely generated, 123 generalized maximal, 97 generated by, 76 irreducible, 98 local, 284 maximal, 96 normalized, 285 of maximal chain, 285 primary, 108 prime, 100 proper, 82 R-semiprimary, 117 reducible, 98 residual quotient, 148 semiprime, 102 //-intermediate field, 161 L-linearly independent, 334 maximal system, 345 L-point, 2 L-power set, 1 /^-primary L-representation, 217

irredundant L-representation, 217 reduced L-representation, 217 L-prime spectrum, 219 /-purely inseparable, 172 L-radical, 104 L-representation irredundant, 210 reduced, 210 L-set of generators, 121 L-singleton, 2 L-subfield, 161 generated by, 162 L-subfield in a ring, 242 L-subgroup, 7 p-primary, 48 Abelian, 20 basic, 61 conjugate, 11 cyclic, 61 divisible, 50 factor, 15, 17 generated by, 9 normal, 10, 15 pure, 52 quotient, 15, 17 reduced, 52 solvable, 20 torsion, 49 L-submodule, 134 7r-primary, 154 basis for an, 143 free, 140 generated, 139 primary, 149 primary decomposition of, 157 residual quotient, 148 L-submonoid, 23 L-subring, 71 difference, 87 generated by, 162 localized, 286 polynomial, 253 quasi-local, 229 quotient, 87, 89 L-subsemigroup, 23 L-subset, 1

404 Abelian, 10 cardinality of an, 339 carrier of an, 347 closed, 312 complete direct product of, 4 contained in, 2 contains, 2 dense, 313 difference of, 70 finite, 2 generating, 139 image of, 2 infinite, 2 intersection of, 2 inverse of, 6 negative of an, 70, 131 normal, 2 open, 235, 312 product of, 6, 70 properly contained in, 2 properly contains, 2 sum of, 70, 131 support of an, 2 union of, 2 unitary, 2 L-subspace, 316, 321 basis, 324 basis of an, 317 dimension of an, 338 finite dimensional, 319 fuzzy finite dimensional, 338 infinite dimensional, 319 L-subspaces isomorphic, 345 L-topological space, 235, 312 irreducible, 313 Noetherian, 313 L-topology, 312 neutrally closed, 313 left coset, 12 left L-coset, 59 linear space, 131 linearly dependent, 47 linearly disjoint, 177 linearly independent, 47, 323 localized L-subring, 286

localized ring, 286

M maximal, 48 maximal element, 96 maximal L-ideal, 96 generalized, 97 maximally p-independent, 64 Metatheorem, 34 metricizer, 256 minimal generating set, 42, 57, 200201, 283 minimal L-set of generators, 125 minimal set of generators, 321

N neutral, 166 neutral closure, 170, 312 neutrally closed, 170 normal, 356-357 normal L-subgroup, 10, 15 normal L-subset, 2 normal fuzzy subgroup, 10 normalizer, 12

p p-basic, 62 p-basis, 63, 325 p-independent, 63, 325 maximally, 64 p-primary L-subgroup, 48 component, 49 p-pure, 62 pairwise comaximal, 106 polynomial L-subring, 253 power series, 254 power set crisp, 30 pre-image, 4 primary L-ideal, 108 L-submodule, 149 primary decomposition irredundant, 157 normal, 157

405 of L-submodules, 157 prime L-ideal, 100 element, 100 prime L-ideal associated, 154 prime L-ideal divisor, 212 isolated, 212 minimal, 212 product of L-subsets, 6 proper L-ideal, 82 pseudo-Hausdorff, 236 pseudo-metric, 236 pure L-subgroup, 52 purely inseparable, 171

semiprime L-ideal, 102 separable algebraic, 170 separable /-algebraic, 172 solvable L-subgroup, 20 solvable by radicals, 362 space, 316 split, 193 strongly compatible, 170, 331 Subdirect Product Theorem, 34 subspace, 316 sup property, 2 support of an L-subset, 2

Q

torsion L-subgroup, 49 closure, 50 torsion independent, 326 torsion-free basis, 326 transcendence basis, 168, 325 transcendental, 166

quasi-local ring, 228 quotient, 137, 357 quotient L-subgroup, 15, 17 quotient group, 14 quotient L-subring, 87 quotient ring, 84

R ^-primary, 208-209 L-components, 215 L-ideal, 115 L-representation, 210 isolated L-components, 215 ^-radical, 111 7£-semiprimary L-ideal, 117 i?d/-primary, 207 reduced, 383 reduced L-subgroup, 52 V-reduced element, 303 reducible L-ideal, 98 regular, 1 relative p-basis, 200, 324 relatively p-independent, 200, 324 residual quotient, 145 right coset, 12 ring quotient, 84

s saturated, 286

T

u union of L-subsets, 2 unitary L-subset, 2 upper well ordered, 335

v vector space, 131

w weak L-direct sum, 90, 95 basis, 354 direct product, 22, 26 homomorphism, 17, 89, 137 isomorphism, 17, 89, 137 minimal generating set, 200-201 product, 22 sum, 132 weakly free, 354 homomorphic, 17, 89, 137 isomomorphic, 17 isomorphic, 89, 137