Lukasiewicz-Moisil Algebras

LUKASIEWICZ-MOISIL ALGEBRAS ANNALS OF DISCRETE MATHEMATICS General Editor: Peter L. HAMMER Rutgers University, New B...

Author: V. Boicescu

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LUKASIEWICZ-MOISIL ALGEBRAS

ANNALS OF DISCRETE MATHEMATICS

General Editor: Peter L. HAMMER Rutgers University, New Brunswick, NJ, USA

Advisory Editors: C. BERGE, Universite de Paris, France R.L. GRAHAM, AT&T Bell Laboratories, NJ, U.S.A. M.A. HARRISON, University of California, Berkeley, CA, USA V: KLEE, University of Washington, Seattle, WA, USA J.H. VAN LINT, California Institute of Technology, Pasadena, CA, USA G.C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, USA T: TROTTER, Arizona State University, Tempe, AZ, U.S.A.

49

LUKASIEWICZ-MOISIL ALGEBRAS

V. BOICESCU A. FlLlPOlU Polytechnical Institute Bucharest, Romania

G. GEORGESCU Institute of Mathematics Bucharest, Romania

S. RUDEANU Faculty of Mathematics University of Bucharest Bucharest, Romania

1991 NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, NY 10010, USA

Library o f Congress Cataloging-in-Publication

Data

Lukasiewicz-Moisil algebras / V . Boicescu ... Let a l . 1 . p. cm. -- (Annals o f discrete mathematics ; 49) Includes bibliographical references and index. ISBN 0-444-66444-0 1. Cukasiewicz algebras. I. Boicescu. V. (Vlad). 194511. Series. PA10.L85 1991 512,3'24--dc20

91-10061

CIP

ISBN: 0 444 88444 0 Q ELSEVIER SCIENCE PUBLISHERS B.V., 1991

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B. V. /Academic Publishing Division, /?O.Box 703, 1000 AC Amsterdam, Netherlands. Special regulations for readers in the USA - This publication has been registered with the Copyright Clearance Center Inc. (CCCI, Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the copyright owner, Elsevier Science Publishers B.V. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.

PRINTED IN THE NETHERLANDS

V

PREFACE

The first system o f many-valued logic was introduced by J. Lukasiewicz in 1920. Independently, E. Post created in 1921 an n-valued propositional calculus distinct from that o f Lukasiewicz.

The development of various systems of logic has always been accompanied by t h e development o f their algebraic counterpart (the associated Lindenbaum-Tarski algebras). In many cases the interest for the purely algebraic aspects has become preponderant and t h e corresponding domain has got the status o f a chapter of algebra having an intrinsic interest. This is precisely t h e case o f Lukasiewicz-Moisil algebras. Gr.C. Moisil introduced in 1940 the 3-valued and 4-valued Lukasiewicz algebras, w i t h a view to obtain the algebraic counterpart of the corresponding logics of Lukasiewicz, and in 1941 the n-valued Lukasiewicz algebras. Then Moisil developed the theory o f Lukasiewicz algebras f r o m an algebraic point o f view. He generalized the Stone representation theorem for Boolean algebras t o the case of Lukasiewicz algebras and studied the axled algebras and the centered algebras, which have remarkable properties. Then followed a standstill of almost two decades. After 1960, when Moisil found applications of Lukasiewicz algebras to the study o f electric circuits, there was a new advance o f the theory from t h e algebraic point of view as well. A n important role has been played by t h e Bahia-Blanca school created by A. Monteiro. Another line of research concerns the relationships w i t h pseudocomplemented lattices. New results were obtained by Moisil himself, who

also directed to this field t h e interest o f some of his pupils. T h e membership to a fuzzy set (in t h e sense of L. Zadeh) is an example

of an infinitely-valued proposition and Moisil found i n this example t h e motivation he had looked for in order t o legitimate t h e introduction and study

of infinitely-valued Lukasiewicz algebras. So in 1968 Moisil defined t h e 29-

Preface

vi

valued Lukasiewicz algebras, where 29 is the order type o f a chain. Then these algebras became the subject of several doctoral theses a t t h e University of Bucharest, written by pupils o f Moisil. Whereas the 3-valued and 4-valued Lukasiewicz algebras are actually the Lindenbaum-Tarski algebras associated w i t h t h e 3-valued and 4-valued Lukasiewicz logics, respectively, it was noticed by A. Rose t h a t this does not hold for n

25

because the Lukasiewicz implication cannot be defined in terms

of disjunction, conjunction, negation and endomorphisms i n n-valued Lukasiewicz algebras, n 2 5 . Using a Gentzen-like technique, Moisil constructed

a propositional calculus for which the corresponding Lindenbaum-Tarski algebras are precisely t h e n-valued Lukasiewicz algebras. O n t h e other hand,

R. Cignoli proved in 1969 t h a t t h e centered n-valued Lukasiewicz algebras (introduced by Moisil in 1941 for n = 3,4) coincide w i t h t h e n-valued Post algebras, introduced by P. Rosenbloom in 1942 as the algebraic counterpart o f t h e n-valued Post logics. Finally t h e relationship between Lukasiewicz logics and Moisil logics was clarified in 1980-1984 by Cignoli, who introduced t h e proper n-valued Lukasiewicz algebras and proved t h a t they are the Lindenbaum-Tarski algebras associated w i t h the Lukasiewicz logics. All these facts establish t h e relative position o f the Lukasiewicz, Post and Moisil logics as various models o f many-valued logics. T h e above sketched picture can be summarized by saying t h a t while the Lukasiewicz algebras originate in and have close connections w i t h the Lukasiewicz logics, these algebras were created by Moisil and the lot of research which led them to t h e position of being one of the important classes of lattices in algebraic logic, is due t o Moisil and his followers. T h a t is why we feel t h a t these algebras should be more properly called Lukasiewicz-Moisil algebras (as a matter o f fact, t h e name “Moisil algebras” has been already suggested in the literature). This monograph is devoted mainly, b u t not exclusively, to t h e algebraic side o f the theory of Lukasiewicz-Moisil algebras, presented i n Chapters

3-

8 and classified as indicated by the very titles of these chapters. Aspects pertinent to logic and applications t o switching theory are treated i n Chapter

9 and t h e Appendix, respectively, while the first t w o chapters establish all

Preface

vii

prerequisites necessary t o ensure selfcontainedness of t h e book. For the work of Moisil in logic see the survey

E. Radu [1978] and

also Onicescu and E.

Radu (19751.

We have aimed at a detailed presentation, with full proofs; some of them are new, without explicit mention of this fact. Objective limitations of space and time have obliged us t o select certain papers for a more attentive presentation, while others have been only summarized in a few words. We apologize for t h e inherent subjectivity of such a selection and for possible involuntary omissions. Topics such as Post algebras, pseudocomplemented lattices and Stone algebras, Lukasiewicz logics, partially studied in our book due t o their connections with Lukasiewicz-Moisil algebras, would deserve separate monographs. The posthumous paper A. Monteiro [1980] may be viewed as a monograph on symmetric Heyting algebras.

The main statements of this book, no matter of their nature, are numbered in the form n.p, which means the p-th statement of section n; they are referred t o in this form within each chapter and in the form m.n.p for statements belonging t o another chapter m. The same rule applies t o the num be rs of for m uI as . We express our deep gratitude t o Peter L. Hammer, who encouraged us t o write this book and accepted it for publication in the series Annals of Discrete Mathematics; t o Roberto Cignoli, Gabriel Ciobanu, Luiza lturrioz and Jules Varlet, who provided us with reprints of their works or copies of other papers; t o our colleague Afrodita lorgulescu, who provided us with technical aid and whose results have raised t h e quality of this book; t o Olga Vitu and Anita Klooster for their careful and competent typing.

We owe our formation as mathematicians and the subject of this book to our much beloved teacher, GRIGORE C. MOISIL.

This Page Intentionally Left Blank

ix

TABLE OF CONTENTS

PR EFAC E LIST OF SYMBOLS Ch. 1. LATTICES, UNIVERSAL ALGEBRA AND CATEGORIES $1. Posets and lattices

$2. $3. $4. $5. $6.

Distributive lattices, De Morgan and Boolean algebras Filters, ideals and congruences in lattices Monadic and polyadic Boolean algebras Universal algebra Categories and functors

V

...

Xlll

1 1 18 31 43 54 73

Ch. 2. TOPOLOGICAL DUALITIES IN LATTICE THEORY

83

$1.The Stone duality of distributive lattices

83

$2. The Stone duality of De Morgan algebras $3. The Priestley duality of distributive lattices 54. The Priestley duality of De Morgan algebras

92 95 102

Ch. 3. ELEMENTARY PROPERTIES OF LUKASIEWICZ-MOISIL ALGEBRAS

105

$1.Basic concepts $2, Axiornatics of &valued and n-valued algebras $3. Axiofiratics of three-valued algebras $4. lrredundant n-valued Moisil algebras

105 124

Ch. 4. CONNECTIONS W I T H OTHER CLASSES OF LATTICES

165

$1. Post algebras $2. Axled Lukasiewicz-Moisil algebras $3. Heyting algebras

165 177 201

131 155

Table of contents

X

$4. Pseudocomplemented lattices

213

$5. Completeness properties and atomicity

228

$6. m-algebras

241

Ch. 5. FILTERS, IDEALS AND 19-CONGRUENCES

247

$1.&filters, &ideals and 19-congruences

247

$2. Prime filters

265

$3. The Cignoli problem

279

Ch. 6. REPRESENTATION THEOREMS AND DUALITY FOR LU KASl EW I CZ-M OlSl L-ALGEB RAS

285

$1.T h e representation theorem of Moisil $2. Applications of t h e representation theorem $3. T h e representation of Lukasiewicz-Moisil algebras by continuous functions

285 300 317

$4. T h e representation o f Lukasiewicz-Moisil algebras by Moisil fields of sets

326

$5. The Stone duality o f &valued Lukasiewicz-Moisil algebras

$6.T h e

332

Stone duality o f &valued Lukasiewicz-MoisiI

algebras w i t h negation

337

$7. The Priestley duality o f d-valued Lukasiewicz-Moisil algebras

341

$8. T h e Priestley duality o f d-valued Lukasiewicz-Moisil algebras w i t h negation

345

$9.The representation o f n-valued Moisil algebras by three-valued algebras

Ch. 7. CATEGORICAL PROPERTIES OF LU KASlEW I CZ-MOISIL ALGEBRAS

349

359

$2. Injective Lukasiewicz-Moisil algebras and injective hulls

359 371

$3. Free Lukasiewicz-Moisil algebras

380

$4. Epimorphisms and projective Lukasiewicz-Moisil algebras

389

$1.Some adjoint functors

Table of contents

xi

$5. Direct sums $6. Free Post and m-Post extensions

397 410

Ch. 8. MONADIC AND POLYADIC LU KAS I EW ICZ- M 0IS IL A L GEB RAS

417

$1. Monadic Lukasiewicz-Moisil algebras

417

$2. Modal operators on Lukasiewicz-Moisil algebras $3. A construction of a three-valued Moisil algebra from a monadic Lukasiewicr-Moisil algebra $4. Polya d ic L ukasiew icz- MoisiI a Igebras

428

Ch. 9. LUKASIEWICZ LOGICS

459

$1.The Wajsberg axiomatization of the three-valued Lukasiewicz logic $2. The Cignoli axiornatization of the n-valued Lukasiewicz logic $3. The d-valued propositional calculus $4. Analytic tableaux for the 6-valued propositional calculus $5. The d-valued predicate calculus $6. Kripke-style semantics for 8-valued predicate logics

441 452

459 471 487 505 513 531

Appendix APPLICATIONS T O SWITCHING THEORY

539

REFERENCES

551

AUTHOR INDEX SUBJECT INDEX

575 579

This Page Intentionally Left Blank

...

Xlll

LIST OF SYMBOLS

Elements: 0, 1 3 ci 117,166 di 168 2+, 2- 3 a: 24

532 k

It

534

f

Operations: Relations:

5, <, 1,> 1

k

489,501

k

+

# 3 E 8,75 N

+ , ++

262,350

M

313,365,464

A, V 4

v

N F- 72

A,

AL 38

+ 350,459,460,476

3

*

201

K , 48 ker f 37,269

3

210,308

m o d F 38

e

pi 349

H 478

p

251

t 461,480 /= 467 I- 490,491,522 k

k

488,519

k

I= (4 k

536

210 210

Pi

106

n,LI 76 3 43,417 'd 44,418 ! 318 Fij 472 J; 478 N 22,93,102,459

xiv

List of Symbols

Sets and algebras:

AB 183 At(L) 235 Af, At 326

L, 117 L,j 107,113 L,j 113

L["] 183

B [ q 107

LIF 38,254

B(X) 317 B X ( w ) 516 B, 429

MR 179 PFl(L) 39 PF129(L) 265 P ( V ) 501

C ( L ) 24,26

C, 308 Cs(X) 263 CT(X) 317 D ( B ) 111

D n ( L ) 304 Ds(L) 216,304

Ds(L)

216

F * 248,265

F(X", L) 454 F X ( w ) 516

Fl(L) 33,249 F119(L) 249 FK(c)381 I 105 I(L) 196

IdA, IdK 63 IdL 84 Hom(A,B) 58 J 105 k-Axm 490 ker p 269

Kerf 270 320 i; 353

P,j(X,R) = P,j 516 Predd(X, R ) 513

Prop(V) 487 Rad(F), Rad(L) 273

Rg(L), Rgo 216 S ( L ) , S'(L) 302 Spec(L) 39 S, T, 472 X+, X - 8

[W,(XI

32

[ X ) , 249 X ( w ) 513 D(A) 326 C(X,L) 399

M ( S ) 528 PF(L) 329 O ( L ) , Oo(L) 251 s.19-rn. 534 s.s.19-rn. 535

Functions: c 400

d 110 fa

317

List of Symbols ker 251,269

Ker 270

-

mod, mod 251

P* 429 PL 169 r , 479

x

xv

73 St 84 St6 332 StN6 337 Top 73 3LM6, VLM9, 3VLM6 418 Set

399 Functors:

Classes and categories:

A M n 376 B 74 BS 90 DO1 38,74 1st 92 I< 209 K, 308 L M n 121 LMn' 369 LM6 121 LMB' 402 LMNd 122 LMNO' 402 Mg 74 Mn 122 M o n B 425 M o n d 418 Pn 175 Pr 95 PrMg 102 PrNd 345 PrN6' 402 Prd 341 Prd' 402 P9 360

C 359,360,453 c3,

cv, c3v 425

F, 355 G 96,104,343,347 H 98,104,343,347 I< 88,90,94,335,339 Spec 88,90,93,334,338 T 359,360 U 368,409 N a t u r a l transformations:

90,91,94,335,340 t , r 101,104,344,348 s, o

This Page Intentionally Left Blank

1

CHAPTER 1 LATTICES, UNIVERSAL ALGEBRA AND CATEGORIES

The first t w o chapters of this book establish all prerequisites necessary

to t h e study of Lukasiewicz-Moisil algebras. Most important is the latticetheoretical background, presented w i t h full proofs: $1deals w i t h lattices i n general, $$2,4 are devoted to the classes of lattices relevant t o our theme, while the theory of filters, ideals and congruences and t h e representation theory i n terms o f topological dualities are treated in $3 and in Chapter 2, respectively. T h e other tools, provided by universal algebra and category theory, are dealt w i t h in t h e last t w o sections of this chapter.

$1. Posets and lattices The main topics treated in this section are: lattices as partially ordered sets and as algebraic systems, morphisms, sublattices, closure operators, the

MacN eille corn pletion. 1.1. Definition. A p a r t i d y ordered set or poset is a couple (f',

5 ) where P is a non-empty set and 5 a relation of partial order on P, i.e. 5 is reflezive (x 5 x) antisymmetric (x 5 y and y 5 x +- x = y) and transitive (x 5 y and y 5 2 * x 5 2). T h e relation

z

<

o f strict partial order associated to

< y -e(x 5 y and x # y).

Then

2

is defined by

< is transitive and fulfils x < y + y f

f x for every x ; moreover, x 5 y Thus in a poset we can use both relations 5 and

which implies

5

tj

<

(x <

y or

2,

x = y).

as well as t h e dual

relations 2 and >, defined by x 1 y H y 5 x and x > y (j y < x, respectively. The duality principle for posets enables one to duplicate

2

Lattices, universal algebra and categories

theorems by interchanging

that

2

5

with

is also a partial order and

2

and

> a strict

< with >; it is based on the fact partial order. We say t h a t

( P ,2)

is the poset dual t o ( P ,I). In the sequel we shall denote various partial orders by the same symbol < whenever

this can be done without risk of confusion. A similar convention

will apply t o the operations and constants dealt with in this book.

1.2. Examples.

The direct product of a family of posets (Pi,
(

II P;, I), iEI

where

I is defined componentwise, i.e.

If t h e index set I is also endowed with a total order (cf. Definition 1.5), then the lezieographic product o f the above family is the poset ( II P , , < D ) , i€I

where

If, moreover, PinPj = 8 whenever i family is the poset

(

# j,then t h e ordinal s u m of t h e above

U P,, I),where ieI

(1.3)

xIy@(3ix

5i

y ) or ( 3 i 3 j i < j and x ~ P and i YEP,)

1.3. Example.

If ( P ,5 ) is a poset and X an arbitrary set, the set Px = {f I f : X -+P } is endowed with the partial order defined pointwi3e, i.e.

1.4. Definition.

Let (P ,<) be a poset and S 2 P . By a lower bound (an upper bound) of S is meant any x E P such t h a t x 5 s (s 5 x) for every s E S. If

3

Posets and lattices

x of S belongs to S then z is said to be the least or f i r s t (greatest or l a s t ) element of S . If t h e set P itself has a least (greatest) element, this is usually denoted by 0 (by 1) and called the zero ( o n e ) of P . A poset having both 0 and 1 is said to be bounded. The infimum ( s u p r e m u m ) of S is its greatest lower bound (least upper bound): g.1.b. S or inf S or A S or A z (1.u.b.S or s u p s or V S or V x) for

the lower bound (upper bound)

X€S

XES

short. A m i n i m a l ( m a x i m a l ) element o f

S is an element rn E S such that there is no element s E S satisfying s < m (rn < s ) or, equivalently, such that for every s E S : s 5 rn + s = rn (rn 5 s + s = rn). For a given subset S o f

P some or all of the above elements may not

exist. O n t h e other hand t h e infimum (supremum) is unique whenever it exists. Moreover, if the least (greatest) element o f S exists, it is unique and is also the infimum (supremum) of S and t h e unique minimal (maximal) element o f S.

1.5. Definition.

x 5 y or y 5 x (or, equivalently, if 5 < y or y < 2 or z = y); otherwise we say t h a t x and y are incomparable and write x # y. A totally ordered s e t or t o s e t or c h a i n

T w o elements z, y of a poset are said t o be comparable if

is a poset in which every two elements are comparable.

1.6. Examples. T h e lexicographic product and t h e ordinal sum of a family o f tosets is a toset

(cf. Example 1.2).

1.7. Definition. Suppose the elements z, y of a toset fulfill z

and there is n o element z

x < z < y . Then we say t h a t y is t h e successor of x and write or, equivalently, t h a t x is the predecessor of y and write x = y - .

such that

y = I+

The successor (predecessor) o f an element is unique provided it exists,

Lattices, universal algebra and categories

4

because if both y and y‘ are successors of

x then e.g. x < y 5 y’ which

is

possible only for y = y’. Also,

(1.4)

x 5 y H x- < y H x < yt

x 5 y and x- # y would imply x 5 y 5 x-, while would imply x- < y < 2. for

5-

and

x $y

1.8. Definition.

A m e e t semizattice (join semilattice) is a poset ( L , 5 ) such that every two-element subset { x , y } has an infimum (a supremum); this element is usually denoted by

x A y (by x V y) and called t h e m e e t ( j o i n ) of x and

y. A Zattice is a poset which is both a meet semilattice and a join semilattice. The following identities are valid in a meet semilattice and in a join semilattice, respectively.

I

(1.5’) (1.5”)

xAy=yAx x V y =y V x

(1.6’) (1.6’)

(x A y) A z = x A (y A z ) (x v y) v z = x v ( y v z )

(1.7’) (1.7”)

xA x =x

x V x =x

I

(commutativity);

I

(associativity);

(idempotency );

and in a lattice

(1.8’) (1.8’)

x A (x V y) = x x V (x A y) = x

(absorption).

Conversely, suppose A, V are binary operations on t h e non-empty set

L. If A fulfils (1.5’)-(1.7’) then L can be made into def ining

(1.9’)

x5y

H x A y =x

and if V fulfils (1.5”)-(1.7”) definition

a meet semilattice by

then

L becomes a join semilattice via the

Posets and lattices (1.9”)

5

5

< y *x v y =y

In other words, meet semilattices and j o i n semilattices can also be defined as algebraic systems satisfying axioms (1.5’)-(1.7’) and (1.5”)-(1.7”), respectively, while lattices can be defined as algebraic systems ( L ,A, V) satisfying (1.5)-(1.8) (as a matter of fact idernpotency follows from absorption). T he lattice ( L , V , A ) is called the dual o f (L,A,V). T he following properties are useful and easily proved:

(1.10’)

z < x A y ~ z < xand z < y ,

(1.10”)

x V y < z ( ~ x < z and y < z ,

(1.11’)

z < yJxA z< y A z,

(1.U’)

x
(1.12’)

x < y and t < v + x A t < y A v ,

(1.12)

x s y and t < v + x V t < y V u ,

(1.13)

~ = y H x A y = x V y .

T he elements 0 and 1, provided they exist, fulfil

(1.14’)

x A 0 =0 ,

(1.14“)

2

(1.15”)

v 1= 1 , x A 1= z , x V 0 =x ,

(1.16’)

xAy =lH z=y =l,

(1.16’)

xvy=o*x=y=o,

(1.15’)

and each of t h e above properties (1.14’)-(1.16”) is characteristic for the corresponding distinguished element. Thus e.g. if L is a lattice and

a

*

2

xAy =

= y = a then since a A (z V a) = a by (1.8’) it follows t h a t x V a = a

for every x, i.e. a = 1.

According to a general usage, posets, semilattices and lattices will sometimes be designated only by their underlying sets.

6

Lattices, universal algebra and categories

1.9. Examples. Suppose Li, i E I , are lattices (meet semilattices, join semilattices); then so are their direct product and their ordinal sum. For the direct product the operations are defined componentwise, i.e.

(1.17')

( 2 i ) i c l A (9i)iG.I = (zi

(1.17")

(2i)icl V ( y i ) i c l

Ai

Yi)icl

7

= (xi V i Yi)icl

7

while for the ordinal sum t h e meet (join) is defined by (1.9') (by (1.9')) if z

E L; and y E Lj where i

# j , and coincides w i t h

the meet (join) in Li if

both z,y E L; (cf. Example 1.2).

1.10. Example.

If L is a lattice (meet semilattice, join semilattice), so is t h e poset Lx constructed in Example 1.3. Here the meet and j o i n are defined pointwise, 1.e.

Lattices can be also characterized as posets in which every finite nonempty subset has an infimum and a supremum. Therefore the following concept is stronger than t h a t o f a lattice.

1.11. Definition. Let rn be an infinite cardinal number. A n rn-complete lattice is a poset in which every subset o f cardinality at most rn has an infimum (also called

m e e t ) and a supremum (also called join); cf. Definition 1.4. B y a complete lattice is meant a lattice which is rn-complete for every rn, or equivalently, a poset in which every subset has a meet and a join.

1.12. Remark. In particular A 0 = 1 and V 0 = 0. It is useful to note t h e somewhat surprising fact t h a t the existence of meets for arbitrary subsets, or equivalently the existence of joins for arbitrary subsets, is a sufficient condition for a poset to

7

Posets and lattices

be a complete lattice. The reason is t h a t the 1.u.b. (g.1.b.) o f a set S is the

g.1.b. (1.u.b.) of the set o f all upper bounds (lower bounds) of S. From a practical point of view we check separately t h e existence of 1and of meets for arbitrary non-empty sets, or the dual conditions.

1.13. Examples. Every finite lattice and t h e lattice

( P ( X ) ,c) of

all subsets of a set

X are

typical examples of complete lattices; in t h e latter lattice meets and joins coincide w i t h set-t heoretical intersections and unions, respectively. Si milarly, the family of all countable subsets of an infinite (uncountable) set is an Nocomplete lattice (which is not complete).

1.14. Definition.

A closure operator o n a poset P is a mapping : P t P such t h a t x 5 x', ''2 = x' and x 5 y + x c 5 y'. A Moore f a m i l y o f a poset P is a non-empty subset M C_ P such t h a t for every x E P the set {y E M 1x 5 y} has a least element.

1.15. Remark. There is a bijection between t h e closure operators and t h e Moore families of

P: it maps a closure operator to t h e Moore family { z E P I x = xc} = {x' I x E P } and conversely, a Moore family M is sent to the closure operator ' defined by xc = least element of t h e set {y E M I x 5 y}. See

a poset

e.g. Balbes and Dwinger [1974], Theorem 2.4.11.

1.16. Remark. In view of Definition 1.14 and Remark 1.12, a Moore family plete lattice

L

is itself a complete lattice

the order inherited from

(1.19')

M of a com-

( M ,infM, supM) w i t h respect t o

L:

infMX = inf X

,

(1.19') supMX = inf {y E M I x 5 y (Vx E X)} , for every X

c M . A n alternative characterization makes use of t h e operator

Lattices, universal algebra and categories

8 from Remark 1.15: SUPMX = (SUPX)"

(1.20)

1.17. Examr.de.

X c P define X + = { y E L I x 5 y V x E X} 5 x V x E X}. Then

Let P be a poset. For every

I

and X - = {y E L y

(ii) X 5 Y

+ Yf

(iii) the maps

+-

s X + & Y-

and -

X-;

are closure operators.

Property (i) is easily checked. From X+ therefore X

Y-

cY

c X + and (i) we get X C X + - ,

implies X C_ Y+- hence Y+ C X + by (i) and similarly

X-. Now (ii) implies th a t X C_ Y

+ X + - c Yt-.

Finally f r o m

c

X+X + - we obtain X+ C X t - + by (i),hence X t - + - C - X+- by (ii), therefore X + - + - = X + - . Similar proof for t h e second half o f (iii). 1.18. Definition. Let P and

P'

be posets. A mapping f

:

P + P' is said to be iso-

t o n e or increasing ( a n t i t o n e or decreasing) if z 5 y f(x) 5 f(y) (if x 5 y + f ( y ) 5 f(x)). T h e mapping f is called an i s o m o r p h i s m (a dual i s o m o r p h i s m ) provided it is a bijection and b o t h f and titone). In this case the posets write P

f-'

are isotone (an-

P and P' are said to be isomorphic and we

P'.

1.19. Comment. If f is an isomorphism so is

f-l;

as a matter o f fact the relation of being

isomorphic is an equivalence relation in th e class of all posets. Since isomorphic posets can be viewed as being identical, it is a natural idea to associate with each class of isomorphic posets a symbol called the order t y p e of all the posets in th a t class. As a matter of fact a theory has been developed which associates order types to classes of isomorphic tosets. In this theory

Posets m d lattices each n E

lN

9

is chosen as the order type of the set (1, ...,n} endowed with

the usual order. Then the class of all order types is equipped with an order relation which extends the order of lN.

1.20. Definition. Let L and L’ be meet (join) semilattices. A mapping f : L -+ L‘ is said to be a meet (join) homomorphism if f ( x A y) = f(x ) A f ( y ) (if f(x V y) = f(x) V f(y)). Meet homomorphisms and join homomorphisms are also designated under the common name of semilattice homomorphisms. If L and L’ are lattices, then f : L + L’ is called a lattice homomorphism (lattice dud homomorphism) if it is both a meet and a join homomorphism

(if f(. A Y) = f ( 4 v f ( Y ) and f ( x v Y) = fb)A f(Y)). 1.21. Definition.

Let

P

A

P’ be posets and f : P -+ I”. If S is a subset of P such that x exists ( V z exists) then f is said t o preserve the meet A x and

XES

XES

provided

A

XES

f ( x ) exists and

f( A

XES

join

V x XES

x) =

XES

provided

V f(x)

exists and f (

XES

L e t rn be an infinite cardinal. Then

A

f(x) (to preserve the

XES

V XES

f

x) = V

f(x)).

XES

is called an m-complete homo-

morphism or simply an rn-homomorphism provided it preserves all existing meets and joins of sets of cardinality a t most m . We say that f is a complete

homomorphism if it is an m-homomorphism for every m, i.e. if is preserves all existing meets and joins. The map f is said to be an rn-complete dual homomorphism (a complete dual homomorphism) provided it is an rn-complete homomorphism (a complete homomorphism) from P t o the dual of PI. If P and P’ are lattices it may be convenient t o replace, in the above terms, t he word “homomorphism” by the words “lattice homomorphism” (this is consistent with Definition 1.21).

Lattices, universal algebra and categories

10 1.22. Remark.

Every semilattice (lattice, m-complete, complete) homomorphism is isotone and every semilattice (lattice, m-complete, complete) dual homomorphism is antitone, because

and dually. In the case of mappings between tosets, isotone (antitone) map-

pings coincide with lattice homomorphisms (lattice dual homomorphisms) by an argument similar t o t h e above one.

A t this point it seems a good idea to introduce the concepts of meet, j o i n , lattice, rn-complete lattice and complete lattice isomorphism, according t o a general usage in algebra. Thus e.g. a meet isomorphism is a bijective

meet homomorphism, which implies t h a t th e inverse mapping is also a meet homomorphism; etc. However, the novelty is illusory: 1.23. Proposition. (i)

Every isomorphism in the sense of Definition 1.18preserves all ezisting meets and joins.

(ii) T h e concepts of meet, j o i n , lattice, m-complete lattice and complete lattice isomorphism reduce t o the concept of isomorphism in Definition 1.18, while the dual concepts reduce to the concept of dual isomorphism in Definition 1.18.

Proof. (i) Suppose f : P + P’ is a poset isomorphism, S C P and a =

A x XES

exists. Then for every

5

E S, from a

Ix

we get f(a)

5 f(x). If

b‘ E P’ fulfils b’ 5 f(x) for every x E S then since b’ = f(b) for some b E P , we apply f-’ t o f ( b ) 5 f(x) and get b I 5 for every 2 E S,

Posets and lattices

11

(ii) Take e.g. a meet isomorphism f : L -+ L'. Then f and f-' are meet homomorphisms, hence they are isotone, therefore

f

isomorphism. The converse follows from (i).

is a poset 0

In view of Proposition 1.23 th e terms isomorphism and dual isomorp-

h i s m can be unambiguously used without further specification, as well as the notation L

L'.

1.24. Definition.

P into a poset P' is a mapping f : P + P' such that the corestriction f : P -+ f ( P ) is an isomorphism. An embedding of a poset

1.25. Remark.

The map f is an embedding iff

because (1.21) implies f(x) = f(y)

+ I = y.

Necessity is trivial.

According t o a general usage in mathematics we are interested in embedding an arbitrary poset into a poset having a much more particular structure.

A typical example is the next theorem, known as the MacNeille completion by cuts: 1.26. Theorem. For every poset P , the m a p (1.22)

z

H

I(.

=

(2

EP

12

5 x}

is an embedding of P into the complete lattice

M

(1.23)

={X

P I X = X+-}

(6. Example 1.17). This embedding preserves all meets and joins existing in P . Proof.

M

is a complete lattice by Remark 1.16. For every z E P we have

Lattices, universal algebra and categories

12

{z}+ =

{t E P 1 z 5 t } , hence (z] = { z } + - E M . Clearly z 5 y

H (z] C_

(y], therefore the map (1.22) is an embedding by Remark 1.25. For the second part of the theorem we compute meets and joins in

1.16 w i t h ’ M =

M

and

M

via Remark

L = P(P).Note that inf and sup in L are

set-

theoretical intersection and union, respectively, hence it remains t o prove that

(1.24”)

provided the left-hand sides exist. But (1.24’) is a mere translation of the

5 z (Vz E X ) . Further set u = V X and Y = U{(z] 12 E X } . Then X Y , hence Y + E Xt and since a E Xt it follows t h a t every y E Y+ fulfils a 5 y, therefore z 5 y for every y E Y + and every z 5 a ; in other words ( u ] C Y + - . On t h e other hand every y E Y fulfils y 5 z for some z E X , hence y 5 a ; it follows that a E Y + ,therefore 0 Yt-E { u } - = (u]. Thus ( u ] = Y + - completing the proof. obvious property z

5

AX H z

1.27. Definition. Every meet (join, lattice, dual lattice) homomorphism of the form f : L -+

L is called a meet (join, lattice, dual Eattice) endomorphism. Every (dual) isomorphism of t h e form f : L -+ L is called a (dual) automorphism. 1.28. Definitio n. An inuolutive operation or simply an involution of a set X is a mapping

f :

X

+ X such that f ( f ( z ) ) = 2 for all z E

X.

1.29. Remark.

A mapping f f-l

:

X

--f

X

is an involution if and only if f is a bijection and

= f. An increasing (a decreasing) involution on a toset is an automorp-

hism (a dual automorphism).

Posets and lattices

13

From now on we shall often use notations like f z instead of f(z). 1.30. Proposition.

Let N be a decreasing involution on a meet semilattice ( L ,A) (join semilattice ( L ,V ) ) . If one defines V b y (1.25’)

z V y = N ( N z A Nu)

then L becomes a lattice ( L ,A, V ) in which

holds. (If one defines A b y (1.25”) then L becomes a lattice ( L , A , V ) in which (1.25’) holds). Proof. Suppose ( L ,A) is a meet semilattice and show that N ( N z A Ny) = 1.u.b. { z , y } . But Nz A Ny 5 Nz implies z = N N z 5 N ( N z A Ny) and similarly y 5 N ( N z A N y ) . Further if z 5 z and y 5 z then N z 5 N z and

N z 5 N y , hence N z 5 Nz A Ny, therefore N ( N x A Ny) 5 N N z = z . Thus ( L ,A, V ) is a lattice and (1.25”) follows from Remark 1.29.

0

Let further ( L ,A, V ) be a lattice. According to Definitions 1.27 and 1.28, an involutive dual endomorphism of L is a mapping N : L --t L such that

,

(1.26)

NNz =x

(1.27’)

N ( z A y) = Nz V Ny

,

(1.27”)

N ( x V y) = N X A Ny

.

1.31. Proposition. Let ( L , A , V ) be a lattice and N : L equivalent:

(i)

L. The following conditions are

N is a n involutive dual endomorphism;

(ii’) N fuIfils (1.26) and (1.27’); (ii”) N fulfils (1.26) and (1.27”);

Lattices, universal algebra and categories

14

(iii’) N fulf;Zs (1.25’); (iii”) N fulf;ls (1.25”). Proof.

(i) j (iii’): Immediate. (iii’) j (ii’): N N z = N ( N s A N c ) = x V z = x, then NxVNy = N(”z A NNy) = N ( e A y). (ii’) j (i): N z A N y = N N ( N c h N y ) = N(NNsVNNy)= N ( z V y ) . 0

The proof is completed by duality.

1.32. Proposition. L e t P be a p o s e t and d :

P

+ P a decreasing m a p p i n g .

(i) If P h a s least e l e m e n t 0 (greatest e l e m e n t 1) a n d d i s surjective, t h e n dO is greatest e l e m e n t ( d l i s least e l e m e n t ) . (ii)

If P =

{al, ..., anel} with al

da; = a,-i

< ... <

a n d d is injective, t h e n

(i = 1 , ...,n - 1).

Proof.

(i) Every x E

P

can be written z = dy for some y E P , hence 0

implies z = dy

y

5 do.

(ii) It follows from the hypotheses that the single possibility is da,-l = al, an-1.

5

< ... < daz < d a l , hence = az, ..., da2 = dul = 0

The next point of this section is the general concept o f subalgebra applied t o lattice theory.

1.33, Definition.

A subsemilattice of a meet semilattice ( L ,A) (join semilattice ( L ,V)) is a subset S C L such that c,y E S + z A y E S (c,y E S + s V y E S).A sublattice of a lattice ( L ,A, V) is a subset S L that is both a subsemilatV). tice of ( L ,A) and of (L,

15

Posets and lattices

1.34. Remark. A subsemilattice (sublattice) S of L is itself a semilattice (lattice) with respect t o the restriction(s) t o S of the operation(s) of L . However, a subset S 5 L which is a semilattice (lattice) with respect t o t h e partial order inhe-

L

rited from

L ; see e.g. be a subsemilattice of (L,V).

need not be a subsemilattice (sublattice) of

1.16 in which M need not

Remark

We conclude this section with an alternative description of the MacNeille construction in Theorem 1.26. This variant, obtained with the aid of an isomorphism, justifies t h e term “completion by cuts” by analogy with the Dedeking construction of the irrationals as cuts of the set of rational numbers. Consider again the completion

M

such that

M

P C L. The bijection y

of P and construct a bijective copy :

M

L of

+ L (where ~ ( ( x ]= ) x) makes

L into a lattice isomorphic t o M via th e definition y ( X ) 5 y ( Y ) e X C_ Y. For every x E L set A, = ( a E P 1 a 5 x}. 1.35. Lemma.

FOT every X E M and x E L:

(i)

y ( X ) = supLX ;

(ii) cp-’(s) = A, ; (iii) z = supLA, ; (iv)

A,+ = { b E P 1 z

Ib}.

Proof. The obvious equivalence ( u E

X+-E M (1.28)

A- e ( a ] C A - ) applied t o X =

yields

X = U

(u]

,

aEX

(1.29)

X

= {u E

P I ( a ] C X} .

p(X) = x. Then we obtain (i) by applying y to (1.28). Further, since ( u ] C X @ u 5 x we can write (1.29) in the form X = A,, which i s Set

Lattices, universal algebra and categories

16

(ii). From (i) and (ii) we obtain (iii). Finally we prove (iv): if x 5 b E P then b E A,+ by the definition of A,; conversely, if b E A,+ then b E P and 0 5 5 b by (iii). 1.36. Lemma. If P = T is a chain then A, U Af = T for every x E L. Proof. If a E T

- A,

then a $ x, hence x

< a , therefore a € Af

1.35(iv).

by Lemma 0

1.37. Definition. A cut of a chain T is a pair of sets ( A , B ) such that

(i) a

< b (Va E A ) (Vb E B ) ,

(ii) A U B = T , (iii) A has no sup in T and (iv)

B has no inf in T.

1.38. Proposition. Let L be the MacNeille completion of a chain T . Then: (a) L is a chain;

(b) the map x H (A,, A f ) establishes a bijection between L - T and the cuts of T , in which the inverse image of a cut (A, B ) is x = supLA = infLB, characterized b y a < 2 < b (Vu E A ) (Vb E B ) . Proof. (a) Let x,y E

L. If A, C A,

then

x 5

y. Otherwise there exists a E

A,-A,, hence a E A, and a E A: by Lemma 1.36, therefore y 5 a 5 z. (b) Let (i)-(iv) stand for the properties in Lemma 1.35.

L - 2'. For every z 5 b by (iv); but

First we prove that (A,,Af) is a cut for every x E

a E A, and b E A: we have a

5x

by (iii) and

17

Posets and lattices

a , b E T therefore a < x < b. Further A, U A: = T by Lemma 1.36. The existence of a = supT A, would imply t h e contradiction x = a E T by (iii) and Theorem 1.26. The existence of b = infTA: would imply a 5 b (Va E A,) (because a 5 b‘ (Vb‘ E A:)) i.e. b E A:, hence b would be the least element of A: i.e. b = supT A,, which is impossible. The map x

H

(A,,Af)

is injective by (ii).

If (A,B) is a cut set 2 = supLA. Then 2 E L - T . If a E A, then a E A, because otherwise a E B hence a’ 5 a (Va’ E A ) , therefore 2 5 a , consequently x = a E T , a contradiction. Thus A, E A and since the converse also holds by (iii) it follows that A = A,. If b E B then b E T and a < b (Va E A ) , hence 5 5 b by (iii),therefore b E A,+ by (iv); we have thus proved that B C A .: To prove the converse take b E A: i.e. b E T and x 5 b by (iv); then b E B because otherwise b E A hence b 5 2 therefore x = b E T , a contradiction. Therefore Af C B , consequently B = A:. We have thus proved that (A,B) = (A,,A,+) for 5 E L - T given by x = supL A, = infL A:; clearly a < x < b (Vu E A) (Vb E B ) .

L satisfies a < y < b then by x = supL A and y 5 x by x = infL B therefore y = x. 0

Finally if ( A , B )= (A,,AZ)

x 5y

is a cut and y E

Lattices, universal algebra and categories

18

$2. Distributive lattices, De Morgan and Boolean algebras T h e s t a c t u r e s indicated i n t h e title are successive specializations of lattices. We introduce them here in view of their importance and as prerequisites t o t h e study of Lukasiewicz-Moisil algebras, which are situated between De Morgan algebras and Boolean algebras.

2.1. Proposition.

The following identities and implications are equivalent in a lattice: (2.1’)

x A ( y V z ) = (x A y ) V ( a : A z ) ;

(2.1”)

x V (y A Z) = (X V y) A (X V Z) ;

(2.2’)

z A (y V z )

(2.2”)

x V (y A z ) 2 ( x V y) A ( x V z ) ;

(2.3)

(x A y ) V (y A Z) V (x A z ) = (z V y ) A (y V z ) A ( x V z ) ;

(2.4)

x Az 5 y &x

(2.5)

xAz 5 y A z &xVz 5 y V z j x

(2.6)

x A z =~

5 ( x A y) V ( x A z ) ;

I y V z J x 5 y;

A &z x V z = y V z

j

5 y;

x =y.

Proof.

(2.1’) H (2.2’): For (x A y) V (X A z ) 5 x A (y V z ) in any lattice. (2.1’)

J

(2.3): Calculate successively

(x

v 9) A (Y v 2) =

((x

v y) A Y) v

((x

v Y) A 2) =

= yV(zAz)V(yAz)=yV(xAz) (x V y) A (y V

Z)

A

(X

,

V Z ) = ((x V z ) A y) V ((x V z ) A z A z ) =

= ( x A y) V ( z A y) V (x A z ) .

(2.3)

j

and x V z

(2.4): Suppose z A z 5 y and z 5 y V z. Then z A z I yA z

5 y V z , hence

19

D i s t r i b u t i v e lattices, D e Morgan and Boolean algebras

=

(2

A y) V (x A z ) V (y A Z) =

-

= ( ~ A Y ) V ( Y A ~ ) I Y

(2.4)

+ (2.5):

If x A z 5 y A z and x V z 5 y V z then x A z

5 y and

x
(2.5) =+ (2.6): Obvious. (2.6) +- (2.1’): We first prove that the lattice is modular, i.e. (2.7’)

x

(y V Z) = (x A y) V z

,

(2.7”)

x 5 z + x V (y A 2) = (x V y) A z

.

2 z jx A

Take x, y, z such that x 2 z and set a = x A (y V z ) , b = (x A y) V z , c = y.

Then b 5 a, hence bAc
1.e.

aAc=bAc.

Also

aVc> bvc

1.e.

aVc= bVc.

2 z V y =y V z V y 2 a V c ,

Therefore a = b; i.e. (2.7’) holds and by duality we get (2.7”). Now take arbitrary 2, y, z and set a = ( X V y) A (Z V

b = (9 V z ) A

(x A y)) = ((x V y) A z ) V (x A y) ,

(X V (y A 2))

= ((9 V z ) A x) V (y A Z)

,

where we have used (2.7), and c = y. Then ~ A c y= A ( z V ( x A y ) ) = ( y A z ) V ( z A y ) ,

bAc= yA(zV(yAz))

=(yAx)V(yAz),

hence a A c = b A c and similarly a V c = b V c . It follows that a = b, therefore x A b = x A a i.e. x A (y v z > = x A ( z v (x A y)) = (x A z ) v (x A y). Thus (2.1’), (2.2’) and (2.3)-(2.6) are equivalent. The proof is completed by duality. 0

Lattices, universal algebra and categories

20 2.2. Definition.

A lattice satisfying the equivalent conditions i n Proposition 2.1 is said

to be

distributive. 2.3. Example. Every toset is a distributive lattice. Here

tA

y = min(z, y) and t V y =

rnax(z, y), i.e. the least and greatest element of t h e set

(5,y},

respectively.

As a matter of fact all th e lattices studied in this book are distributive. However, non-distributive lattices do exist, e.g.:

1) t he pentagone, i.e. th e five-element lattice (0, a , b, c, l} where a # b and b # c while a < c; 2) t he diamond, i.e. the five-element lattice having three pairwise incomparable elements;

3) t he lattice of normal subgroups of a group, where t h e partial order is set-t heoretical inclusion. Note t hat t he pentagone is not even modular, i n contrast with t h e other t w o lattices. T h e next tw o short characterizations of distributive lattices t u r n o u t t o be very useful.

2.4. Proposition (Sholander [1951]).

Let L be a set and A, V : L2 L. Then ( L ,A , V) is a distributive lattice i f and only if the following identities are satisfied: --f

V 8) =t

,

(2.8)

2

A

(2.9)

2

A (y V z ) = ( z A

(Z

t)V

(y A z) .

Proof. Necessity is trivial. To establish sufficiency we first prove idempotency in the following steps:

21

Distributive lattices, De Morgan and Boolean algebras z A z = (z A z) A

((2

A z) V (z A z)) = (Z A z) A

5

,

zAz = z A ((zj\z)V(~Az)) =

= ((z A X ) A Z) V ((z A

z v 2 = ( X A X ) v (ZA 3) = 5

Z)

A z) = (Z A X ) V (z A Z) = z

,

.

Further we prove commutativity and the dual absorption law in four steps:

z A Y = Z A ( Y V Y ) = ( Y A Z ) V ( ~ A Z )= y~

5 ,

(zAy)Vz =(yAz)V(a:Az) =z A ( z V y ) =z xA(yVz)=

=

,

(zAz)V(yAz)=~V(zAy) = 2

V ((z A y) A ((z A y) V

3))

=

= (z A X ) V ((z A y ) A x ) = = ( z A ((zAy)Vz) = z A z = z , z V y = (z A (y V z)) V (y A (y V z))

=

= (yVz)A(yVz)=yVz. The steps for associativity are as follows: zA((zVy)Vz) z

v (Y v 4

(ZV y) V

=

= (zA(zVy))V(sAz)=zV(zAz)=z,

(.

A

v

(2 A

((. v Y) v 2)) v ((Y A ((Y v 4 v 2)) v

v Y) v 2)))

=

v Y) v 2)) v

(((z

((z

=

(. A ((.

=

(bv Y) v z ) (. v (9 v 4)

v Y) v z )

(Y v 4 ) =

9

z = z V (3 V Z) = ((z V y) V z) A =

A

((3

(Z V (y V z))

=

V y) V z ) A (z V (y V z)) = z V (y V z )

and the other associative law follows dually.

0

Lattices, universal algebra and categories

22

2.5. Proposition (Ferentinou-Nicolacopoulou [1968]). Let L be a set, 0 E L and A,V : L2 + L. Then (L,A,v,O)i s a

with zero if and only if it satisfies identities (2.8) and d i s t r i ~ u t i ~lattice e (2.10)

x A (y V z ) = ( z A (x V 0)) V (y A (x V 0))

.

Proof. Follows from Proposition 2.4 provided we succeed to show that xV0 = x. But x V z = (xA(xVO))V(XA(XVO))

=zA(zV~)=z,

XAX=XA(XVX)=X,

xAy=xA(yVy)

= (yA(zV0)) V (yA(zV0))

=

= yA(zVO), x

v o=(zvO)A(X

vO) = X

A ( X

vO) = 2 .

2.6. Defi n ition. An algebra ( L ,A, V, N , 0 , l ) where ( L ,A, V, 0 , l ) is a bounded distributive lattice and N an involutive dual endomorphism of L, is said to be a D e Morgan algebra. N is called the negation or the involution of L . See Moisil (19351 and Kalman [1958]. See 1.27-1.31 for the previously defined concepts and alternative characterizations of N . See also Propositions 2.9-2.10. 2.7. Example. , U, 0,X ) Let X be a set and f : X + X an involution. The lattice ( P ( X ) n, endowed with the operation N X = X - f ( X ) , is a De Morgan algebra.

2.8. Example Every toset endowed with a decreasing involution is a De Morgan algebra; cf. Example 2.3 and Remark 1.22. As a matter of fact many lattices studied in this book are particular cases

of De Morgan algebras.

Distributive lattices, De Morgan and Boolean algebras 2.9. Proposition (Maronna [1964]). Let L be a set, A : L2-+ L and N : L (2.11)

tVy

=N(Nt A Ny)

-+

23

L. Define

.

T h e n ( L , A , V ) is a distributive lattice and N an involutive dual endomorphism of L if and only if the following identities are satisfied: (2.12)

z A N ( N t A Ny) = z

(2.13)

zA

N(Ny A N z ) = N ( N ( z A

Z)A N(y A z)) .

Proof. In view of (2.11), Axioms (2.12)-(2.13) reduce to (2.8)-(2.9), therefore Proposition 2.4 implies that (2.12)-(2.13) are necessary and sufficient for ( L , A , V ) to be a distributive lattice. Now N is an involutive dual endo0 morphism by Proposition 1.31, condition (iii'). 2.10. Proposition. Let L be a set, A : L2 + L, N : L --t L and 0 E L. Define V b y (2.11) and 1 = NO. T h e n ( L , A , V , N,O, 1) is a De Morgan algebra if and only if L satisfies identities (2.12) and (2.14)

tAN(NyANz) = = N(N(. A

N ( N A~ N O ) )A ~ ( A 3N ( N A~ N O ) ) ).

Proof. Similar to the proof of Proposition 2.9 but using Proposition 2.5 cl instead of 2.4, plus Proposition 1.32(i). 2.11. Definition. An element t of a bounded lattice is said t o be complemented or chrysippian if there is an element y such that t A y = 0 and t V y = 1; every element y with these properties is called a complement of 5. Note that if y is a complement of 5 then t is a complement of y. An element may have several complements, a single complement or none. Thus e.g. the element b of the pentagone has the complements a and c , while

Lattices, universal algebra and categories

24

each of the elements a and c has b as unique complement. Also, each of the three incomparable elements of the diamond has the two other elements as complements. It is important to note that in a bounded toset every element different from 0 and 1 has no complement.

2.12. Proposition.

(9

In every bounded lattice 1 is (0 i s ) the unique complement of 0 (of 1).

(ii) Every element of a bounded ddstrzbutive lattice has a t m o s t o n e complement. Proof.

(i) Obvious. (ii) Let y and y' be complements of z. Then

2.13. Definition. Let L be a bounded distributive lattice. We denote by C ( L ) the set of all complemented elements of L and call it the center of L . For each z E C ( L ) we denote by ii i t s unique complement. Thus

(2.15') z A 5 = 0 ,

(2.15") z V 5 = 1 . 2.14.Proposition. Let L be a bounded distributive lattice. T h e n 0,l E C ( L ) and f o r every x,y E C ( L ) it follows that x A y, x V y, 3 E C(L);namely,

25

Distributive lattices, De Morgan and Boolean algebras (2.16')

Sny = 5 V g ,

(2.16') (2.17)

=5A

-

x

X=

g,

.

Proof. 0 , l E C ( L ) by Proposition 2.12(i) and it remains to prove (2.16)(2.17). Using (2.1') and (2.1") we get ( X A Y ) A ( Z V ~ ) = ( X A Y A A ) V ( ~ A Y A ~=)

= (oAY)v(xAO)=OVO=O, ( ~ A y y ) v ( % V @=) ( z V 2 V y ) A ( y V S V j j )

=

= ( 1 ~ ~ ) ~ ( 1 ~ ~ ) = 1 ~ 1 = 1 ,

which implies (2.16'), while (2.16') follows by duality. Finally (2.15) yield D 5 A 5 = 0 and 5 V z = 1, that is (2.17). Proposition 2.14 minus (2.16)-(2.17) can be restated t o the effect that

C ( L ) is a sublattice of ( L ,A, V, 0 , l ) ; cf. Definition 1.33. 2.15. Remark. In a bounded distributive lattice

L , the following properties hold

for every

z E C ( L ) and y E L:

(2.18')

z A (5 V y) = z A y

,

(2.18')

.1:V

(5 A y) = 2 V y

,

(2.19')

x

2y

2 A y =0

(2.19')

t

5y

* 5 v y = 1,

,

because e.g. x A (3 V y) = (X A 2) V (z A y) = z A y and z 2 y implies 0 = Z A X 2 f A y , while i A y = 0 implies y = ( s V 5 ) A y = ( z A y ) V ( z A y ) = XAY. 2.16. Definition. A Boolean algebra is a bounded distributive lattice L such that L = C ( L ) ,

Lattices, universal algebra and categories

26

i.e. all the elements are complemented.

2.17. Examples. a) The two-element toset (0,1} is a Boolean algebra; cf. Example 2.3 and

Proposition 2.12(i), b) For every set X , the lattice ( P ( X ) ,n, U, 0 , X ) is a Boolean algebra. c) For every bounded distributive lattice

L , the sublattice C(L)is a Boolean

algebra by Proposition 2.14. This example is essential i n our book. 2.18. Remark. In view of Proposition 2.12(ii) and Definition 2.16, to every element x of a Boolean algebra L corresponds a unique complement 3 , so that taking complements is a unary operation - : L --t L. Therefore Definition 2.16 can be restated to the effect that a Boolean algebra is a De Morgan algebra

( L ,A, V,; 0 , l ) satisfying identities (2.15). Beside the properties established so far in this and the previous section, Boolean algebras have of course specific properties. Thus e.g. (2.20’)

x =y

(2.20”)

5

=y

($

( 5 A y) V

(x A

v) = 0 ,

( 3 V y) A (x V jj) = 1 ,

follow from (2.19). Also

(2.21’)

((u A x)V(bA

5)) A ( ( c A x) V ( d A 3 ) ) =

= ( U A C A ~ ) V ( ~ A ~ A Z ) ,

(2.21”)

( ( u A x)V(b A 5 ) ) V ((c A

x) V ( d A 3)) =

= ( ( u V c) A x) V ( ( b V d ) A Z) ,

(2.22)

( u A ~ ) v ( ~ A ~ ) = ( s ~ A ~ ) v ( ~ A ~ ) ,

are easily checked and can be generalized to several variables.

The following properties hold only in the Boolean algebra {O,l}:

27

Distributive lattices, De Morgan and Boolean algebras (2.23’)

xA y =0 e

x

(2.23”)

x v y = 1 e x = 1 or y = 1 .

= 0 or y = 0 ,

Recall that;according t o Definition 1.21, if L and L‘ are lattices, a mapping f : L + L’ is called a lattice homomorphism provided it preserves A

and V, i.e. f(x A y) = f(s)A f(y) and f(z V y) = f(x) V f(y); the Same concept applies in particular t o distributive lattices.

2.19. Definition. If L and L’ are bounded lattices, a lattice homomorphism f : L

-+

L’ is

said t o be a bounded-lattice homomorphism provided it preserves 0 and 1, i.e. f(0) = 0 and f(1) = 1. If, moreover, L and L‘ are De Morgan algebras and f preserves

N , i.e. f ( N s ) = N f ( s ) , then f is termed a De Morgan

homomorphism.

2.20. Remark. Let L and L‘ be De Morgan algebras. Every meet (join) homomorphism

f :L

+ L’ that preserves N , 0 and 1 is a De Morgan homomorphism. For

2.21. Remark.

L‘ be bounded distributive lattices. Every bounded-lattice homomorphism f : L -+ L‘ preserves complements whenever they exist, i.e.

Let L and f(5) =

m.

For it is easily seen that f ( x ) A f ( Z ) = 0 and f(x)Vf(.)

= 1.

2.22. Definition.

Let L and L’ be Boolean algebras. A mapping f : Boolean homomorphism provided it preserves A, V,;

L

L’ is called a 0 and 1. In other +

words, a Boolean homomorphism is a De Morgan homomorphism between Boolean algebras.

Lattices, universal algebra and categories

28

2.23. Proposition. Let L and L‘ be Boolean algebras and f conditions are equiwalent:

:

(i)

f is a Boolean homomorphism;

(ii)

f is a bounded-lattice homomorphism;

L + L’. T h e following

(iii’) f is a meet h o m o m o r p h i s m that preserves complements; (iii”) f is a j o i n h o m o m o r p h i s m that preserves complements. Proof.

(ii) j (i): f(z)Af(5) = f(zA5) = f(0) = 0 and similarly f(z)Vf(5) = 1, hence f(5) =

fo.

(iii’) + (i): f is a j o i n homomorphism as in Remark 2.20. Then f(0) = f ( z A 5 ) = f ( z ) A f o = 0 and similarly f(1) = 1. T h e converse implications are trivial.

0

W e conclude this section w i t h a few words about subalgebras.

2.24. Definition. A 0-1-sublattice of a bounded lattice (L, A, V, 0 , l ) is a sublattice S of L such t h a t 0 E S and 1E S. A D e Morgan subalgebra of a D e Morgan algebra ( L ,A, V, N , 0 , l ) is a 0-1-sublattice S of L such t h a t z E S

+ N z E S.

A Boolean subalgebra of a Boolean algebra ( B ,A, V,; 0 , l ) is a D e Morgan subalgebra of B viewed as a Boolean algebra. Clearly a 0-1-sublattice of a bounded (distributive) lattice is itself a bounded (distributive) lattice, while a De Morgan (Boolean) subalgebra of a De Morgan (Boolean) algebra is itself a De Morgan (Boolean) algebra. T h e converses do not hold, as can be seen e.g. f r o m the following example, which is important in many respects.

29

Distributive lattices, De Morgan and Boolean algebras 2.25. Example.

L be a

Let

(distributive) lattice and take a, b E L with a

5 b.

The segment

or closed interval [a,b] defined by (2.24)

[a, b] =

is a sublattice of

{z E L I a _< z _< b}

L

,

and a bounded (distributive) lattice with respect t o the

L. However, if L is bounded but a # 0 or b # 1 then [a,b] is not a 0-1-sublattice of L. In particular if L is a Boolean algebra then

order inherited from

[a,b]is not a Boolean subalgebra, yet if we define (2.25)

d = (2 V a ) A b = ( 3 A b) V a

(vz E [a,b ] ) ,

then ( [ a ,b],A, V,', a, b) is a Boolean algebra; the proof is easy.

2.26. Proposition. Let L be a De Morgan algebra and ( u , ) , ~ sC L. Then (2.26')

N

v

A a, =

sES

(2.26")

N

Nus,

SES

V a, = A N u , , SES

SES

to the effect that the existence of one side of the equality implies the existence of the other side and the equality itsey. Proof. We suppose a =

A a,

and check t h a t N u fulfils the definition of

sES

V Nu,.

But a

8ES

(Vs E S). Then a,

I a,

hence N u

2 Nb

Thus the existence of

2 Nu,

(Vs E

(Vs E S), hence a

S). Further t a k e b 2 N a ,

2 Nb, therefore b 2 N u .

A a, implies (2.26')

and by duality the existence

SES

V a,

of

implies (2.26"). These results can be applied t o

V Nu,, sES

A Nu,

and

,€S

s€S

completing the proof.

0

Lattices, universd algebra and categories

30

Proposition 2.27 below, as well as Remark 2.13, generalizes well-known properties of Boolean algebras. 2.27. Proposition. Let L be a bounded distributive lattice, (a,),€, G L and c E C ( L ) . Then

v

(2.27')

a, exists

=$

cA

A

(2.27")

V a, SES

SES

a, exists

+

cA

V

( c A a,)

,

(cV a,)

,

SES

V a, = A SES

SES

=

SES

to the effect that the right sides exist and the equalities hold. Proof. Suppose a =

V

a , exists. Then a 2 a , hence c A a

2 c A a,

SES

S ) . Now take b 2 c A a , (Vs E S ) ; then E V b 2 C V a, (Vs E S) by (2.18'), hence C V b 2 C V a , therefore b 2 c A b 2 c A a by (2.18'). Thus (Vs E

cAa=

V

(cAa,).

0

SES

2.28. CorolIa ry .

I n a n y Boolean algebra Properties (2.27) hold for arbitrary elements c, a, (s E S).

Filters, ideals and congruences in lattices

31

$3. Filters, ideals and congruences in lattices

The dual concepts of filter and ideal t u r n o u t to be a powerful t o o l o f lattice theory, w i t h applications i n other fields of mathematics. T h e main points of this section are t h e complete lattices of all filters and of all ideals, the characterization of prime filters and o f prime ideals, the relationship between prime and maximal filters (ideals), t h e congruences associated w i t h filters. W e refer t o distributive lattices and t o Boolean algebras; t h e results will be generalized in Chapter 5 @l-Z to the framework of Lukasiewicz-Moisil algebras.

3.1. Definition. Let

L

b e a lattice. A non-empty subset

F C L ( I 2 L ) is called a fiEter (an

i d e a l ) provided for every z,y E L ,

+x A y E F , & x 5 y +y E F

(3.1)

z,y E F

(3.2)

z EF

(for every z,y E

L,

(3.1')

z , y ~ l + ; c V y ~ I ,

(3.2')

x ~ l & y l z + y ~ I ) .

3.2. Remark. The concepts o f filter and ideal are dual t o each other. Therefore w i t h each theorem and definition referring to filters is associated a dual theorem and a dual definition, respectively, referring to ideals. In view of t h e duality principle we can restrict to one proof for each pair of dual propositions.

3.3. Examdes. a) The lattice L itself is b o t h a filter and an ideal, called t h e i m p r o p e r filter

( i d e a l ) , while the other filters (ideals) are said to be proper. b) For every u E L, the set [u ) = {z E L I u 5 z } is a filter (the set (a]= { z E L I z 5 u } is an ideal) called t h e p r i n c i p a l f i l t e r ( i d e a l )

Lattices, universal algebra and categories

32

L we denote by generated b y a. More generally, for each subset X [ X )(by ( X I ) the filter ( i d e a l ) generated b y X , i.e. the least filter (ideal) which includes X ; the term "least" refers t o set inclusion G as partial order. Example 3.3.b requires a proof which we provide below. In the sequel we shall generally leave t o the reader the statements of the dual theorems and, of course, their proofs; cf. Remark 3.2. Caution: the duality principle refers to the lattice-theoretical

concepts and not t o t h e set-theoretical

ones; see

e.g. the dual definitions in Example 3.3.b. 3.4. Proposition.

Let L be a lattice and X G L.

If X # 0 t h e n

and if L has greatest e2ement 1 t h e n

[0) = 1.

Proof. Let Y stand for the right side of (3.3). Clearly X Properties (3.1) and (3.2) are easily verified, therefore

E Y ,hence Y # 0. Y is a filter. Sup-

F . If y E Y then z1A ... A z, < y for some zl,...,5, E F ,therefore z1A ... A t, E F by (3.1) and this implies y E F by (3.2). Thus Y E F completing the proof of Y = [ X ) . The reader is pose

F is a filter and X

urged t o check that (1) is the least filter, which i s equivalent t o (1) = [0).0 Numerous filters and ideals are dealt with in this book, where they play an important role. 3.5. Remarks. a)

A non-empty subset F G L is a filter if and only if (3.4)

z E F & y E F + z A y E F .

b) Let F be a filter of a lattice L. If L has greatest element 1then 1E F . If L. has least element 0 then 0 E F if and only if F = L.

Filters, ideals and congruences in iat tices

33

3.6. Theorem. If the lattice L has least element 0 (greatest element 1) t h e n t h e ideals (filters) of L f o r m a complete lattice (Id(L), ((Fl(L), G)) and the m a p x H (x] (x H [x)) is a n injective lattice (dual lattice) homomorphism.

c)

Proof of the filter part. FL(L) is a Moore family on

( P ( L )5 , ) by

Proposi-

tion 3.4; cf. Definition 1.14. Now apply Remark 1.16. Finally it is easy t o check that

[xV y) = [x)n [y) and [x A y) = [x)V [y).

0

3.7. Definition.

A proper filter F (proper ideal I) is said to be p r i m e if for every 2,y E L , (3.5)

X V ~ E F J Z EorFy e F

(for every z , y E L , (3.5')

xAyEI+zEI

or y E I )

For example every proper filter (ideal) in a chain is prime. Further examples will be the object of various propositions of this book.

3.8. Remark. If P is a filter (proper filter, prime filter) of a De Morgan algebra L , then N P = {NxIx E P} is an ideal (a proper ideal, a prime ideal) of L. 3.9. Proposition. T h e following conditions are equivalent f o r a subset F of a lattice L:

(i) F is a prime filter; (ii) F is a proper filter and f o r every x,y E L ,

(iii) L - F is a prime ideat (iv) there i s a surjective lattice h o m o m o r p h i s m h : L -+

that F = h-'({I}).

{0,1} such

Lattices, universal algebra and categories

34 Proof.

(i)+ (ii): By Definition 3.7 and (3.2). (ii) + (iv): Set h ( z ) = 1 for z E F and h ( z ) = 0 for z E L - F. Then h is surjective because 0 # F # L and it follows from (3.4) that

h ( z A y ) = 1 @ h ( z ) = 1 & h ( y ) = 1 M h ( z )A h ( y ) = 1 , therefore h ( z A y ) = h ( z ) A h ( y ) .Similarly one proves h ( z V y ) = h ( z ) V h ( y ) , using (3.6). (iv) =+ (ii):

0# F # L

because

h is surjective. Then

X Ay E F M h ( x ) A h ( y ) = h ( z A y ) = 1

5

E

F &yE F ,

therefore F is a proper filter by Remark 3.5.a and one proves similarly (3.6).

(ii) j (i): Trivial. Thus (i) @ (ii) M (iv) and it follows by duality t h a t (iii) is equivalent t o t he existence of a surjective homomorphism h :

L

+

{0,1} such that

L - F = h-’({O}); but the latter property is equivalent t o (iv).

0

3.10. Theorem. Let L be a distributive lattice, F afilter and I an ideal of L. If F n I = 8 then there is a prime filter P such that F P and P f l I = 0.

c

F’ which satisfy F c F’ and F’ n I = 0. It follows from t h e Zorn lemma that G has a maximal element P . Since P E G it remains to prove t h a t the filter P is prime. P is proper because P n I = 0. Suppose P is not prime. Then there exist a , b E L such that a V b E P , a # P and b # P . Let X = P U { a } . Then [ X )n I # 0, otherwise P c [ X ) E G contradicting the maximality of P . Take z E [ X ) n I .Then Proposition 3.4 implies easily the existence of p E P such that p A a 5 z and since z E I it follows that p A a E I . Similarly there is q E P such that q A b E I . Then ( p A a ) V ( q A b ) E I and on t h e other Proof. Let G be the family of those filters

hand

Filters, ideals and congruences in lattices therefore I n

P # 0, a

contradiction.

In Corollaries 3.11-3.14

3.11. Corollarv. If I is an ideal and a E L and P n I = 0. Proof. Take

F = [a)

35 0

L is a distributive lattice.

-I

there is a prime filter P such that a E P

in Theorem 3.10.

0

3.12. Corollary. If F is a filter and b E L - F there is a prime filter P such that F C P and b # P . Proof. Take

I = (b] in Theorem 3.10.

0

3.13. Corollarv.

If a, b E L are such that a

b there is a prime filter P such that a E P

and b # P . Proof. Take I = ( b ] in Corollary 3.11.

0

3.14. Corollarv.

In a distributive lattice every proper filter is included in a prime filter and is an intersection of prime filters.

F be a proper filter. Denote by 3 the non-empty (cf. Corollary 3.12) family of those prime filters that include F . We set F' = n { P I P E 3} and prove that F = F'. Otherwise there is a E F ' - F , hence Corollary 3.12 yields a prime filter P E F s u c h that a # P,which contradicts a E F'.O Proof. Let

3.15. Definition. By a mazimal filter (mazimal ideal) is meant a maximal element of the family of all proper filters (proper ideals) ordered by set-inclusion. A maximal filter is also called a ultrafilter.

Lattices, universal algebra and categories

36 3.16. Proposition.

In a distributive lattice every maximal filter is prime. Proof. Let F be a maximal filter. Then F is included in a prime filter Corollary 3.12, therefore

P

F = P by the maximality of F .

by 0

Now let us turn t o Boolean algebras. First we extend Remark 3.8 and the notation N P . For every subset A of a Boolean algebra B set A = {5 I z E A } .

3.17. Lemma. T h e following conditions are equivalent f o r a filter F of a Boolean algebra

B: (i)

F is proper;

(ii) F n F = 8 ; (iii) f o r every z E B at m o s t one of the elements

5, 5

belongs t o F .

Proof.

(i) + (iii): Otherwise 0 = z A 5 E F hence F = B. (iii) + (ii): Otherwise there is z E F n F , hence 2 E F and some y E F , therefore a: = y E F . (ii) + (i): Trivial.

2

=

for 0

3.18. Proposition.

T h e following conditions are equivalent f o r a filter F of a Boolean algebra

B: (i) F is mazimal;

(ii) F is prime;

(iii) f o r every z E B , exactly one of the elements Proof.

(i) + (ii): By Proposition 3.16.

2,

5 belongs t o F .

Filters, ideals and congruences in lattices

37

(ii) + (iii): x V ii! = 1 E F hence x E F or ii! E F , but not both because of Lemma 3.17. (iii) + (i): Note first that F is proper. If G is a filter such that F C G then there exists x E G - F , hence x E G and 2 E F c G, therefore 0 0 = x A f E G showing that G = B . Proposition 3.18 has t h e following converse:

3.19. Theorem (Nachbin [1947]). A bounded distributive lattice L is a Boolean algebra if and only if every prime filter of L is maximal.

if” part is included in Proposition 3.18. For the “if” part suppose t he lattice L contains an uncomplemented element a . Consider the filters FO = {x E L I a V x = 1) and Fl = {x E L I a A y 5 x for some y E Fo}. Then 0 4 F1 otherwise a would be complemented; hence there is a prime filter Pl such t h a t Fl C Pl.Let I = ( ( L- Pl)U { a } ] . Note that L - PI C I because a E I and a E Fl Pl implies a # L - Pl.Further we prove that Fo n I = 8. Otherwise t a k e x E Fo n I. Then x E Fo and since L - PI i s an ideal by Proposition 3.9, the dual of Proposition 3.4 implies easily that x 5 a V y for some y E L - Pl.Then 1 = a V x 5 a V y therefore y E Fo G Fl E P,a contradiction. Thus Fo n I = 0 and Theorem 3.10 yields a prime filter P such that Fo E P and P n I = 0. It follows that 0 P C L - I c Pl, therefore P is not maximal. Proof. The “only

The concept of congruence in a lattice is a specialization of t h e general notion in Definition 5.11: a congruence on ( L ,A, V) is an equivalence relation p on

L such that x p y & x’py’ =+x A x’py A y’ & x V x‘py V y’

(3.7)

For example, the relations ker

h :L (3.8)

--t

h, L2 and AL

L’ is a homomorphism, z ker h y H h(x) = h(y)

,

.

are congruences, where

Lattices, universal algebra and categories

38

(3.9)

x L2y vc, y E

L

,

(3.10)

XALY x = y

.

*

3.20. Definition. Let DO1 be the class of bounded distributive lattices. For every L E DO1 and every filter F of L , let m o d F be the relation on L defined by (3.11)

xmodFy@3uEFxAu=yAu

3.21. Proposition.

Let L E D01. T h e n m o d F is a congruence o n L for every filter F of L, and F = { x E L I z m o d F l } . Proof. Easy via the following remark.

If x A u = y A u and x' A u' = y' A u'

withu,u'€ F t h e n x A v = y A v a n d x ' A v = y ' A v w h e r e v = u A u ' € F . Forthesecond p a r t x E F ~ x A x = i A x + x m o d F l a n d x m o d F l ~

3u E F x A u = 1 A u

+ x 2 U , u E F + x E F.

0

3.22. Corollary. Suppose L E DO1 and F is a proper filter of L. Let L/F stand f o r LfmodF. T h e n L / F E DO1 with least and greatest elements 6 and i = F , respectively, and card(L/F) > 1.

3.23. Proposition. T h e following conditions are equivalent f o r L E D01, card L > 1, and a filter F of L:

(i) t h e only filters of L / F are the trivial filters

{i}and L / F ;

(ii) F is mazimal. Proof. l(ii)

+ l(i): Suppose F is not maximal.

Let F' 3 F be a proper

{i E L / F 15 E F'} is a filter of L / F . Moreover, G # {i}because a E F' - F implies i? E G and i? # i. Also, G is proper because 6 E G would imply 6 = i for some 5 E F' hence 0 A u = x A u for some u E F , therefore 0 = z A u E F', a contradiction.

filter. It is easy to check that G =

Filters, ideals and congruences in lattices l(i)

+ +):

t o check that

Let G be a proper filter o f L / F , G

F’ = {x E L 1 i E G }

tz E ( L / F - G ) shows that x E F +f = t h a t F c F’.

iE

a

4 F’

G. Now t a k e

39

# {i}. It is

easy

is a filter of L . The existence of

i.e. F’ is proper. Then F

E F’

because

L E G, L # i; then b E F‘ - F , showing 0

3.24. Definition. Let PFl(L) (let Spec(L)) denote the set of all prime filters (prime ideals) of the lattice

L.

3.25. Proposition. If L E DO1 t h e n (3.12)

fl {mod F I F E PFl(L)}= A,

.

E L , x # y. Then we have e.g. x $ y. In view of Corollary 3.13 there is F E PFl(L) such that x E F and y # F . Then x A u # y A u for every u E F , otherwise y A u = x A u E F , which would imply the contradiction y E F. Therefore (x,y) $ mod F and this proves the non-trivial Proof. Take x,y

part of (3.12).

0

In the case o f Boolean algebras the above results can be strengthened.

3.26. Proposition. In a Boolean algebra B every congruence p as of the f o r m m o d F f o r

F = {x E B I x p l } . Com me nt . According t o Definition 5.11, the congruences p of a Boolean algebra satisfy also zpy

+ zpy. F

(3.1) and also (3.2) because if x p l and x 5 y then y = z V y p l V y = 1. Thus F is a filter. If xpy then xAypyAy = y and Zpy hence ZAYpy. therefore u = (xAy)V(ZAg)pyVjj = 1. Thus u E F and x A u = x A y = y A u , proving that x mod F y. Conversely, Proof.

Clearly

is non-empty, satisfies

40

Lattices, universal algebra and categories

the latter relation means x A u = y A u for some u E because

F ,therefore

xpy

3.27. Definition. Let A be an algebra in the sense of Definition 5.2, c a r d A > 1. Then A is said to be: semisimple if the intersection of all its maximal congruences is the identity AA, and simple if the only congruences of A are the trivial congruences AA and A2. 3.28. Proposition. Every Boolean algebra is semisimple. Proof. From Propositions 3.21, 3.26, 3.18 and 3.25.

0

3.29. Proposition.

T h e following conditions are equivalent f o r a Boolean algebra B : (i) B is simple;

(ii) the only filters of B are (1) and B ;

(iii) B 2 {0,1). Proof.

(i) ($ (ii): From Proposition 3.21, 3.26 and he remark that AB = mod{ 1) and B2= mod B. (ii) + (iii): If a E B - ( 0 , l ) then [ u ) is a proper filter distinct from (1). (iii) +-(ii): Trivial. 0 3.30. Proposition. A filter F of a Boolean algebra B i s maxima2 and only if B / F 2 { & I ) . Proof. By Propositions 3.23 and 3.29.

(OT

equivalently, p r i m e ) if

0

41

Filters, ideals and congruences in lattices

3.31. Remark (cf. lorgulescu (19841). Let m be an infinite cardinal. Following the same idea as in Definition 2.31,

F

it is natural t o define an m-filter as a filter

for every subset S &

satisfying

L such that cards 5 m and A

x exists. A

proper

XES

m-filter (prime m-filter, mazimal m-filter) is defined as a proper filter (prime filter, maximal filter) which is also an m-filter. By an m-prime f i l t e r is meant an m-filter

for every subset S

F such that

CL

such t h a t cards

5m

d

v

x exists. A

m-

XES

mazimal f i l t e r is defined as a maximal element of t h e family of all proper m-filters. A complete f i l t e r (proper complete filter, prime complete filter,

mazimal complete filter, completely-prime filter) is then defined as a filter which is an m-filter (proper m-filter, prime m-filter, maximal m-filter, m-prime filter) for all m. Also, a completely-masimal f i l t e r is defined as a maximal element of t h e family of all proper complete filters. For example every finite filter is complete, every proper principal filter is complete but not

{ X E P(E)Icard(E-X) 5 m} is an m- filter on P ( E ) , while the filter { X E P(E)I X - E is finite} is necessarily prime. If E is an infinite set then

not an m-filter. Clearly every m-prime (completely prime) filter is a prime m-filter (prime complete filter) and every maximal m-filter (maximal complete filter) is an m-maximal (completely maximal) filter. It is conjectured

that the converse implications do not hold in general. The reader is urged t o specify the results of this section t o the corresponding m-concepts.

The following result will be needed in Chapters 7 and 9.

Lattices, universal algebra and categories

42

3.32. Lemma (Rasiowa-Sikorski).

c

Suppose B is a Boolean algebra and let X,, Y, B such that inf X, and sup Y, exist (Vn E I?). Then for every a E B - (0) there is an ultrafilter F of B such that

(i) a E F and for every n E I?: (ii) infX, E F

* z E F (Vz E X,), and

Proof. Let 2, = (jj I y E Y,} ( n E

IN). It follows from

the De Morgan

formulas and Proposition 3.18 t h a t (iii) is equivalent t o property (ii) for 2,. Therefore it suffices t o prove t h e existence of an ultrafilter satisfying (i) and

(ii). First we set a, = inf X, (Vn E I?)and construct by induction a sequence of elements z,

(Vn E $0

E X , such that b, = a A

IN). For

a0

... A (a, V Z), # 0 (a0 V 5 0 ) # 0 for some

V Z0)A

n = 0 the assertion is that a A

E X O :otherwise a A

hence a

(a0

= 0 and a A Z = 0, i.e. a

5 ao, therefore a = a A a.

5 z, for

every z E

Xo,

= 0, a contradiction. The inductive step

is similar, with b, instead of a .

Further we apply Proposition 3.4 to the decreasing sequence X = (€1-1 =

a, bo,bl ,...,b,, ...} : it follows easily that y E [ X ) ++ b, 5 y for some n E IV U (-1) and since all b, # 0 this implies 0 # [ X ) . Thus X is a

[ X )E F , by Corollary 3.14 and Proposition 3.18. Then a E F and for each n E N ,from b, E F we obtain a , V 2 , E F , therefore either a, E F ,in which case X, F , or 0 a, # F ,which implies 2 , E F hence z, # F . proper filter hence there is an ultrafilter F such that

c

43

Monadic and polyadic Boolean dgebras $4. Monadic and polyadic Boolean algebras

T h e monadic Boolean algebras and potyadic Boolean algebras are algebraic structures imposed by t h e predicate calculus (see Halmos [1962]). In this section we shall present in an outlined form some basic results on these structures. See also Daigneault [1967].

4.1. Definition. A n ezistential quantifier on a Boolean algebra such that, for any p , q E

A , the following

(4.1)

30 = 0 ;

(4.2)

PI3P;

(4.3)

3 ( p A 3q) = 3 p A 3q .

A

is a function 3 :

A +A

properties hold:

A monadic Boolean algebra or m o n a d i c algebra for short is a pair ( A , 3 ) , where A is a Boolean algebra and 3 is an existential quantifier o n A . 4.2. Definition. Let ( A , 3 ) , (A’,3) be two monadic algebras. A m o r p h i s m of m o n a d i c algebras f : ( A , 3 )+ (A’,3)is a Boolean morphism f : A t A‘ such t h a t

f ( 3 p ) = 3 f ( p ) , for any p E A . 4.3. Example.

B be a Boolean algebra and X a non-empty set. Suppose A is a Boolean B X such that, for any p E A , there exists i n B t h e following supremum: V { p ( t ) It E X } . Suppose A is closed w i t h respect t o the Let

subalgebra of

operation 3 defined by

(3p)(y) = V { p ( t ) I t E X }

,

for any p E A and y E X

Then one can prove t h a t 3 is an existential quantifier on called a B-valued functional m o n a d i c algebra.

A . ( A , 3 ) will

be

Lattices, universal algebra and categories

44

4.4. Definition. A u n i v e m a l quantifier on a Boolean algebra A is a function V : A -+ A such that, for any p , q E A , we have:

(4.4)

v1 =1 ;

(4.5)

VPSp ;

(4.6)

V ( pV V q ) = V p V V q

4.5. Lemma. In a m o n a d i c algebra ( A , 3 ) t h e following properties are t r u e : (4.7)

31=1;

(4.8)

33=3;

(4.9)

p E 3(A) H 3 p = p ;

(4.10)

P5 9

(4.11)

3(3p)=3p;

(4.12)

3 ( A ) is a Boolean subalgebra of A ;

(4.13)

3(Pvq)=3Pv39;

(4.14)

3P

+

+- 3P 5 39 ;

S 3(p +9)

-

4.6.Remark. In a monadic algebra ( A ,3) we can define a universal quantifier V by V p = 3, for any p E A . It is obvious that a monadic algebra can be also defined using a universal quantifier.

4.7. Definition. A m o n a d i c subalgebra of a monadic algebra ( A ,3) is a Boolean subalgebra A’ of A which is closed under 3. A m o n a d i c ideal of ( A , 3 ) is an ideal M

M . By (4.14), the quotient Boolean algebra AIM has a canonical structure of monadic of the Boolean algebra A such that p E M implies 3 p E

Monadic and polyadic Boolean algebras

45

algebra and t he natural Boolean morphism h : A + A / M is a morphism of monadic algebras. 4.8. Proposition.

Let ( A , 3 ) be a monadic algebra and B = 3 ( A ) . FOT a n y ideal M of the Boolean algebra B we denote by M the Boolean ideal of A generated by M . Then: monadic ideal of A ;

(a)

M is

(b)

M nB

(c)

If N is a monadic ideal of A, then N r l B = N ;

a

= M ; M = 3-'(M);

(d) T h e m a p M H M gives a n isotone bijection between the ideals of B and the monadic ideals of A; (e)

If M is a m a x i m a l ideal of B , t h e n M is a m a x i m a l monadic ideal of A.

4.9. Definition.

A monadic algebra ( A , 3 ) is simple if (0) is the unique monadic ideal of A; ( A , 3 ) is semisimple if the intersection of i t s monadic ideals is (0). An existential quantifier 3 is simple if p # 0 implies 3 p = 1. One can prove that a monadic algebra ( A , 3 ) is simple iff t h e quantifier

3 is simple. 4.10. Proposition.

(i) A n y monadic algebra is semisimple. (ii) A n y monadic algebra is isomorphic t o a subdirect product of simple monadic algebras. Proof.

(i) By Proposition 4.8 and th e semisimplicity of Boolean algebras.

Lattices, universal algebra and categories

46

0

(ii) Straightforward. 4.11. Definition.

A constant of a monadic algebra ( A ,3) is a Boolean morphism c : A + A such t h a t c3 = 3 and 3c = c . A monadic algebra ( A , 3 ) is rich if for any p E A there exists a constant c of A such that 3 p = c ( p ) . For any constant c and every monadic ideal M of ( A , 3 ) the following properties hold: (4.15)

p E ](A)

(4.16)

c ( A ) = 3 ( A ) ; cc = C ; CV = V; VC= c ;

(4.17)

p E M implies c ( p ) E M ;

(4.18)

there is a unique constant C of

c(p) =p ;

A / M such that hc = Ch .

4.12. Lemma.

A n y rich monadic algebra ( A , 3 ) is isomorphic t o a functional monadic algebra. Proof. Let B = 3 ( A ) and map f : A -+

X th e

set of constant of

( A , 3 ) . Consider the

BX defined by

f ( p ) ( c )= c ( p )

,

for any p E A and c E X

( A ,3) and th e properties of constants one can show that f ( A ) is a B-valued functional monadic algebra and f is an isomorphism Using the richness of

between A and f(A).

0

4.13. Lemma.

A n y simple monadic algebra ( A , 3 ) is rich. Proof. If p E A, p

#

0, then there exists a Boolean morphism c : A -+ {0,1} such that c ( p ) = 1; c can be considered a Boolean morphism c : A -+ A such that c(A) = {O,l}.

Monadic and polyadic Boolean algebras

47

Since 3 is simple, we can show that c is a constant of

A such t h a t

3 ( p ) = c ( p ) = 1. 4.14. Lemma. A n y direct product of rich m o n a d i c algebras i s a r i c h m o n a d i c algebra. 4.15. Theorem (Halmos). A n y m o n a d i c algebra is isomorphic t o a f u n c t i o n a l m o n a d i c algebra. Proof (given by Leblanc [1967]). By 4.10(ii), 4.13, 4.14 and 4.12.

0

4.16. Def inition. A polyadic (Boolean) algebra is a structure ( A ,U,S, 3), where A is a Boolean algebra,

U

a non-empty set, S is a function from

endomorphisms of

Uu to

t h e set of

A and 3 is a function from P(U)t o t h e set of existential

quantifiers of A such that the following axioms hold:

(4.19)

S(lu) = 1~;

(4.20)

S(OT)= S(O)S(T),

(4.21)

3(0) = 1~;

(4.22)

3 ( K U K ' ) = 3(K)3(A7'),

(4.23)

S ( a ) 3 ( K )= S ( ~ ) 3 ( 1 <,) such t h a t

(4.24)

~

o ~ U -=~

for any

1

E 'U

for any

;

~

,

;

K,K'C U

for any I< C

3 ( K ) S ( a )= S(o)3(0-'(1{)) such that oI.-I(K)

O,T

;

U and O , T

E

Uu

~ for any K

U and

a E Uu

is injective.

The notion of m o r p h i s m of polyadic algebras or polyadic m o r p h i s m for short is defined in a natural way. Card(U) will be called the degree of the polyadic algebra ( A ,U,S, 3). A subset I< o f U will be called a support of an element p of A if 3(U - IC)p = p ; p is i n d e p e n d e n t of K if 3 ( K ) p = p . A polyadic algebra is locally f i n i t e if every element has a finite support.

Lattices, universal algebra and categories

48 4.17. Lemma.

FOTany polyadic algebra ( A ,U,S, 3) such that Card(U) 2 2, the following properties are equivalent:

(i) K is a support of p ;

All the polyadic algebras considered i n the sequel are supposed to be locally finite and of infinite degree. 4.18. Remark.

( A ' , U , S , 3 ) be two polyadic algebras and f : A t A' be a Boolean morphism such t h a t S(a)f = f S ( u ) ,for any u E U u . Then f is a polyadic morphism iff f ( 3 ( i ) p ) = 3 ( i ) f ( p ) ,for any p E A and i E U. Let ( A , U ,S,3),

4.19. Example. Let B be a complete Boolean algebra, set.

U

an infinite set and X a non-empty

Denote by Set(Xu,B) the set of all functions

Xu

t

B . For any

IC 5 U and x,y E Xu put xK*y w

If IC

U

and

X1U-K

= Y1U-K

.

E 'U define

3 ( ~ ): Set(Xu,B ) t Set(Xu,B )

S(U)

:

Set(Xu,B ) + Set(Xu7B )

X u + B , u E'U and K E U. It is straightforward to show t h a t Set(Xu, B ) becomes a polyadic algebra.

for any p :

49

M o n a d i c a n d p o l y a d i c Boolean algebras 4.20. Lemmq. F o r a n y p : X u + B and K

s U t h e following properties are equivalent:

(i) K i s a support of p in Set(Xu, B ) ;

W e shall denote by F ( X U ,B ) the set of the elements of ving a finite support. lt is obvious that

S e t ( X u , B ) ha-

F ( X U ,B ) is a locally finite polyadic

algebra. 4.21. Definition.

A polyadic subalgebra of F ( X U ,B ) will be called B-valued f u n c t i o n a l polyadic algebra. 4.22. Lemma.

In a polyadic algebra ( A , U,S, 3 ) t h e following properties hold: (i)

If p E A, t h e n {I< [ p i s i n d e p e n d e n t of I<} is a n ideal of P(U)a n d the set {I< I I< i s a support o f p } is a filter of P(U);

(ii) I f p is independent of K a n d K' C U , t h e n 3 ( K ' ) p = 3(K' - I+;

(iii) If I<' is a support o f p a n d K E U ,t h e n 3 ( K ) p = 3 ( K n Ir'')p a n d K' - I( is a support of 3 ( K ) p . 4.23. Lemma.

Let ( A ,U, S, 3 ) be a polyadic algebra, p E A; u,r E U u a n d I<, Is7' such that (i)

K i s a f i n i t e support

(ii)

ulKnKr

ofp;

is injective;

(iii) u ( K n K ' ) n r(IC - K ' ) = 0;

U

50

Lattices, universal algebra and categories T h e n S(r)3(IC')p= 3(u(IP n K))S(a)p.

4.24. Proposition. L e t ( A ,U , S, 3 ) be a polyadic algebra. F o r a n y p E A, r E U u and K' C U w e have t h e equality: (4.27)

S(r)3(K')p= V {S(a)pI ~ I { : T }.

Proof. It is easy to see that S(r)3(K')p2 S(a)p,for any uI<:r. Consider q E A such that q 2 S(o)p, for any uI
j ,

ifk=i

k,

ifkfi.

(j/i)(k) =

4.25. Corollary. In a n y polyadic algebra ( A ,U , S , 3 ) w e have: (4.28)

3(i)p = V {S(j/i)p 1 j E U } .

4.26. Theorem (Halmos [1962]). A n y polyadic algebra ( A ,U , S, 3 ) is i s o m o r p h i c t o a f u n c t i o n a l polyadic algebra. Proof. Let B be the MacNeille completion of the Boolean algebra A . Consider the map

defined by

@ ( p ) ( a )= S(a)p ,

for any p E A and u E U'

Monadic and polyadic Boolean algebras Using Proposition 4.24 we can show that

51 is an injective polyadic mor-

phisrn.

0

4.27. Remark. Let ( A ,U,S, 3) be a polyadic algebra. For any p E A we shall denote by K P the minimal support of p . Consider p E A and let of

U

such that

KP C U’. For any r E UtU and I<’

U’ be a countable subset

U

we have a modified

form of (4.28):

(4.29)

S(r)3(K‘)p= V { S ( a ) p1 0 E U”, aI<;r} .

The following representation theorem is t h e main result of this section. 4.28. Theorem (Halmos 119621).

Let ( A ,u,s, 3) be a polyadic algebra and M a proper filter of the Boolean algebra

T h e n there exists a n o n - e m p t y s e t X and a polyadic m o r p h i s m @ : A F ( X U , L z )s u c h that @ ( p ) = 1, f o r a n y p E M . Proof (Potthoff [1971]). Let any

Sw(A)be the set of finite subsets of A and for

F E Sw(A)we denote

k = {F‘ E Sw(A)I F Consider an ultrafilter

{ kI F

D

F’}

.

of the Boolean algebra

E Sw(A)}C

P ( S w ( A ) )such that

D.

F E Sw(A),set ICF = U {KP I p E F } and choose a countable p E F and IT f Sw(U) subset UF of U such that KF C UF. If F E Sw(A), For any

then we can prove that the set

( ( S ( 0 ) p 1 0 E u;, aK*r}17- E is countable. Since

F

u:}

is finite, the following set is countable:

Lattices, universal algebra and categories

52

Applying t he Rasiowa-Sikorski Lemma 3.32 we get, for any F E Sw(A),an ultrafilter N F of A having the following properties: (4.30)

F n M E NF ;

(4.31)

for any p f A , (T E

UF

and

K

c Iir~we have:

S ( a ) 3 ( K ) pE NF H there is T E UF, such that TIC*(Tand S ( r ) p E

II

Consider t he ultraproduct

NF .

U p / D (cf. Definition 5.46) and for any

FESu(A)

rE

( II

UF)'

define

F

by putting

( r / D ) ( i )= r ( i ) / D and r ~ ( i=) r ( i ) ( F )

)

Define the map @ : A

--$

for any i E U

.

F ( ( II V,/D>")Lz) by: F

for any p E A and r E

( II

UF)'. Using the properties of t h e ultraproduct

F

one can prove t h a t i9 is well-defined. It is straightforward t o show that i9 is

Uu. Now we shall prove that 3(IC)@(p)= @ ( 3 ( K ) p ) ,for any p E A and K U. It is enough to consider the case IC C Kp. It is easy t o see that {F I K 5 K F } E D ,therefore we have: a Boolean morphism which commutes with S(a),for any

(T

E

Monadic and polyadic Boolean algebras

53

Q ( 3 ( K ) p ) ( r / D= ) 1 ++ {F I S ( r F ) 3 ( K ) pE

NF}

E

D

H

++ {F I S ( r ~ ) 3 ( K )EpN F , K G K F } E D

I

By (4.31), t he latter condition is equivalent t o (4.33)

{F I there is a~ E Ug, S ( a ~ )Ep N F , a ~ l ( t r K ~ , ICF}ED .

Using again (4.34)

{F I K G K F } E D ,(4.33)

is equivalent t o

{ F 1 there is OF E V g , S ( a ~ )Ep N F , O F K J F }E D

.

But

3 ( W Q ( p ) ( r / D )= v { @ W ( s / D ) I ( s / D ) K * ( r / D ) } therefore 3(11)Q(p)(r/D)= 1 is equivalent t o (4.35)

there is ( s / D ) K ( r / D ) ,such that { F I S ( s ~ ) Ep N F } E D

.

That (4.34) is equivalent t o (4.35) is straightforward, therefore @ is a polyadic morphism. The rest of the proof is obvious.

0

4.29. Remark.

This representation theorem is th e algebraic counterpart o f the completeness theorem o f predicate calculus.

Lattices, universal algebra and categories

54 $5. Universal algebra

In this section we sketch those ,dndamental concepts of universal algebra t h a t will be used in t h e remainder of this book. For a complete introduction see e.g.

Cohn [1965] or Pierce [1968]; for an encyclopedic treatment the

reader is referred t o Gratzer [1968]; cf.

also Burris and Sankappanavar

[1981]. T h e basic concepts and results presented below are abstractions o f similar concepts and results encountered i n various “concrete” fields of algebra such as group theory, module theory, lattice theory etc.

5.1. Definition. If A is a set, n E nV and f : A” + A , we say t h a t f is an n - a r y operation on A . We also say t h a t n is the arity o f f . It is convenient t o distinguish between constant operations of a r i t y n

>0

(i.e. f(xl, ...,s,) = a for some a

E A and all x l , ...,x, E A ) and operations of a r i t y zero (i.e. f : A’ = 0 + A or simply f E A w i t h no arguments).

5.2. Defin ition. A type of algebras is a map r : A + nV where A is a non-empty set. A n algebra of type r or simply a r-algebra is a pair

A is a set and each f~ is an operation of arity .(A) o n A . W e refer to fx(A E A) as the basic operations of the algebra. T h e same notation A is often used for the underlying set A of A and t h e algebra A itself. If r is a f i n i t e type, i.e. t h e set A is finite, say A = {l,..,m}, then r can be

where

, ~(rn)). represented by the vector ( r ( l ) ..., T h u s e.g. a lattice (L, A, V) is an algebra of type (2,2), a group (G, -,- l , e )

is an algebra of type (2,1,0) etc.

55

Universal algebra

5.3. Definition. A subalgebru of an algebra ( A , { ~ A } A ~ isA )a subset S

A that is closed

with respect to the basic operations, i.e. (5.2)

51,**.,ZT(A) E

s * fA(Z1,

..a,

&(A))

E

s (vx E A)

5.4. Remark.

A subalgebra S of a 7-algebra (5.1) is itself a 7-algebra with respect t o the restrictions of th e basic operations fA of A t o t h e elements of S. Note for example that a subsemilattice of a semilattice

( L ,A) (6. Defi-

nition 1.33) is a subalgebra of ( L , A ) in the sense of Definition 5.3; similar remarks hold for the concepts of sublattice, 0-1-sublattice, De Morgan subalgebra, Boolean subalgebra and monadic subalgebra introduced in Definitions 1.33, 2.24 and 4.7.See also Remark 1.34 and Example 2.25. 5.5. Definition.

X of an algebra A we denote by X t h e intersection of all subalgebras of A that include X and we call X the subalgebra generated b y X. For every subset

The justification of the above definition originates in the obvious fact that the subalgebras of an algebra A form a Moore family of the complete lattice P ( A ) ;cf. Example 1.13 and Definition 1.14. In view of Remark 1.15, the map which sends every subset X C A to

X

is a closure operator and X

is the least subalgebra including X . We give below an intrinsic characteri-

zation of the elements of

R.

5.6. Definition.

Let ( A , { ~ A } A ~ Abe ) a .r-algebra, X C A and a E A . By a f o r m a t i v e cons t r u c t i o n of length n , of the element a from the elements of X w e mean a finite sequence a l , ...,a , of elements of A such that a, = a and for each

i

E (1, ...,n } one of the following situations hold:

(a) ai

E X, or

Lattices, universal algebra and categories

56

(b) there exist X E A and

kl,

...,k , ( ~ < ) i such that

ai = fx(xkl,

...,~ k , ( ~ ) ) .

5.7. Proposition. The subalgebra X generated b y a subset X of an algebra A coincides with the set of all elements of A having a formative construction from the elements of X .

Y be the set of all elements having a formative construction. Then X s Y because every x E X is a formative construction o f length 1. The set Y is a subalgebra of A because, for each X E A, if q ,...,xc,(x)E Y Proof. Let

then we can concatenate the formative constructions of these elements and

add a t the end the element y = fx(xl, ...,x.(~)) obtaining in this way a new formative construction, therefore y E that

X

S, then

s

Y

Y. Finally if S

is a subalgebra such

S follows easily by induction on t h e length of a

formative construction of an arbitrary element y E

Y.

0

5.8. Corollary (Definitions by Algebraic Induction). A subset S A coincides with the subalgebra X generated b y X if and only i f the following conditions hold:

s

(i)

if x E X then x E S;

(ii) if

...,z,(x) E S then fx(z1, ...,z,p)) E S (VX E A);

z1,

(iii) every element of S is obtained b y applying rules (i) and (ii)finitely many times. Proof. This is a paraphrase of Proposition 5.7: rules (i) and (ii) state that

the elements of the formative constructions belong to S while rule (iii) says t h a t every element of S has a formative construction.

5.9. Corollary (The Compactness Property). For each element z E X there is a finite non-empty subset

X such that

...,zn}.

z E (21,

0

...,x,}

(51,

57

Universal algebra Comment.

We mark this fact by writing z = ~ ( 2 1..., , 2,).

Proof. Let 2 1 , ..., z, E X

be the elements occurring in a formative construc-

tion of z .

0

Another corollary establishes a method for proving properties of elements in an algebra. The term "property" can be understood either intuitively, or according t o the following formal definition: a property defined on a set simply a map

:

7r

A

A

is

-+ {0,1};for every a E A , the property is interpreted is 1or 0.

intuitively as being true or false according as .(a) 5.10. Corollary (Proofs by Algebraic Induction).

Let X be a subset of an algebra ( A , {fx}xE+,).

(i)

?r

is true f o r all

2

7r

7r

satisfies

E X , and

(ii) (VX E A) (Vzl, ...,".(A) E A ) if fx(z1, ...,z.(q) satisfies ?r; then

If a property

$1,

...,x.(x)

satisfy

7r

then

i s true f o r all z E X ,

Proof. The set

Y

of all elements of A satisfying

a subalgebra by (ii),therefore X

7r

includes X by (i) and i s

CY.

0

5.11. Defi nitio n.

A congruence on a .r-algebra (A,{fx}xE~) is an equivalence relation

-

on

A such that

E A. The quotient algebra of the algebra A by the congruence is the pair (A, { j x } ~ , = ~ ) ,where A = A / and fx : AT(x)--+ a (A E A)

for every X

-

are defined by

N

Lattices, universal algebra and categories

58

5.12. Proposition. The quotient algebra of a r-algebra is a r-algebra. Proof.

For every X E

z1 N

and

...

and

A, if 21 = 2; and ... and ST(x) = then x : ( ~ )therefore , (5.3) implies fx(zl, ...,z,(x))

-

N

fx(zi,...)x : ( ~ , ) i.e. , the operations fA are well-defined.

0

T he construction of th e quotient algebra is well known i n each "con-

H is a normal subgroup of a group G t h e construction of t h e quotient group G / H obeys t h e general scheme of Definition 5.11, taking as th e congruence associated w i t h H , i.e. z y iff H z = Hy. crete" field of algebra. Note e.g. th a t if

-

N

5.13. Remark.

A congruence relation on t h e algebra A is a subalgebra of t h e direct product A x A. (The direct product o f a family {(Ai,{ f i x } x E ~ ) } i E ~ of r-algebras is II A ; , { f x } x E A ) , where for every X E A and every the r-algebra

(

$1

5.14. Definition. Let

A = ( A , { f x } x E ~ ) and 0 = ( B ,{ g X } X E A ) be r-algebras. A

T - ~ O ~ O ~ O T -

phism (also called a 7-morphism or simply a homomorphism or a morphism when no confusion is possible) from A to B is a map 'p : A + B which commutes with the operations f x , i.e.

A + B designates t h e fact t h a t 'p is a homomorphism, while Hom(A, B) or simply Hom(A,B ) denotes t h e set of

for every X E A. The notation

'p :

59

Universal algebra

all homomorphisms from A t o B. If one works simultaneously with various types of algebras it is convenient t o provide Hom with an index designating the type. The term 7-isomorphism designates a 7-homomorphism which is also a bijection. We say that two 7-algebras A and B are isomorphic

A M 0 provided there is a 7-isomorphism between them. The 7-homomorphisms and 7-isomorphisms from A t o A are also referred t o as the (T-) endomorphisms and (T-) automorphisms of A, respectively. and we write

The various concepts introduced in Definitions 1.20, 1.21, 1.27, 2.19 and 2.22 are particular cases of the universal-algebra concepts in Definition 5.14. Also, it will be tacitly assumed that the homomorphisms (isomorphisms, endomorphisms, automorphisms) corresponding t o the various classes of algebras to be studied in this book are given by Definition 5.14 applied t o the types of those algebras. 5.15. Remark. Several well-known properties of homomorphisms are easily transferred from various “concrete” fields of algebra t o the above general concepts. Thus e.g. the identity map 1~ is an automorphism. The composite of two homomorphisms (isomorphisms) is a homomorphism (an isomorphism). The relation

M

is reflexive, symmetric and transitive. If cp

E Hom(A,B), S is a subalgebra of A and T a subalgebra of B , then cp(S) is a subalgebra of B and cp-l(T) is a subalgebra of A . If is a congruence of A and A = A / -, the canonical map^ : A + A is a homomorphism.

-

5.16. Proposition (The Homomorphism Theorem).

Let cp : A + B be a homomorphism. T h e n : (i) (5.6)

T h e relation kerA,B cp

x ker cpx’

OT

simply ker 9,defined by

cp(x) = cp(z‘)

i s a congruence o n A .

(ii) FOT every algebra

C and every homomorphism II,

:

A

+

C such

Lattices, universal algebra and categories

60

that ker cp ker II,there is a unique homomorphism x : A / ker cp C such that II,= x o *.

(6) The map

x

:

---t

A / ker cp -+ cp(A) defined b y

(5.7)

is an isomorphism. Proof. Essentially th e same as i n each “concrete” field o f algebra.

0

5.17. Corollary.

If cp

:

A

t

B is

a

surjective homomorphism then B

M

A / ker cp.

Next we turn to classes of algebras and free algebras. T h e t e r m class will be given t h e usual meaning in set theory. 5.18. Defi n itio n . Let IC be a class of r-algebras and {Ai}iEIa family o f IC-algebras, i.e. A; E

K

(Vi E I ) . A subdirect product in K: of these algebras is a subalgebra S of the direct product ( I I A i ) E K such th a t S E IC and for every i E I and every xi E Ai there is x E S such that 1~iz= x i , where 7ri : IIAi -t A; are the canonical projections. A subdirect (direct) decomposition in K of a K-algebra A is an isomorphism between A and a subdirect product S (the direct product) of a family { A i } i E Io f K-algebras. W e usually express a subdirect (direct) decomposition i n th e form f : A + II A;, where f = L o g , g :

A

(where

--t

f

S is th e isomorphism and is t h e isomorphism).

L

: S --f IIAi is t h e inclusion mapping

T h e decomposition is said to be proper if

no factor Ai is isomorphic to A . T h e algebra A is said to be subdirectly

K: if cardA > decomposition in K.

(directly) irreducible in (direct)

1 and A has n o proper subdirect

5.19. Theorem (Birkhoff). Every r-algebra has a subdirect decomposition into subdirectly irreducible factors (in t he class of all r-algebras).

61

Universal algebra Proof. See e.g. Birkhoff

0

(19671or Pierce [1968].

5.20. Definition.

X , we denote by X * the set of all finite sequences of elements of X , including the "empty sequence" A. We then refer t o X as an alphabet, i t s elements are called letters and those of X' are termed words. If 2,,..., xm E X , the word (21,..., xm) will be denoted simply by For every non-empty set

concatenation: zl...x,, and said t o be a word of length m. The words ~ ~ . . . x ~ +(k~ = z k1, ...,m - 1) are called the segments of 21. .AT,.

More ge-

nerally, given the words wl,...,wn, we denote by wl...wn the word obtained by their concatenation.

5.2 1. Definition. Let X be a non-empty set.

F n X' = 0. (W,{Fx}AEA), where for

such t h a t

Take a A-indexed set

The set W =

(X

F = {FA1 X

E

A)

U F)' is made into a T-algebra

E A the operation denoted FA maps every sequence (wl, ...,w,(l)) E W'@)t o the word FAw ,...w,(X) E W . By the Peano (T-) algebra on the set X of free generators we mean the subalgebra P ( X ) of (W,{ F A } A ~generated A) by X . each X

We will prove that the Peano algebra is a free algebra in the following sense.

5.22. Definition. Let: R be a class of T-algebras, A a r-algebra and X a set. Then A is said t o be the free algebra i n the class R (or simply t h e free R - a l g e b r a ) on the set X of free generators (or equivalently, freely generated by X ) provided the following conditions hold: (a) A E

R;

(b) there is an injection i (b,)

i ( X ) = A and

:

X +A

such that

Lattices, universal dgebra and categories

62 (b2) for every algebra B. E

R

unique homomorphism (p : A

If-as

is usually th e case-X

i ( z )= z (Vz) then (p

o

B

and every map cp : X -+ t

5 A

B

such t h a t (p o

there is a

i = cp.

and i is the inclusion mapping

i = cp reads (p I X = cp.

5.23. Lemma.

If w E P ( x ) there is no segment of w belonging to P ( x ) . Proof by induction of th e length n of w. If n = 1 (i.e. w E X or with .(A)

w = FA

= 0) then w has no segment. For the passage from n t o n + l take

w E P ( X ) of length n + l >_ 2. Inasmuch as P ( X ) is generated by X , w is of

w = FAwl...w,(~) for some X E A and wl, ...,w,(A) E P ( X ) where r ( X ) 2 2. If w' is a segment of w and w' E P ( X ) then w' = FAW; ...w:(~) for some wi,,..,w:(~) E P ( X ) . This implies there is an index k E (1, ...,.(A)} the form

such that wI, = segment of

Wk ,

Wh

( h = 1,...,k

- 1) and

WE, # wk.

Therefore

which contradicts the inductive hypothesis.

WE,

is a 0

5.24. Lemma.

x

FOT every w E P ( x ) either w = x E or w is uniquely represented in the f o r m w = F A W...~w,(A) with w1,...,w,(x) E P ( X ) . Proof.

For n = 1 the statement is trivial.

For n

>

1 suppose there

is w E P(X) of length n and having two distinct representations w =

FAW ...w,(x) ~ = Fpw~...w~(,~ with A, p E A and w1, ...,w,(A), wi, ...,w&) E P ( X ) . It follows that FA= F, therefore X = p and w1...w,(x) = wi ...w', ( P I . This implies further th e existence of an index k E {1,..., r(X)}such t h a t w; = wh ( h = 1,..., k - 1) and w6 # Wk. Therefore wi i s a segment of W k , which contradicts Lemma 5.23.

0

5.25. Theorem.

FOT every non-empty set X , the Peano r-algebra P ( X ) on X is the free algebra freely generated b y X in the class of all r-algebras. Proof. Conditions (a) and (bl) in Definition 5.22 are fulfilled for the inclusion mapping

i

: X

--f

P ( X ) . To prove (b2) let ( A , { ~ A } x ~be A )a 7-algebra

Universal algebra

63

and y : X + A. Define a relation (p

c P(X) x A

by algebraic induction

in the algebra P(X) x A:

(a) ( x , y ( z ) ) E p for every z E X,

(7)every element ( w ,y) E (p is obtained by (a)and (p). In view of Lemma 5.24 every w E P ( X ) appears as the first component of an element ( w , y ) E (p exactly once, therefore (p is a function

A . Now conditions (a)and (p) read F I X = y and (p(Fxwl...w,(x)) = fx ((p(wl), ...,(p(wT(x))), i.e. (p is a homomorphism. Finally if 1c, E H o m ( P ( X ) , A ) and 1c, ] X = cp it is easily proved by algebraic 0 induction in P ( X ) that $(w) = (p(w) for every w E P ( X ) . (p

: P(X)

---t

Our next step is t o introduce equational classes and their free algebras. The basic concept is that of identity o f an algebra. To understand Definition 5.26 below take the simple example of the identity

of a semigroup

(G,.). Let (P(X),F) be t h e

Peano algebra of type (2) and

take wl,wz E P(X) defined by w ~ ( ~ , x ~ = ,F zF x~1 )x 2 x 3 and

w2(x1, ~

2 ~ x = 3 )

Fq

F5223,

respectively (cf. the comment t o Corollary

5.9). Then for every cp E H o m ( P ( X ) , G ) such that c p ( z k ) = a k (k = 1,2,3) we have y(wl)= (al . u 2 ) a3 = al - (a2 . a3) = cp(w2), so t h a t (5.8) can be viewed as expressing the property cp(wl) = y(w2) (Vcp E Hom(P(X), G)). 5.26. Definition. Let P(X) be a Peano r-algebra and set

(5.9)

I d A = n {ker y I y E H o m ( P ( X ) , A)} ,

(5.10)

IdK = n {IdA I A E K )

Lattices, universal algebra and categories

64

for every r-algebra A and every class K of r-algebras. If (wl, w2) E I d A we say t h a t (w1,w2) is an identity of A or the identity w1 = w2 holds in A;

the pairs in IdK are called the identities of the class

K.

5.27. Proposition.

Let P(X) be a Peano r-algebra and

K

a class of r-algebras. T h e n

(i) P(X)/IdK: is a r-algebra and

(ii) f o r every algebra A E K and every m a p v : X --f A there is a unique h o m o m o r p h i s m V E Hom(P(X)/Id K ,A) saeh that v = V 0-0i, where i : X + P(X) i s the inclusion m a p a n d ^ : P ( X ) + P(X)/IdK i s the canonical surjection. Proof.

(i) IdK: is a congruence in view of Proposition 5.16(i) and the easy remark that the congruences of an algebra form a Moore family. Therefore P(X)/IdK is a r-algebra by Proposition 5.12.

i

Fig. 5.1.

(ii) In view of Theorem 5.25 there is a unique cp E Hom(P(X),A), such that v = cp o i. Inasmuch as ker^= IdK kercp by (5.9) and (5.10), Proposition 5.16(ii) (applied for A := P(X), B = C := P(X)/IdK:, cp := *, C := A, 1c, := cp) yields the existence of a unique V E Hom(P(X)/IdK:,A) such that cp = V o It follows that v =

i = V 0 - 0 i. To prove the uniqueness of fi with the latter property, take 1c, E H o m ( P ( X ) / I d K , A ) such that v = 11, 0 - 0 i. Then 11, o A = cp by the uniqueness of cp, therefore 11, = E by the uniqueness of V. 0 cp o

65

Universal algebra 5.28. Definition.

A class

K

of .r-algebras is called a variety or an equational class provided

A x K C_ P ( X ) x P ( X ) such that

there is a set

A E K e A x K 2 IdA

(5.11)

for any r-algebra A (cf. Definition 5.26). Many important classes of algebras, e.g. semigroups, groups viewed as algebras of type (2,1,0), lattices, Boolean algebras a.s.o., are defined by systems of identities; such a system of axioms can be viewed as establishing an equivalence of the form (5.11) so that the corresponding classes of algebras are equational in the sense of Definition 5.28. 5.29. Lemma.

T h e following conditions are equivalent f o r a class

(i)

K

K

of r-algebras:

is a n equational class;

IdK G Id A) f o r a n y .r-algebra A;

(ii) (A E K

ij

(iii) (IdK

IdA 3 A E K ) f o r a n y r-algebra A.

Proof.

(i) + (iii). It follows from (i) that A x K G

(l IdA = IdK therefore AER

I d R C IdA

+ A x K G IdA + A E K.

(iii) =$ (ii). Because th e implication converse t o (iii) is trivial. (ii) =+ (i). Trivial. It is easy t o see that if

K

is a variety and

0

A E K then every subalgebra

of A is in K ,every homomorphic image of A (or equivalently, every quotient algebra of A) is in

K

and every direct product of algebras of

K

is in

K ;cf.

Definitions 5.3, 5.11, 5.13, 5.14 and Corollary 5.17. (Remark 1.1.34 and the remark following Definition 1.2.24 are particular cases of the above remark.)

The converse also holds, i.e. a class K of r-algebras is equational if and only

Lattices, universal algebra and categories

66

if it is closed with respect t o taking subalgebras, homomorphic images and direct products. This is the Birlcho$ T h e o r e m which we quote here without proof.

5.30. Theorem. Let K be a n equational class of r-algebras and P(X) the Peano r-algebra

o n X. T h e n P(X)/IdK is the free K-algebra o n the set X. Proof.

let i

: X + P(X) be the inclusion map and

: P(X) -+

A

P(X)/IdK the canonical surjection. Then- o i : X -+- P(X)/IdR is an injection. From P(X) = X we obtain P(X)/IdK = X = o i)(X), i.e. condition (b,) from Definition 5.22, while (b,) holds by Proposition 5.27. It remains t o prove (a), i.e. P(X)/IdK E K. In view of Lemma 5.26 this is equivalent t o IdR Id(P(X)/IdR). Thus we take (w1,wZ) E IdK and according t o Definition 5.26 we must prove that cp(w1) = cp(w2) for every cp E H o m ( P ( X ) , P ( X ) / I d K ) . Set (p(wk) = ziji ( k = 1,2). Further we t a k e A E K: and 1c, E H o m ( P ( X ) , A ) ; then ker^ = IdK I d A hence by Proposition 5.16(ii) there exists x E H o m ( P ( X ) / I d K , A ) such t h a t 1c, = x o inasmuch as x o cp E H o m ( P ( X ) , A ) it followsthat IdK ker(Xocp) hence(wl,w2) E ker(Xocp)

C

c

c

A.

and

(wi,wi) E ker4 for every 1c, E H o m ( P ( X ) , A ) and every A E K , therefore (wi,w:) E IdR or equivalently ziji = G;, that is cp(wl) = cp(w2). Thus 0

The remainder of this section is based on the following remark. Roughly speaking, t h e object of universal algebra is the study of the general concept of algebra, while other fields of algebra deal with “more concrete” algebraic structures, like groups, rings, modules, lattices etc. Note however that there may be several ways of thinking of a “classical” algebraic structure as a .r-algebra; thus e.g. a group can be viewed not only as an algebra of type

67

Universal algebra

(2.1,O) but also as an algebra (G,.) of type (2). We are going t o concentrate a bit more on this point.

5.31. Definition.

Let (A,{f,}xE~) be a 7-algebra and X a non-empty set. The set AAX=

{f 1 f

: AX

4

A} is made into a 7-algebra ( A A X{f,,},,~) ,

X E A and every

01,

where for every

...,o , ( ~E) AAX,fx(ol,...,o,(,)) is defined by

we usually write simply

fx instead of

f,,.

5.32. Definition. Let A be a 7-algebra and

X

a non-empty set. By A[X] we denote the

subalgebra of AAX generated by the set (5.13)

T ~ ( V= ) W(Z)

{7r,},E~

of projections, defined by

(VW E Ax)

for each z E X. The elements of A [ X ] are said t o be t h e polynomials of A in the variables from X. An alternative description of polynomials is given in Definition 5.33 jus-

tified by Proposition 5.34. 5.33. Definition. Let P(X)be a Peano 7-algebra and A a 7-algebra. The canonical m o r p h h m N

: P(X) 4 AAXis the homomorphic extension of the map T : X

defined by ~ ( x=) 7rz

AAX (Vx E X). For each w E P(X), 6 is called t h e poly-

nomial generated by w. 5.34. Proposition.

P p )= A[X]. Proof. By double inclusion, each of them by algebraic induction:

4

Lattices, universal algebra and categories

68

5.35. Proposition.

Homomorphisms commute with polynomials (2.e. if cp E Hom(A, B ) and w E P ( X ) then ~ ( 6 =)

cpF)).

Proof. Algebraic induction.

The canonical morphism

0

N

yields an alternative description of t h e iden-

tities o f an algebra.

5.36. Lemma.

Let P ( X ) be a Peano .r-aEgebra, A a r-algebra, V E Hom(P(X),A) the homomorphic extension of a m a p v E AX, and w E P ( X ) . T h e n G ( v ) = v(w).

w. For x E X we have 2 ( v ) = ~ ( x ) ( v=) a,(v) = v(z) = C ( x ) . If w1,...,wT(x) E P ( X ) satisfy the property then Proof by algebraic induction on

-

Fxwl*"wT(x)(v)

=

= f x ( G 1 > *'.,GT(x))(v)=

fx(61(v),"',tZT(x)(v))

= fx(v(wl)>

.*.?V(wT(X)))

=

.

= ij(F~Wl...W,(~))

0

5.37. Proposition.

Let P ( X ) be a Peano algebra, A a r-algebra and w1,w2 E P ( X ) . T h e n

Proof. Note that every cp E Hom(P(X),A) is of the form

v E AX, namely v = 'pix. Therefore Lemma 5.36 implies

'p

= 6 where

69

Universal algebra

@ GI(.)

(V.)

= tq.)

.

0

At this point note that when we study a specific algebraic structure we are in fact interested i n its polynomials rather t h a n in t h e choice of t h e basic operations. Thus e.g., no matter whether an Abelian group is viewed as an algebra of type (2,1,0) or of type (2), i ts polynomials are t h e operations of the form p ( t l , ...,zn) = t y ...22,where al,...,a, E

Z.

In this example th e basic operation of type (2) is one of t h e basic operations of type (2,1,0), b u t this feature is irrelevant, as it is missing in the next example. It is well known th a t every Boolean algebra becomes an idempotent (i.e., x 2 = t for all

(5.15')

5

t) ring

+ y = (2 A y) V ( 5 A y) ,

w i th unit if one defines

zy = 2 A y

and conversely, every idempotent ring w i th unit is a Boolean algebra under the operations

moreover, this correspondence is one-to-one.

Let us set this point i n t o a

general framework.

5.38. Definition. T w o algebras

A and A' o n th e same underlying set A (but not necessarily

of t he same type) are termed equivalent if they have the same polynomials.

5.39. Definitio n . A class K: of algebras of type T and a class K:' of algebras of type r' are said to be equivalent if there exist tw o maps @ : K + K' and !D : K:' --t K such that:

(i) every A E

K: is equivalent

to

@ ( A )and every A' E K' is equivalent to

@(A'),and (ii) * @ ( A )= A for every A E K: and @@(A') = A' for every A' E K:'.

70

Lattices, universal algebra and categories Now we can present a construction which is a sufficient condition for the

equivalence of two equational classes o f finite types. 5.40. Remark. Let

T

: {1,...,m } +

IV

be a finite type and

X

a non-empty set. Every

w E P ( X ) has a formative construction which uses (some of) t h e symbols Fl,...,F,; we indicate this by writing w = w(Fl,..., Fm).Further let ( A ,{ f i } + ~ ) be a T-algebra. Inasmuch as'is a homomorphism, it follows easily by algebraic induction that the polynomial G has a formative construction which uses fi, ..., f, in t h e same way as the construction of

Fl,...,F, have been used in

w ;we indicate this fact (which can be given an obvious

technical definition) by writing G = G(f1,

...,f m ) .

5.41. Notation. Let

T

and

(T

be two types o f algebras and X a non-empty set. Let P o ( X )

and P T ( X )denote the corresponding Peano algebras; their elements are denoted by w and w' respectively, possibly provided with indices. Finally let

K' be equational classes of .r-algebras and : K: +K:'and Q : K : ' + K : .

and

K:

o-algebras, respectively. Let

The next proposition uses the above notation. 5.42. Proposition.

Suppose T : (1, ...,m } + IV and (T : (1, ...,n } + IV are finite t y pes and Ax K: = {wlh, w z h ) / h E H } , Ax K:' = { w i k ,w i k ) / k E I<} f o r finite H and I<. Suppose further there exist w1, ...,w, E P o ( x ) and w;, ...,w:, E P T ( X )such that:

(i) f o r a n y A = ( A , { f ; } i = ~ E)

K: the operations

are of arities a(l), ...,( ~ ( n respectively, ), and satisfy (5.17)

a' W,

(gl,..., g,) = fi

(i = 1,...,m ) ,

71

Universal algebra

(5.18)

m’ Wlk

where” and

m

(Sl,...,gn) =EL. (g1, ...,gn)

(Vk E I.’)

denote the canonical morphisms of the algebras A and

@(A)= ( A , {gj}j=F), respectively, and (ii) for a n y A’ = ( A , {gj}j=F) E K’ the operations (5.19)

fi

= Gi(g1,

...,gn) (i = 1,...,m )

are of arities .r(l), ...,~ ( r n )respectively, , and satisfy

where- and k: denote the canonical morphisms of the algebra A’ and Q(A‘)= ( A , { f ; } i = ~ )respectively. ,

Then the classes K and K’ are equivalent via the transformations @ and Q. Proof. For every therefore @ :

K

A E Ic, @(A)is --f

a .r-algebra and

K’. It follows easily from (5.16)

@ ( A )E K’ by (5.18), that the polynomials of

+ ( A ) are polynomials of A, while (5.17) implies t h a t t h e polynomials of A are polynomials of @(A), therefore A and * ( A ) are equivalent. One proves similarly that Q : K’ + K and A’ and Q(d’)are equivalent for every A’ E K’. It follows readily that Q @ ( d=) A and @*(A‘)= A’. 0 5.43. Definition.

The situation described in Proposition 5.42 will be expressed simply in the following terms: every algebra of K is equivalent t o a n algebra of K’ via the transformations (5.16) and (5.19). Thus the example before Definition 5.35 can be reworded t o the effect that every Boolean algebra is equivalent to a Boolean ring with unit via the transformations (5.15’) and (5.15”). in the subsequent chapters.

We will actually use Proposition 5.42

Lattices, universal algebra and categories

72

We conclude this section with a universal-algebraic construction which is important in logic. 5.44. Definition.

be a family of .r-algebras (or simply sets) and E a filter of the Boolean algebra ( F ( H ) , s ) . Let further A = II Ah be t h e direct Let

(&,)hEH

hEH

product of t h e algebras Ah and -F- the relation on A defined by

5.45. Lemma. -F- is a congruence on

A.

Proof. Routine.

0

5.46. Definition.

If

is an ultrafilter on

( P ( H ) ,c)then t h e quotient algebra A / -F, -

denoted

also A / E , i s called t h e ultraproduct of the algebras Ah with respect t o E.

73

Categories and f u n c t o r s

56. Categories and functors In this section we list those category- heoretical concep s and results that will be used in the book. Proofs are omitted.

6.1. Definition. A category C is a class ObC endowed with the following structure: (i) For any A, B E ObC there is a set C(A, B ) such that (A,B)#(A‘,B’)

+ C(A,B)nC(A’,B‘)

=8;

(ii) For any A, B , C E ObC and for any f E C(A, B ) , g E C(B, C ) there is a unique element g o f of the set C(A,C) such that the following conditions are verified:

f E C(A,B), g E C(B, G ) , h E C(C, 0) f) = ( h o g ) o f ;

(a) If A, B, C,D E ObC and

then h o (g o

(b) For any A E ObC there is an element 1~ in C(A,A) such that 1~ 0 f = f and g o 1 A = g, for any B,C E ObC, f E C ( A , B ) and g E C(A,C). The elements of ObC are called objects of C and an element f of C ( A , B ) is called a m o r p h i s m from A t o B. If f E C(A,B) then we shall write f : A + B or A & B ; A is the d o m a i n o f f and B t h e codomain of f ; g o f is called the composite o f f and g. Thus composition is a partially defined associative operation. The element 1~ is called t h e identity of A . We shall write A E C instead of A E ObC and sometimes g f instead of g o f . It is easy t o verify the uniqueness of the identity. A category is usually described by mentioning first i t s objects, then i t s morphisms. Thus e.g.:

6.2. Definition. In what follows Set will be the category of sets and functions and Top the

Lattices, universal algebra and categories

74

category of topological spaces and continuous functions. If K: is a class of similar algebras then K will be the category whose objects are t h e algebras in K and whose morphisms are all t h e homomorphisms f

: A -+ B , where

A,B E K . K will be called an algebraic category. Thus Po is the category of posets and isotone mappings, DO1 is th e category of bounded distributive lattices and bounded-lattice homomorphisms, Mg is the category of De Morgan algebras and De Morgan homomorphisms, B is the category of Boolean algebras and Boolean homomorphisms. An equational category is an algebraic category K where

K is a variety. K is trivial if cardA

= 1 for any

A E K. 6.3. Definition. The d u d category Co o f a category C is defined by ObC' = ObC, C o ( X ,Y )= C ( Y , X ) for any X , Y E Co and for any morphisms f : X 4 Y and g : Y 4 2 , t h e composite g o f in C" is the composite f o g in C. 6.4. Definition.

A subcategory of C is a category C' such that (i)

ObC' E ObC;

(ii) C'(A,B)5 C(A,B)for any A,B E C'; (iii) For any A E C' the identity of A in C' coincides with the identity of

A in C . (iv) The composition of morphisms in C' is the same as in C.

C' is a fuZ1 subcategory of C if to conditions (i)-(iv) we add (v)

C'(A,B ) = C(A,B ) for any A, B E C'.

6.5. Definition. A morphism f of a category C is a monomorphism (an epimorphism) if f o g = f o h implies g = h (if g o f = h o f implies g = h ) for any morphisms g , h in C.

Categories and functors A morphism f : A

75

B of C

isomorphism if there is a morphism g : B + A in C such that g o f = 1 A and f o g = 1 B ; the morphism g is unique and is denoted by f - l . Then f-’ i s also an isomorphism. The 4

is an

composite of two isomorphisms is an isomorphism. Every isomorphism is a monomorphism and an epimorphism, but the converse is not true. Two objects A , B are isomorphic if there is an isomorphism f : A -+ B ;

in this case, we will write A

E

B.

B E C if there exist two morphisms f : A --t B and g : B 3 A such that g o f = 1 A . A variant of this definition i s to say that A is a retract of f : A 4 B if there i s g : B -+ A such that g o f = 1 A . An object A E C is a retract of an object

Note t h a t if g o

f = 1~

then

f is a

monomorphism and g is an epimorp-

hism.

6.6. Proposition (Balbes and Dwinger [1974]). Let K be a n equational category and f E K(A, B ) . T h e n f is a m o n o -

morphism ;$ f is injective.

6.7. Definition. Let A be an object of a category C. A subobject of A is a monomorphism

f

B + A ; sometimes we will abusively say t h a t B is a subobject of A . f2 : B” -+ A are isomorphic if there is an isomorphism g : B’ -+ B” such t h a t f2 o g = f l , :

Two subobjects f1 : B‘ + A,

6.8. Definition. An eztension of an object A is a monomorphism f : A -+ B. An extension f’ : A + B‘ is included in (isomorphic to) an extension f ” : A --t B” if there is a monomorphism (an isomorphism) g : B’ -+ B” such that g o f‘ = f”. An extension f : A --t B is called proper if f is not an isomorphism and essential if for any morphism g : B -+ B’, if g o f is a monomorphism then g is a monomorphism.

76

Lattices, universal algebra and categories

6.9. Definition. Let {Ai}iE~ be a family of objects in a category C. A direct product of

{Ai}iEl is a family {ri : A + A;}i,g of morphisms in C with the property t h a t for every family {f; : B + A i } i E l of morphisms in C there is a unique morphism f : B + A such that ri o f = fi for any i E I . The morphisms r; are called projections. 6.10. Remark. In any equational category K the direct product of a family { A i } i € ~is the Cartesian product

II Ai endowed with the canonical structure of an algeiEI

bra in

I<.

6.11. Definition. Let {Ai}iEI be a family of objects in a category C. A direct sum (coproduct) of { A i } i E if ~ a family {ji : Ai + A}i,I

of morphisms in C with the property

Ai + B};,I of morphisms in C there is a unique morphism f : A --t B such that f o j ; = fi for any i E I . The morphisms ji are known as injections (although they need not be injective functions). that for every family

{fi:

6.12. Remarks. (a) The direct product of a family {Ai}iEI is unique up t o an isomorphism:

if {xi : A + Ai}iEl, {r: : A’ + Ai}iEl are two direct products of { A i } i E then ~ there is an isomorphism f : A + A’ such t h a t r: o f = ri for any i E I . Sometimes we shall say that the direct product of {Ai}iE~ is A and write A = II A;. &I

(b) The direct sum of a family ( A i } i € ~is unique up t o an isomorphism. If

{ji : A; + A}iEl is the direct sum of { A i } i € ~we also say that the Ai.

direct sum of {Ai};€l is A and we write A =

i€Z

Proving the existence of direct sums is in general a difficult problem. For

77

Categories and functors algebraic categories we have the following result:

6.13. Theorem (Pierce [1968], Corollary 4.2.8). If K is a n algebraic category closed t o direct products and subalgebras and {A;}iE1 is a f a m i l y of objects in K such that f o r each i E I , Ai is a subalgebra of B;, f o r s o m e B; E K and there exist morphisms h;,j : A; t Bj f o r each pair i , j , t h e n the coproduct of

{Ai}iEl

ezists.

6.14. Definition.

If C, D are two categories, then a covariant f u n c t o r or simply a f u n c t o r F : C -+ D is a map which assigns t o any A E ObC an object F ( A ) of D and t o any morphism f : A t B in C a morphism F ( f ) : F ( A ) 4 F ( B ) in D, such t h a t

(b) F ( g o

f) = F ( g ) o F ( f ) , for

any morphisms f : A

+ B,

g : B +C

in C.

A contravariant f u n c t o r F : C t D is a map which assigns t o any A E UbC an object F ( A ) of D and t o any morphism f : A -+ B in C a morphism F(f) : F ( B ) + F ( A ) in D such t h a t (a')

F ( ~ A=) ~ F ( A )for , any A E ObC; ,g : B

t

C

For every category C we will denote by idc the identity f u n c t o r idc : C

--t

C

(b') F ( g o f) = F ( f ) o F ( g ) for any morphisms f : A in C.

defined by idc(A) = A and idc(f) =

f E C ( A , B ) . If F

D

f

and G

+B

for any A , B E ObC and any

D

E are two covariant (contravariant) functors, their composite G o F : C t E is defined by (G o F ) ( A )= G(F(A)) and (G o F ) ( f )= G(F(f)) for any A , B E ObC and any f E C(A,B). : C

-+

:

t

It is easy t o see that: the composite of two covariant (contravariant) functors is a covariant functor, composition is associative, F o idc = F for

Lattices, universal algebra and categories

78

F : C

any functor

--t

D

and idc o G = G for any functor G :

D + C.

6.15. Defi nitio n . A covariant functor F : C + C' is fulZ (faithful, fully faithful) if, for any

A , B E C , the map

F : C ( A ,B ) + C ' ( F ( A ) , F ( B ) ) is injective (surjective, bijective). A similar definition is given for t h e contra-

F preserves (reflects) a property P if F(f)(if f ) satisfies P whenever f (whenever F ( f ) ) has property P.

variant functors. A covariant functor

6.16. Definition. A covariant functor F : C + D is an isomorphism if there is a covariant functor G : D + C such t h a t G o F = idc and F o G = idD. This concept being t o o restrictive, one usually prefers t h e concept of

equ ivaIe nce.

6.17. Defi nitio n . A covariant functor F : C + D is an equivalence of categories if it is fully faithful and for any B E D there is A E C for which F ( A ) is isomorphic to B. We say t h a t the categories C and D are equivalent. 6.18. Remarks. (a) Any equivalence

F : C + D preserves and reflects monomorphisms,

epimorphisms, direct products and direct sums.

(b) A contravariant functor F : C + D defines in a natural way a covariant functor

F* : C + Do. F is a coequivalence of categories if F* is an

equivalence o f categories. In this case, t h e category D is equivalent t o t h e dual category o f

C.

6.19. Definition. Let F

:

C + D, G

:

D

--f

C be t w o covariant functors. A

na-

79

Categories and functors

tural transformation or a functorial m o r p h i s m X : F 4 G is a family {XA : F ( A ) t G ( A )I A E C} of morphisms of D such that for any morphism f : A -+ B in C the diagram in Fig. 6.1 is commutative. We say that X is natural in A.

6.20. Notation. Let C be an arbitrary category and X E the covariant functor defined by:

C. We will denote by hX : C -+ Set hX(Y)= C(X,Y), Y E C and for

Y + 2 , h x ( f ) is the function given by h x ( f ) ( g ) = f o g, for E C(X,Y). The contravariant functor h , : C + Set is defined by: h x ( Y )= C(Y,X), Y E C and for f : Y 4 2 , h x ( f ) is the function given

f

:

any g

by hx (f>(s) =9

0

f,for any 9 E C(2,XI.

6.2 1. Def inition. Let F : C -+ D , G : D t C be two covariant functors. F is a Zeft adjoint of G and G is a right adjoint of F if for any X E C ,X’ E D there is a bijective map:

cp(X,X‘): C(X,G(X‘)) + D(F(X),X’) such t h a t for every morphisms

f : X

--f

the following diagrams are commutative:

Y

in

C and f’ : X’+ Y’in D

Lattices, universal algebra and categories

80

6.22.Remarks. X

(a) Let us consider the morphisms qx :

X‘

--+

G ( F ( X ) ) ,EX’ : F ( G ( X ’ ) ) --t

defined by qx = cp(X, F ( X ) ) - * ( l F ( X ) )E,X ! = (p(G(X‘),X’)(lc(x,)).

We can see that qx is natural in X and E X : is natural in X’, therefore we have the natural transformation q : idc -+ G o F and E : F O G4 idD. q and e are called adjointness morphisms associated with F and G. (b) Let C‘ be a subcategory of C. C‘ is reflective if t h e inclusion functor C’ + C has a left adjoint, which is called the refEector.

6.23. Proposition.

If q, e are the adjointness morphisms G

:

of the adjoint functors F : C + D ,

D -+C and X E C , X‘ E D then

81

Categories and functors

6.24. Proposition. Let F : C + D be a covariant functor and G a right adjoint of F . Then F preserves direct sums and epimorphisms, while G preserves direct products and monomorphisms. 6.25. Proposition. Let G : D + C be a right adjoint of F : C (a)

-4

D. Then:

The functor F as faithful iflprlx is a monomorphism for each X E C.

(b) G is fully faithful i f

EXI

is an zsomo~phismfor each X’ E D.

6.26. Proposition. For any covariant functor F

:

C + D the following assertions are

equivalent (i)

F is an equivalence of categories;

(ii) F is fully faithful and it has a fully faithful left adjoint; (iii) F is fully faithful and it has a fully faithful right adjoint. The proofs of Propositions 6.24-6.27 can be found in MacLane I19711 or Popescu and A. Radu [1971]. 6.27. Definition. An object A of a category C is called injective if for any monomorphism f : B + C and for any morphism g : B + A there is a morphisrn h : C + A such that h o f = g . 6.28. Definition. An object A of C is called projective if for any epimorphisrn f : C -+ B and for any morphism g : A --t B,there is a morphism h : A --f C such that f o h = g .

32

Lattices, universal algebra and categories

6.29. Remarks. (a)

A retract of an injective (projective) object is also injective (projective).

(b) A n injective object A is a retract of any of its extensions A (take B = A , C = A, f : A + A and g = 1~in Definition 6.28). (c) Any direct product (sum) of a family of injective (projective) objects is also injective (projective).

6.30. Definition. An

injective hull of A E C is an injective essential extension of A .

6.31. Remark. In every equational category any t w o injective hulls of an object are isomorphic.

83

CHAPTER 2 TOPOLOGICAL DUALITIES IN LATTICE THEORY

There is a wide literature dealing w i t h t h e characterization of various classes o f lattices as categories dual to certain categories of topological spaces. In this chapter we present t h e Stone duality and t h e Priestley duality for bounded distributive lattices and De Morgan algebras, which will be used in Chapter 6. See also Brezuleanu and Diaconescu [1969] and Brezuleanu

[1972], [1978]. $1.The Stone duality o f distributive lattices In this section we present the categories dual to

DO1 and B i n terms of

Stone spaces.

1.1. Definition.

A S t o n e space is a topological space X satisfying t h e following conditions: (i) X is a compact To space; (i) the set K X of compact open subsets of X is a ring of s e t s (i.e., closed w i t h respect to finite intersections and unions) and a basis for the topology of

X;

(iii) if C C K X is closed w i t h respect t o finite intersections and F C X is a closed set such that

F n Y # 0 for every Y E C then F n

n

Y

# 0.

Y €C

1.2. Definition.

A mapping f : X -+ Y between t w o Stone spaces X , Y is said to be strongly c o n t i n u o u s provided f-l(A) E K X for every A E KY.

Topological dualities in lattice theory

84

Clearly every strongly continuous function is continuous.

1.3. Notation. (a) We denote by S t the category of Stone spaces and strongly continuous

functions.

(b) For every L E of

DO1 we

denote by Id L the complete lattice of all ideals

L (cf. Theorem 1.3.6),by Spec L

and by

k the

Id L the set of prime ideals of L

C

mapping

P(SpecL) ,

(1.1)

k : Id L

(1.2)

k(1) = { P E Spec L 1 I

---t

g P} .

1.4. Lemma.

The following properties hold:

(1.3)

k({O}) = 0 ,

(1.5)

k(l1 n 12)= k ( I l )n k(Iz),

k ( L ) = SpecL

,

Proof. (1.3) is obvious. The non-trivial part of (1.4) follows from t h e fact

g

I2 then we can take z E Il- I2 and using t h e dual of Corollary 1.3.11 we find P E SpecL such that I2 E P and z # P , hence P E k(I1) - k(I2). Further (1.4) implies the “C” part of (1.5) and the “2’‘ part of (1.6). Conversely, if P E k(I1) n k ( 4 ) then there exist x E Il- P and y E I2 - P , hence x A y 4 P while z A y E I, fl I2 because x A y 5 x,y; this implies P E k(l1 r l I2). Also, if P # U k(1,) then It C P for every that if

Il

tET

t

E

T,hence V It C P , therefore P 4 k( V tET

t€T

It)

0

The Stone duality of distributive lattices

85

1.5. Notation. Let i : L -+ IdL be the embedding i ( u ) = ( u ] . Let s = k o i, that is s : L -+ P(SpecL) is defined by s(u) = { P E

(1.7)

Spec& I a (Z P} .

1.6. Corollary. The following properties hold:

(1.8)

s is

(1.9)

s(0) = 0,s(1) = Spec L

(1.10)

s ( x A y) = s(x) fl ~ ( y ),

(1.11)

s ( 5 v y) = s ( x ) u s(y)

injective ;

,

.

Proof. From Corollary 1.3.13, Theorem 1.36 and Lemma 1.4 applied to and (Yl.

(21

1.7. Corollary. The lattice L as isomorphic to ( { s ( z ) / xE L},fl,U,8,SpecL). 1.8. Theorem. For every L E DO1 the family { k ( l ) / I E IdL} is a topology on SpecL

3UCh that

(1.12)

Spec L is a Stone space ,

(1.13)

K(SpecL) = ( s ( z ) / x E L }

.

Proof. The sets k ( l ) form a topology by Lemma 1.4. Since

I= V

aEI

for every

(1.14)

I E Id L , it follows that

k(1) = U k((a])= U aEI

a d

S(U)

(a]

Topological dualities in lattice theory

86

by (1.6) therefore the family { s ( x )1 x E L } is a basis of t h e topology and moreover it is a ring of sets by (1.10) and (1.11). Every s ( x ) is compact, because if s ( x )

k( tET V It)

by (1.6) hence

2

E

V

c

U k ( I t ) then k ( ( x ] ) G t€T

It by (1.4), therefore t h e dual of

t@

x 5 al V...Va, for some n E lN,tl, ...,t , E T and ah E It,, ( h = 1,...,n ) , so that we can go back and get s(z) E k ( I t , ) U ... U k(Itn). We have thus proved the ''2" part of (1.13). Conversely, if k ( I ) E K ( S p e c L ) then from (1.14) and (1.11) we infer the existence of Proposition 1.3.4 implies t h a t

n E lN and ul, ...,a,, E I such that

k(I) = s(s1) u ... u S ( U , ) = s(u1 v ... v a,) . SpecL is compact by (1.9) and (1.13). If Pz hence there is P1,P2E SpecL and Pl # P2then we have e.g. Pl a E Pl - P 2 implying t h a t Pl 4 s ( u ) but P2 E s(u). Thus SpecL is To and we have verified (i) from Definition 1.1. As (ii) follows from (1.12) and the first part of the proof, it remains t o check (iii). Let C = { s ( u t ) It E SpecL fulfil the hypotheses from (iii). Then T } C K ( S p e c L ) and F F = S p e c L - k ( I ) for some I E IdL. Since C is closed with respect t o finite intersections, it follows from (1.10) that so is the set { a t It E T},therefore the filter A generated by this set in L is A = {x E L I3t E T,at 5 x} by the dual of Proposition 1.3.4. But A n I = 0, otherwise at 5 II: for some x E A n I and t E T ,hence Finally we prove (1.12).

that is F

=

P E SpecL

0, a

contradiction.

Now apply Theorem 1.3.10 t o

such that I C_ P and A

nP

0.

It follows t h a t E SpecL - k ( I ) = F and also P E ~ ( I I : for ) every x E A, therefore

obtain

P

n .(at)

=

1.9. Definition. The topology constructed in Theorem 1.8 is called the Stone topology of

87

The Stone duality of distributive lattices Spec L.

1.10. Notation. For every L , L’ E DO1 and f E Hom(L,L’),

(1.15)

Specf : Spec&’ 3 Spec L

(1.16)

Specf(P) = f-’(P)

define

,

.

1.11. Lemma. FOTevery L , L’,L” E D01, f E Hom(L,L’) and g E Hom(L‘,L”):

(1.17)

Spec f is strongly continuous ,

(1.18)

Spec(g o f ) = Specf o Specg

(1.19)

Spec(idL) = idSpecL

,

.

Proof. Routine check of (1.18) (1.19)and (1.20)

(Spec f)-’ (4.)) = s

(m).

1.12. Notation. Let

(1.21)

Spec : DO1 + St

be the functor pointed out in Theorem 1.8 and Lemma 1.11. Also, for every X, X’ E S t and f E H o m ( X , X‘) set

(1.22)

Kf

(1.23)

K f ( A )= f-’(A)

:

K X ’ --t ICX

,

.

1.13. Lemma. FOT every X , X‘, X ” E St, f E

H o m ( X , X’) and g E H o m ( X ’ ,

(1.24)

(KX, n, U, 0,X ) E DO1 ,

(1.25)

Kf is

a

morphism in DO1 ,

X”):

Topological dualities in lattice theory

88

(1.26)

K(g

(1.27)

K(idx) = i d K X

f) = K f

o

o

Kg

,

.

Proof. Routine.

1.14. Notation. Let

(1.28)

K : St

--t

DO1

be the functor established in Lemma SL(Z)

1.13.

Also, for every

L E DO1 set

= s(z) vx E L.

1.15. Lemma. For every L, L' E DO1 and f E Hom(L, L'):

L + K(SpecL) is a n i s o m o r p h i s m ,

(1.29)

SL :

(1.30)

K(Specf) o

Proof.

(1.29) follows

SL

= sLf o

f

from Corollary

. 1.7 and (1.13), while

for

(1.30) it is

easy to check that

P E (IC(Specf) o s L ) ( x ) H P E ( s L f o f)(x)

1.16. Lemma. For every X E St, formula (1.31)

OX(.)

= {V E ICX

12

$! V }

(VX E X )

establishes a n isomorphism

(1.32)

OX

: X 4SpecICX

in St. Moreover,

(1.33)

~ O every T

SpecICf o

f E Hom(X,X'),

OX = r~xto

f

.

.

0

89

The Stone duality of distributive lattices

K X ; this proves (1.32). Further ax is injective because K X is a basis of the Tospace X . To prove surjectivity take P E S p e c K X and show first that C = ICX-P and F = n ( X - V ) Proof. Clearly OX(.)

is a prime ideal of

UEP

fulfil the hypotheses of (iii) from Definition 1.1. But C is a prime filter of K

X

by Proposition 1.3.9, therefore it is closed with respect t o finite intersections.

P C KX

U E P is open, therefore F is closed. Suppose by way of contradiction that F n V = 0 for some V E C; then using the compactness of V E K X we get From

we see that every

a contradiction. Now use the conclusion of (iii)from Definition

an element every

V

2

E F n (J V , that is 2 v EC

E C. Thus for every

V

E

1.1 t o obtain @ U for every U E P and 2 E V for

KX

i.e. ax(s)= P. We have thus proved that ox is a bijection.

Further we know from (1.13) that the compact open sets of SpecKX

s ~ x ( V with ) V E K X ,therefore the strong continuity ax and as1 would follow from ax'(s~~(V)) = V . But, are of the form

.

E

axl(sh-x(V))e ax(.)

E

of

SKX(V) V

We have thus proved (1.32), while (1.33)follows from the routine proof of

Y

E

(SpecI'f

o OX)(.)

* Y E (OX#

0

f)(x)

.

1.17. Theorem. The dual of the category DO1 is equivalent t o the category St. Proof. Lemmas 1.15 and 1.16 establish the functorial isomorphisms

0

Topological dualities in lattice theory

90

idDol + K

Spec

(1.34)

s :

(1.35)

o : idst -+Spec o K

o

,

.

0

The above results yield a duality theory for Boolean algebras.

1.18. Definition. A topological space is said t o be totally disconnected provided the clopen sets-i.e. t he sets that are both closed and open-form a basis. By a Boolean space is meant a compact totally disconnected TZ space. We denote by

BS t he category of Boolean spaces and strongly continuous functions. 1.19. Lemma. (a) T h e compact open sets of a compact

T2space coincide with its clopen

sets. (b) Every Boolean space is a Stone space. Proof. (a) Follows from the well-known fact that every compact set of a

TZ space

is closed and every closed set of a compact space is compact. (b) Condition (i) in Definition 1.1 holds trivially, (ii) follows from (a) and

(iii) is an immediate consequence of compactness.

13

1.20. Corollary.

T h e categories B and BS are full subcategories o f DO1 and St, respectively. Proof. From Lemma 1.19 and Proposition 1.2.23.

1.21. Theorem. T h e dual of the category B i s equivalent t o the category BS.

0

The Stone duality of distributive lattices

91

Proof. (a) First we show that the dual SpecL (cf. Theorem 1.8) of a Boolean

algebra L is a Boolean space.

If P,Q E SpecL and P

# Q

then

using Proposition 1.3.18 we obtain in turn that P and Q are maximal

ideals, hence there is x E P such that x

# Q,therefore 3 # P , so that

P E s ( 3 ) , Q E s ( x ) and s ( x ) n s(3) = 0. Thus SpecL is a compact T2 space and it is also totally disconnected by condition (ii) in Definition 1.1 and Lemma l.l9(a).

(b) Also, t he dual KX (cf. Lemma 1.13) of a Boolean space X is a Boolean algebra, because K X consists of all clopen sets by Lemma 1.19(a) and the complement of a clopen set is also clopen.

(c) The above results (a), (b) and Corollary 1.20 imply the existence of the restrictions (1.36)

Spec : B -+ BS

(1.37)

I< : B S - t B

,

,

of the functors (1.21) and

(1.28), respectively. Therefore we can restrict

t h e functorial isomorphisms (1.34) and (1.35) t o (1.36) and (1.37):

(1.38)

s : idB

(1.39)

0

-t

Ir' o Spec

: idBs + Spec o

K

,

.

D

Topological dualities in lattice theory

92

$2. The Stone duality of De Morgan algebras In this section we present a category constructed in terms of Stone spaces and equivalent t o the category Mg of De Morgan algebras. The representation theorem

L

S K o Spec L for De Morgan algebras is due t o Byalinicki-

Birula and Rasiowa [1957] and its extension t o a category-theoretical duality was pointed out by Petrescu [1971]. See also Murkavdin [1975].

2.1. Definition.

A couple ( X ,g ) is called a Stone space with involution provided:

(i) X is a Stone space; (ii) g : X + X is a function satisfying g2 = idx and X - g ( A ) E K X for every

A E KX.

2.2. Notation.

1st the category of Stone spaces with involution, in which Horn((X,g),(X’,g’))consists of those morphisms f : X + X‘ in St that fulfil g’ o f = f o g . ~e

denote by

2.3. Lemma.

Let ( L ,A, V, N , 0 , l ) be a De Morgan algebra. For every P E Spec L set g ( P ) = L - N P , where N Z = { N z I z E 2 ) . T h e n (Spec L , g ) is a Stone space with involution. Proof. Spec L E S t by Theorem 1.8 and g : Spec L

1.3.8 and Proposition 1.3.9. From x E N r (2.1)

z E

-, Spec L

u N x E 2 we

that is g 2 ( P )=

P.

L and since g

(2.2)

infer

L - N(L-N P )ux $N(L- N P ) u

u N X$ L - N P U NXE N P H X =NNx a E

by Remark

is

EP ,

A E K(SpecL), hence A = s ( u ) for some a bijection and Q = g ( P ) u P = N ( L - Q ) , we get Now take

Spec L - g ( A ) = { Q E Spec L I N ( L - Q ) $ A } =

93

The Stone duality of De Morgan algebras = {QESpecL)aEN(L-Q)} = = {Q E Spec L I N u

# Q} = s ( N a ) E Ir‘(Spec L ) .

0

2.4. Lemma.

Let f : L (2.3)

-+

L’ be a morphism in Mg. Then

Specf : (SpecL‘,g’) -+ (specL,g)

defined by (1.16), is a morphism in 1st. Proof. S p e c f is strongly continuous by Lemma 1.11. Now for every P’ E

Spec L’, (2.4)

z E (9 o

e+ N z

Specf)(P’)= L - Nf-’(P’) e+

# f-’(P’) * f ( N s ) 4‘ P‘

*

2.5. Notation. Let

(2.5)

Spec : Mg -+ 1st

be the functor pointed out in Lemmas 2.3 and 2.4. 2.6. Lemma. Let ( X , g ) be a Stone space with involution. For every A E K X set N ( A ) = X - g(A). Then ( K X ,n, U, N , 0, X ) is a De Morgan algebra. Proof. Immediate from Lemma 1.13, Definition 2.1 and a routine verification of the De Morgan laws.

(7

2.7. Lemma.

Let f : ( X , g ) + (X’,g’) be a morphism in 1st and K X , K X ‘ the De Morgan algebras constructed in Lemma 2.6. Then the function (1.22)

Kf

:

K X ’ + KX

Topological dualities in lattice theory

94

defined by (1.23) is a m o r p h i s m of D e M o r g a n algebras. Proof. h’f is a rnorphism in DO1 by Lemma 1.13, while N o

Kf = K f o N‘

is proved as in Lemma 2.4, but with L , N , g, Spec and P’ replaced by X , 0

g , N , I< and A’, respectively. 2.8. Notation. Let (2.6)

K : 1st + Mg

be the functor pointed out in Lemmas 2.6 and 2.7. 2.9. Theorem.

T h e dual of the category Mg is equivalent t o the category 1st. Proof. We construct the functorial isomorphisms (2.7)

s : idMg -+

(2.8)

u :

I<

o Spec,

idBt + Spec o

K

by taking the restrictions of the functorial isomorphisms (1.34) and (1.35) from Theorem 1.17. Thus it still remains t o prove that SL and ox are morphisms in

Mg and 1st’respectively.

For every a E L, we get from (2.2)

( N o s L ) ( u ) = SpecL - g ( s L ( a ) ) = s t ( ~ a,) that is N o SL = SL o N . To prove g o u x = ox o g take

5

E X and

A E K X ; then, setting again A = ~ ( a ) , A E g ( u x ( 2 ) ) = SpecICX - ~ u ~ ( e z ) (j

A $ { N B I B E c ~ ( z ) }= { X - g ( B ) 12 $ B E K X )

M

The Priestley duality of distributive lattices

95

53. The Priestley duality of distributive lattices In this section we present briefly the characterization o f the dual of the category

DO1 in terms o f ordered topological

Priestley

119701,119721.

spaces. The results are due t o

We start with the following generalization of the concepts of filter and ideal in lattices. 3.1. Definition.

( X ,5 ) be a poset. A subset E C X is said t o be increasing (decreas i n g ) if V J : , ~E X : J: E E and J: 5 y (y 5 x) imply y E E . Let

3.2. Definition.

( X , 7 , 5 ) such that ( X , I )is a topological space and ( X ,<) is a poset. In such a space we denote by G X the set of all ctopen (i.e. both closed and open) increasing subsets of X . The An ordered topological space is a triple

space is said t o be totally disconnected provided for every J:, y E J:

$ y there exists A E GX which separates J: from y, i.e.

5

X such that

E A and y @ A .

3.3. Remark. Every totally disconnected ordered topological space is separated. (If

$ y or y $ x, say X - A € 7).

then J:

y€

J:

$

y, which implies

x E A E GX

c

J:

#y

7 and

3.4. Definition.

A Priestley space is a compact totally disconnected ordered topological space. We denote by Pr the category o f Priestley spaces and isotone continuous mappings between them.

3.5. Lemma. Let ( X , be a Priestley space. T h e n ( G X ,n, U,0, X ) is a h a n d e d distributive lattice.

7,s)

Proof. Immediate.

0

Topological dualities in lattice theory

96

3.6. Lemma. Let f : X + X I be a morphism in Pr. Then

Gf

(3.1)

:

GX‘+ G X

defined by (3.2)

(Gf)(A’) = f-’(A’)

(VA’ E GX’)

is a morphdsm in D01. Proof. One checks readily that well known that

f-’

A’ E GX’ implies f-I(A’) E G X and it is

commutes with set-theoretical operations.

0

3.7. Notation. Let

G : Pr + DO1

(3.3)

be t he functor pointed out in Lemmas 3.5 and 3.6.

3.8. Lemma. Let L be a bounded distributive lattice. Set

H L = HomDol(L, (0,1}) C (0,l}L.

(3.4)

Let 7 be the topology induced o n H L b y the product topology o n (0, l}L, where {0,1} is equipped with the discrete topology. Denote b y 5 the pointwise ordering of H L . Then ( H L , 7 , < ) is a Priestley space. Proof. To establish compactness, note first that for all functions W,

: (O,l}L

+ (0,1} defined by

x E L the projection

~,(f)= f(z), are continuous

by the very construction of the product topology. The functions W, A 7rY and 7rx

V

7rY

constructed in Example

1.1.10 are continuous too, because the sets

(w, A T ~ ) - ~ ( O=) w;~(o) n ~ ; l ( o ) , (w, A

n

,

~~) =-~ l ; l(( ii) )~ ; l ( i )

are open and similarly for 7r, V w ~ Now . remark t h a t the set X1 of all meet homomorphisms f :

L + (0,l) can be written in the form

97

The Priestley duality of distributive lattices

showing that XI is a closed set of the space (0, l}L.Similarly the set X2 of

all join homomorphisms f : L + {0,1} is closed in (0, l}L,therefore

H L = x1n x2n ril(o) n ~ ; l ( i ) is a closed subset of the Cantor discontinuum (0, l}L. But the latter space is compact by the Tichonov theorem, therefore H L is compact.

We have thus proved that H L is a compact ordered topological space and it remains t o establish total disconnectedness. Take f , g E H L with f $ g. Then f ( u ) = 1 and g(a) = 0 for some a E L . It follows t h a t 0 f E ri'(1) E G ( H L ) while g # r;l(l).

3.9. Lemma. Let f : L + L' be a morphism in D01. Then (3.5)

H f : HL'+ H L

defined by

(3.6)

( H f ) ( h ' )= h'

o

f

(Vh' E HL')

is a morphism in Pr. Proof. Since

f

and h' are morphisms in D01,so is h' o

holds, while the fact that hi

5 hi

+ hi

o

f 5 hi

o

f, therefore (3.5) f shows that Hf is

increasing. The continuity of H f follows from the fact that for every

5

EL

and a E {O,l},

(Hf)-'n,'(a) = {h' E HL' I h' = {h'

3.10. Notation. Let

o

f E T,'(u)} =

E HL' I (h' o f)(z) = a } = ( ~ ; ( ~ ) ) - ' (. a )

0

Topological dualities in lattice theory

98

(3.7)

H : DO1 + Pr

be t h e functor pointed o u t i n Lemmas

3.8 and 3.9.

3.11. Lemma. For every L E D01, formula (3.8)

t ~ ( a=) {f E H L I f ( ~=) 1 )

(VU E L )

establishes an isomorphism

(3.9)

t~

:

L - t GH(L)

in D01. Moreover, for every h E Hom(L,L'), (3.10)

G H ( h ) o t~ = tLt o h

.

Proof. Clearly t L ( a ) E G H ( L ) for every a E L. Then f r o m (3.4), (1.1.16')

(1.2.23") we infer easily t h a t t~ is a morphism in D01. T h e injectivity of t L follows f r o m Corollary 1.3.13 and Proposition 1.3.9. To prove surjectivity take U E G H ( L ) . For every f E U and g E H L - U , since U is increasing it follows t h a t f $ g , hence there is an element a j g E L such t h a t f ( u j g ) = 1 and g ( a j g ) = 0, therefore f E t ~ ( a fand ~ ) g E H L - t ~ ( a f ~Thus ) . for each fixed f E U , t h e compact set H L - U is covered by t h e clopen sets H L - t L ( u f s ) , where g runs over H L - U . W e deduce the existence o f a and

finite covering m

HL -u

c u i=l

m

( H L - t L ( a f g i )=) H L

m

Setting u f =

A i=l

ufgi it follows that

-

n

i=l

t L ( a f g i.)

The Priestley duality of distributive lattices

f

and on the other hand f(ar) = 1, t h a t is

U t L ( a r ) and we find

99 E t ~ ( a f ) Therefore .

U =

again a finite covering

f€U

h E Hom(L,L') and a E L ,

Finally, for every

( G H ( h ) o t L ) ( a )= ( H h ) - ' ( t L ( u ) ) = = {h' E HL' 1 ( H h ) ( h ' )E t ~ ( a ) }=

= {h' E HL' I (h' o h ) ( ~=) I} = t,y(h(a)) = ( t p o h ) ( ~. ) 3.12. Lemma. FOT every X E Pr, formula

(3.11)

I

1

ifzEY

T ~ ( x ) (=Y )

(W E G X ) (Vz E X ) 0

ifz$Y

establishes a n isomorphism

(3.12)

:

TX

X

HG(X)

-+

in Pr. Moreover, f o r every h E Hom(X,X'), (3.13)

HG(h) o

Proof. Clearly

f

E

TX

TX(Z)

E H G ( X ) for every z E X . To prove surjectivity take

H G ( X ) = Hom(GX, {0,1}).

U = {U

E

m

n ui E U i=

Setting

GX I f ( U ) = 1) ,

it follows t h a t for every

1

= T X' o h .

Ul, ...,Urn E U

and

n

and

U Vj E GX -21, j=1

hence

Vl, ...,Vn E GX - U

we have

Topological dualities in lattice theory

100

n

n

n ( x - v ; . ) = x j=1 -u

j=1

therefore the family of clopen sets

U UV

has t h e finite intersection property

X is compact it follows t h a t the intersection 2 of all the sets from U U V is not empty. Now t h e total disconnectedness of X implies and since

immediately that 2 is a singleton. Then 2 = { z } and T X ( Z ) = for every

f

because

Y E GX we have z E Y or z E X - Y according as Y E U or

Y $ U ,therefore

Note further that z

5y

+ ~ x ( z5) rx(y) because the sets Y E GX are

$ y then z E A and y # A for some A E G X , hence TX(Z)(A) = 1and ~ x ( y ) ( A=)0, therefore T ~ ( z$) ~ ( 9 )It. follows increasing. Conversely, if z

from surjectivity and Remark 1.1.25 that T X is an order isomorphism.

To prove continuity take a fixed A E G X and compute

Finally for every h E

H o m ( X , X ' ) and z E X ,

( H G ( h ) 0 ~x)(z)= ( H G ( h ) ) ( ~ x ( z )=)T x ( z ) hence for every

Y'

0

Gh

7

E GX',

3.13. Theorem.

T h e dual of the category DO1 is equivalent t o the category

Pr.

Proof. Lemmas 3.11 and 3.12 establish th e functorial isomorphisms

The Priestley duality of distributive lattices

(3.14)

t

(3.15)

T

Beznea

: idDol :

idp,

--+

--+

Go H

H

(19841 found

o

G

101

,

.

a subcategory of

category of complete distributive lattices.

0

Pr equivalent

to the dual of the

Topological dualities in lattice theory

102

$4. The Priestley duality of De Morgan algebras In this section we present a category constructed by Cornish and Fowler [1977] in terms of Priestley spaces and equivalent t o the dual of the category

Mg of De Morgan algebras. 4.1. Definition.

A couple (X, g ) is called a Priestley space with involution provided:

(i)

X i s a Priestley space;

(ii) g : X idx .

4

X is

a decreasing and continuous function satisfying g 2 =

4.2. Notation.

We denote by PrMg t h e category of Priestley spaces with involution, in which Horn((X,g),(X’,g‘)) consists of those morphisms f : X + X’ in Pr that fulfil g’ o f = f o g . 4.3. Lemma.

Let ( X , g ) be a Priestley space with involution. FOT every A E G X set

g(A) = { g ( a )I a E A ) . Then, ( G X , n, U, N , 8,X) is a De Morgan algebra.

N A =X

- g(A), where

Proof. Note first that if A E G X then g(A) = g-’(A) is clopen and decreasing, hence N A E G X . Then apply Lemma 3.5 and an obvious slight 0 generalization of Example 1.2.7. 4.4. Lemma.

Let f : ( X , g ) t (X’,g’) be a morphism in PrMg. Then (4.1)

Gf

:

G X ’ + GX

defined b y (4.2)

( G f ) ( A ’ )= f-’(A’)

(VA‘ E GX’)

103

T h e Priestley duality of De Morgan algebras

in Mg.

as a morphism

Proof. Apply Lemma 3.6, then note that f-1s'

= f-lg'-l

= (g'f)-'

-

=sGf

g-lf-l

which implies easily

(Gf

o

= (fg)-1

=

,

")(A') = ( N

o

0

Gf)(A').

4.5. Lemma.

Let ( L ,A, V, N , 0 , l ) be a De Morgan algebra. Define H L by (3.4) and y : H L --t H L b y yf(u) = f ( N a ) . Then ( H L , y ) is a Priestley space with involution. Proof. H L is a Priestley space by Lemma 3.8 and it is easy t o verify that

yf E H L for every f E H L , y is antitone and y2 = idHL. The continuity of y follows from the f a c t that y-'({f E H L I f(u) = E } ) = { h E

H L I h ( N a ) = t?}

(E

0

E {0,1}).

4.6. Lemma.

Let f : L

(4.3)

---f

L' be a morphism in Mg. Then

Hf

:

(HL',y') -+ ( H L , y )

defined b y (3.6) is a morphism in PrMg. Proof, H f is a morphism in Pr by Lemma 3.9. Besides, for every h' E HL' and a

E L, (Hf

0

-f')(h')(4= h'h'

0

f > ( 4= y'h'(f(4)

= IL'(f(Na)) = h ' ( f ( N u ) ) =

= r ( H f ( h ' ) ) ( a )= (7

0

=

HfTh"a)=

Hf)(h')W

*

0

4.7. Lemma. FOTevery L E Mg, the mapping tl; from Lemma 3.11 is a n isomorphism in Mg. Proof. It remains t o compute

Topological dualities in lattice theory

104

) I)) = NtLa = H L - ~ ( { Ef H L I f ( ~ = =

H L - {rf I f E H L & f ( ~ = ) 1)

= H L - { h E H L I r h ( a ) = 1) = =

{f E H L I r f ( a ) = 0)

= {f E H L f ( N a ) = 1) =

= tLNa.

0

4.8. Lemma.

FOTevery ( X , g ) E PrMg the mapping morphism in PrMg. Proof. For every (7

o

tE

7~

b o r n Lemma 3.12 is an iso-

X and Y E G X ,

n ) ( z ) ( Y )= 1 H T X ( X ) ( N Y=) 1 H

u TX(.)(X - g(Y))= 0 u z $2x - g(Y) u

u

z E g ( Y )u g ( z ) E

Y H (7x

0

g ) ( z ) ( Y )= 1 .

0

4.9. Notation. Let (4.4)

G

Mg

,

(4.5)

H : Mg + PrMg

,

:

PrMg

--t

be the functors pointed out in Lemmas 4.3-4.4 and 4.5-4.6, respectively. 4.10. Theorem. The dual of the category Mg i s equivalent to the category PrMg. Proof. Lemmas 3.11, 4.7, 3.12 and 4.8 establish the functorial isomorphisms (4.6)

t : idMg + G

(4.7)

T

o

H

,

: idprMg+ H o G

.

0

105

CHAPTER 3 ELEMENTARY PROPERTIES OF LUKASIEWICZ-MOISIL AL-

GEBRAS

The basic definitions and examples of the theory of Lukasiewicz-Moisil algebras are given in $1of this chapter. In Q2,3 we present various systems of axioms for LM-algebras, while in $4 a special (somewhat axiomatic) problem is solved. Basic concepts In this section we introduce the various types of Lukasiewicz-Moisil algebras starting with th e most general and then proceeding t o successive specializations: Definitions 1.2, 1.3 and 1.7. We hope t h a t this approach, converse t o the historical development of the theory, will result in a gain

if conciseness. In a comment following Theorem 1.18 we explain how our terminology tries t o correct certain discrepancies existing in the terminology used so far, via two quire minor changes. Finally we discuss a generalization of the concept of Lukasiewicz-Moisil algebras, as well as morphisms and subalgebras of LM-algebras. Let

Q be the order type of a totally ordered set with

suppose Q

2 2.

(1.00)

J = (0)

least element 0 and

In the sequel we shall fix

+I

(ordinal sum) of type 6,but otherwise arbitrary. Note however that we shall use the same symbol 0 (symbol 1) for the least (greatest) elements of

J and of the algebras L under investigation.

1.1. Definition. A Q-valued Luka~iewicz-Moisilpre-algebra o r Q-valued LM-pre-algebra o r 6-pre-algebra f o r short is a n algebra

Elementary properties of Lukasiewicz-Moisil dgebras

106

(1.1)

( L ,A, V, 0,l)is a

(1.2)

(pi

bounded distributive lattice

is an endomorphism in

.

(Vi E I ) .

DO1

If, moreover, L fulfils the determination principle, i.e. Vx,y E L : c p ; ~= (pig

(1.6) then

L

(VZ E I ) + x = y

,

is said t o be a 19-valued Lukasiewicz-Moisil algebra or a 19-valued

LM-algebra or a ?%algebra for short. We refer t o vi and Cpi as the endomorphisms and dual endomorphisms of L , respectively; cf. Proposition 1.6. below. We denote by 0 both the least element of J and that of L ; a similar convention will apply t o the greatest element of J , if any. The concept of a &algebra is due t o Moisil [1968].In Moisil’s and other papers the axiom

is often adopted instead of (1.5). The idea of considering separately the “equational part” of a d-algebra, i.e. the above concept of a d-pre-algebra,

is due ot Beznea [1981]in the case 19 = n.

The d-algebras constructed in Examples 1.2-1.3 below were essentially given by Moisil [1941a],and [1968]in the case 19 = n and for arbitrary 19, respectively; cf. Georgescu and Vraciu lorgulescu

[1969c],[1970],Boicescu [1984]and

11984~).See the comment t o Example 1.10.

Basic concepts

107

1.2. Example.

J from (1.00) has greatest element 1 and d : I + I is a decreasing map, i.e. i 5 j + dj 5 di, then the totally ordered set J

(a) If the set

can be made into a d-pre-algebra

where the endomorphisms cpf and dual endomorphisms Cpf are defined by

(1.8')

0

ifx
1

ifxzdi,

1

ifx
0

ifx2di.

cpfx =

(1.8")

(Pfx =

Note first that (1.1)holds by Example 1.2.3, while (1.3) is obvious. Now

if x

< y then either cptx = 0 or y 2 x >_ di, in which case cpty = 1;

thus cpf are isotone, hence D-morphisms by Remark 1.1.22.

< d i E I ) , hence (1.2) is verified. The latter equalities also imply (1.4). Finally if i 5 j then dj 5 di hence vtx = 1 + cpjx = 1, therefore q t x 5 cpjx i.e. (1.5) Moreover cpt1 = 1 and ( ~ 4 0= 0 (because 0

holds.

(b) If moreover the map d is also an involution, i.e. ddi = i for all i E I, then Ld is a d-algebra. For suppose cpfx = cpfy for all i E I. If x = 0 then cpty = cpt0 = 0, hence, y < di for all i E I and since d is a bijection by Remark 1.1.29 we have in fact y < j for all j E I,that is y = 0; similarly y = 0 implies x = 0. Now suppose x,y E I . From cp:dy = cpfdx = 1 we get dy 2 da: and similarly dx 2 dy, hence dx = dy, therefore x = y. 1.3. Example. Let

B

(1.9)

= (B, A, V,; 0 , l ) be a Boolean algebra. The set

~ [ q = { f l f : I+B, i<j+fi
Elementary properties of Lukasiewicz-Moisil algebras

108

o f all increasing functions f r o m I t o

B can be made i n t o a 6-algebra

where the operations of t h e lattice (B[q,A, V, 0 , l ) are defined pointwise (6. Example 1.1.10) and

(1.11)

(rif)(j)= fi

(1.12)

(?if)(j)

=

(Vj E I ) (Vi E I ) , ( V j E I ) (Vi E I )

.

1.4. Remark. T h e determination principle is equivalent to the following statement:

For (pix

5

(piy

e pi(z A

y) = (pix A (p;y = v i x implies t h a t (1.6)

=+

(1.6’). The converse is obvious. 1.5. Proposition.

The following conditions are equivalent f o r a n element x of a d-nlgebra L: (i)

x E

C(L);

(ii) 3i E I 3 y E L x = (piy;

(iii) 3i E I x = (pix;

Basic concepts (iv)

Vi E I

109

x = pjx;

Proof.

(i) j (v): Take

5

E C ( L ) and i , j E I , say i

5 j . Then 'pi3 5 'pj3 by

(1.5), hence 'pjx = 'pja: 5 cpia: = 'pix by Remark 1.2.21; using again (1.5) we get (v).

It follows from (v) and (1.4) that 'pjx = cpjcpix, (Vi E I ) (Vj E I ) , therefore (iv) holds by (1.6). (iv) j (iii) + (ii): Trivial. (v)

j(iv):

(ii) j(i): By (1.3).

0

1.6. Proposition. In every d-pre-algebra

==

(1.13)

Cpi~

(1.14)

pi

(1.15)

i 5j

(Vx E L ) I

is a dual endomorphism in DO1

+ Cpj

5 pi

(Vi,j E I )

(Vi E I) ,

,

and in every d-algebra, Vx,y E L:

Proof. Property (1.13) is a mere translation of (1.3) and it enables us to obtain (1.14)-(1.16)

from (1.2), (1.5) and (1.6), respectively.

0

1.7. Remarks. (a) A variant of Proposition 1.5 in terms of the dual endomorphisms

be obtained by taking complements in both sides of (i)-(v) using (1.13).

Fi

can

and then

(b) Proposition 1.5 and i t s variant 1.7.1 show in particular that in every d-algebra

Elementary properties of Lukasiewicz-Moisil algebras

110 (1.17) (1.18) hence (1.19)

1.8. Definition. Suppose the set J from (1.00) has both least element 0 and greatest element 1. A d-valued Lukasiewicz-Moisil (pre-)algebra with negation or d-valued

LM-(pre-)algebra with negation or d-(pre-)algebra with negation for short is an algebra

(1.22)

( L ,A, V, N,0 , l ) is a De Morgan algebra ,

(1.23)

pix = N p ; x

(1.24)

there is a decreasing involution d : I

(1.24.1)

p;Nx = N

(Vx E L ) (Vi E I )

p d i ~

, --+

I such that

(VX E L ) (Vi E I ) .

We say that (1.21) is the negation-free part of C and d is i t s ezternal involution while N is the internal involution or the negation of &. Marek and Traczyk [1969] suggested a slightly more general definition, in which d is only a decreasing bijection and the set 1 is not supposed t o have least or greatest element. For an even more general concept see Definition 1.23 below. The above variant is due to lorgulescu [1984c] and it turns out from her thesis that Definition 1.8 is the most natural framework ensuring the development of a significant theory.

111

Basic concepts 1.9. Remarks. 1. According t o the existence of the greatest element 1, t h e set

I

has also

least element, namely

(1.25)

0

= dl

.

2. It follows from (1.13) and (1.23) that in a d-pre-algebra with negation (1.26)

y3i5= N ~ ; x (VX E L ) (Vi E I) .

3. In every &algebra

this follows by applying (1.6’) t o the inequalities

Examples 1.10 and 1.12 below are taken from lorgulescu [1984c]. The negation-free part of Example 1.10 was given by Moisil as an example of a d-algebra; cf. Example 1.3 and th e comment preceding Example 1.2. The conditions imposed on

I in

Remark 1.11and Example 1.12 are due t o Pon-

asse [1978]. 1.10. Examples. (a) Let

B

(1.28)

= ( B ,A, V;,

0 , l ) be a Boolean algebra. The set

D(B)= {f I f

:

I +B, i 5 j

of all decreasing functions from with negation

I to B

=j

fj 5 fi}

can be made into a d-algebra

Elementm y properties of Lukasiewicz-Moisil algebras

112

where the operations o f the lattice ( D ( B ) ,A, V, 0 , l ) are defined pointwise (cf. Example 1.1.10) and

(1.30)

f’(j) = f o

(1.31)

(Sif)(j)=f(di)

( Vj E I) (ViE I) 9

(1.32)

(&f)(j) = f(dij

( V j E I) (Vi E I) .

Properties (1.1)-(1.4)

(Vi E I) ,

are checked as in Example 1.3. The implications

establish (1.5), while (1.6) follows from the remark that when i runs over I so does

di. Then f’ E D ( B ) because d , f and-are decreasing.

Now (1.22) and (1.23) are immediately checked, while (1.24.1) follows from

(b) The algebra B[q in Example 1.3 is also endowed with t h e negation (1.3). 1.11. Remark. The following properties of the set 1 are equivalent:

I) every element i # CT has a predecessor i- (i.e., ielement j E I satisfying i- < j < i);

11) every element i

#

and there is no

1 has a successor i+ (dual definition).

As a matter of fact it is easy t o prove t h a t (1.33)

i+ = d ( ( d i ) - )

and

i- = d ( ( d i ) + ) .

1.12. Example. Suppose the set I fulfils the equivalent conditions from Remark 1.11. Then

the &algebra Le from Example 1.2.b can be made into a &algebra with negation

113

Basic concepts

where the negation

(1.35)

-

x=

1

N

is defined by

1

ifx=O,

0

ifx=l,

d(x+) = (dx)-

if x E I - {I)

-

Property (1.21) was established in Example 1.2, while the easy remark cptx = cptx implies (1.23) via (1.13). Furthermore we prove that is a decreasing involution (which accomplishes the proof of (1.22) in view of Example 1.2.8) and fulfils (1.24.1). But these one- and two-element identities and implications are readily established for x E {0,1} or y E (0, l}, while for j,k E I - { 1) we get

-

--

and

j = (d(d(j+)))- = (j+)- = j

- (pij = cpf

-

j for every i E

,

I because

cp:ij = 0 H & j = 1H j 2 ddi = i H i

u

-

j = d ( j + ) < di

#cpt

-

<j+

H

j =O

The following result was proved by Sicoe [1967b,c], Cignoli [1969a] and Boicescu [1984] in the case 9 = n and di = n - i. 1.13. Proposition. E v e r y 9-algebra with negation is a Kleene algebra, i.e. (1.36)

2

A Nx

2y VN y

Proof. For every i E

( t l z , y E L) .

I, if i 5 d i then

Elementary properties of Lukasiewicz-Moisil algebras

114

Beside the conditions from Proposition 1.5 and Remark 1.7.a the chrysippian elements of a d-algebra with negation can be given some further characterizations: 1.14. ProDosition.

T h e following conditions are equivalent f o r a n element x of a 9-algebra with negation L:

(i)

x E C(L);

(ii) N x E

C(L);

(iii) x A Na: = 0; (iv) x V N x = 1;

(v) 3 = N s . Proof. (i) j (v): 5 = 'pi" = N5 x = N x by Pro osition 1.5 a d (1.13). (v) + (ii): Trivial. (ii) + (i): x = N N x = % because (i) + (v). (v) + (iv): Trivial. (iii) (iv): By applying N . (iv) + (v): Because both (iv) and (iii) hold.

0

1.15. Remark. The list of computation rules valid in a d-algebra with negation comprises also the properties obtained from (1.13)-(1.16)

and (1.18)-(1.19) by appl-

ying the transformation (1.23). Given a d-algebra, it seems a good idea to try to transform it into an algebra with negation: the latter is easier to handle. This problem is solved

Basic concepts

115

as follows.

1.16. Theorem (Suchori [1972], [1973]). Let C be a 0-algebra and d : I -+ I a decreasing involution. Then:

(i) T h e necessary and s u f i c i e n t condition f o r the existence of a 0 algebra with negation L: having C as negation-free part and d as external involution is (1.37)

(VX

(ii) T h e algebra

E L ) (3y E L ) (Vi E I ) (piy = p

p.

i s unique whenever it exists: N x is y from (1.37).

Proof.

Necessity follows from (1.24.1) and (1.13): (piNx = Nydix = (1.37) holds with y = N x . This also proves (ii) via (1.6). To establish sufficiency note first that the element y from (1.37) is again

(pdix, hence

uniquely determined by x on account of (1.6). follows that (piNx = Then

m.

Now define N x = y ; it

hence N N x = x by (1.6). Furthermore

(piN(x A y) =

(pdi(2

A y) = (pdix A (pdiy = (PdiS v (pdiy =

= 'piNx V (piNy = (pi(Nx V N y )

(Vi E I ) ,

therefore N ( z A y) = N o V N y . Thus L is a De Morgan algebra. Moreover,

hence N q i z = (pi" again by (1.6). It follows that v i N x = cpd;a: = N(pdix. (See Example 1.12). 0 Next we specialize t o the case 19 = n E JV; cf. Moisil [1941a]. 1.17. Definition. Let n E JV, n 2 2. Take

Elementary properties of Lukasiewicz-Moisil algebras

116

J

(1.38) so that

= {0,1,...,

n

- l}, hence I = (1,..., n - 1) ,

the elements 0 and 1of J are now denoted by 1and n - 1, respec-

tively. By an n-valued M o i s i l (pre-)algebra, or n-valued M-(pre-)algebra or Moisil n-(pre-)algebra for short, we mean an n-valued LM-(pre-)algebra with negation. 1.18. Theorem.

The external i n v o l u t i o n of every n-valued M o i s i l (pre-)algebra is

di = n - i

(1.39)

(i = 1, ...,n - 1) .

Proof. Immediate from Proposition 1.1.32(ii). (As a matter of fact, The-

orem 1.18 is valid for the more general algebra suggested by Marek and Traczyk [1969]; cf. the comment following Definition (1.8).)

0

A t this point we insist on terminology. Moisil has introduced both nvalued algebras, which are endowed with a negation, and &algebras (infinite S), which have no negation, under the common name of “Lukasiewicz algebras”. However, we have now &algebras (finite or infinite 9) both with and without negation (in particular n-algebras without negation were introduced in Moisil 119721, 298-310), so that we need different terms t o designate them. So we reserve th e term ‘Walgebras” t o the algebras without negation in order t o observe Moisil’s already established terminology, and we call the other algebras ”&algebras with negation”.

However, this would still

violate t he conventional terminology in the case 19 = n: in that terminology “n-valued algebras” are always supposed t o have a negation. That is why we have resorted t o the subterfuge of duplicating our term “n-valued Lukasiewicz-Moisil algebra with negation” by the shorter term “n-valued Moisil algebra”. Another point is that our 9 is 1+6 in Moisil’s notation. We have chosen t o denote by 29 the order type of of

I,with

J

from (1.00) instead of the order type

a view t o avoid a discrepancy between the finite and the general

case: in our notation a &algebra with 6 = n is simply an n-algebra and not an (n

+ 1)-algebra, as happens in the conventional notation; cf.

lorgulescu

117

Basic concepts [1984c].

1.19. Example. The n-valued Moisil algebras 23[4 and D(L)obtained from Example 1.10 by the specialization 19 = n (hence di = n - i by Theorem 1.18) can be also given the following description:

both with componentwise meet, join, zero, one; then

Note that

(1.45)

C ( B [ q )= C ( D ( B ) ) = ((2,...,S) E B"-l

12

E

B} .

1.20. Example. The set I = { 1, ...,n - 1) and the involution di = n - i fulfil the conditions in Remark 1.11. We shall also use the notation

for the algebra constructed in Example 1.12. The negation and the endomorphisms are given in Table 1.1below.

Elementary properties of Lukasiewicz-Moisil algebras

118 '

X N X

I

0 n-1

9%

0 0

...

...

d (~n-22 d

0 0

Cp%

~p,-,x

1

2

n-2

n-3

0 0 ... 0

0 0

n-1

... n-1 n-1

... n - 3 ... 2 ... 0 ... 0 ... ...

n-2 1 0 n-1

n-11 0 n-1 n-1

...

...

... n - 1

n-1 n-1

n-1 n-1

...

n-1

Table 1.1.

1.21. Example. Every Boolean algebra is a 2-valued Moisil algebra. Here I is a singleton and

the unique endomorphism is the identity mapping. This example is, of course, fundamental because it shows that t h e theory of Lukasiewicz-Moisil algebras is a generalization of t h e theory of Boolean algebras. In particular th e algebra Ln plays a role similar to t h a t of L2 in

the theory of Boolean algebras. Example 1.21 can be sharpened to a chain of equivalences. For Definition 1.17 is a specialization of Definition 1.8 and t h e l a t t e r is in its turn a specialization of Definition 1.1; however, every 2-valued LM-algebra is in fact a Boolean algebra. For there is a single endomorphism (pl and (pl'plx = plx, hence ( P I X = x by (1.6),therefore L = C(L)by (1.17). Roughly speaking, we can state: 1.22. Remark. The following concepts are equivalent:

(i)

2-valued LM-algebra;

(ii) 2-valued LM-algebra with negation = 2-valued M-algebra; (iii) Boolean algebra. We continue this section with a few words about further generalizations of t he concept of LM-algebra.

Basic concepts

119

1.23. Definition. The concept of generalized &valued LM-(pre-)algebra with negation is obtained from Definition 1.8 by relaxing axiom (1.24) t o the effect that d : I -+ I is allowed to be an arbitrary mapping. For 29 = n we obtain the concept o f generalized n-valued M-(pTe-)dgebTU. This definition, due to Moisil [1965] for the case 29 = n, originates in the remark that whereas a &algebra is also a dl-algebra for every 19'

> 29,

this is no longer true for +algebras with negation, even for 29 = n. Now it is easy t o see that every generalized n-valued Moisil (pre-)algebra is also

> n.

( i = l,...,n - 2 ) and 'p: = ( P ~ -(i~ = n - 1,...,rn - 1) and define d' by d'i = d i ( i = 1,...,n - 2 ) and d'i = d(n - 1) (i = n - 1,...,rn - 1). In the opposite direction it is

rn-valued for every m

(Define 'pi by 'pi =

cpi

again easy t o see that every generalized n-valued Moisil (pre-)algebra is also an rn-valued Moisil (pre-)algebra for a suitably chosen rn set

5n

(re-index the

{vll...,vn-l} or cardinality rn - 1 I n - 1 so as t o obtain a collection

of rn - 1 distinct endomorphisms.) 1.24. Proposition.

In every generalized 29-valued LM-pre-algebra with negation:

Proof. Property (1.48) follows from

while 'Pdjx

iI j implies v i N x I p j N z or equivalently N p d i x 5 Ntpdjz, hence

5 (Pdix.

0

1.25. Theorem.

E v e r y generalized 6-valued LM-(pre-)algebra with negation having distinct endomorphisms is a 29-valued LM-(pre-)abgebru with negation. Proof. It follows from (1.48) that ddi = t. Now if

i 2 j then dj 5 di for

Elementary properties of Lukasiewicz-Moisil algebras

120 otherwise di

< dj would imply ( ~ d < i 'pdj in contradiction

with (1.49).

0

1.26. Remark.

The converse of Theorem 1.25 does not hold. Take e.g. the endomorphisms c p 1 , . . . , ( ~ ~ - of 3 an (n - 2)-valued Moisil algebra L and define 'pi = 'pl, ~ p : = (Pi-1

(i = 2, ...,n - 2), cpL-, = (

~ ~ and 3

d'i = n - i ( i = 1, ...,n

- 1).

Then L is made into an n-valued Moisil algebra whose endomorphisms are not distinct. 1.27. Proposition.

The e n d o m o r p h i ~ m sof a generalized Moisil (pre-)algebra are distinct if and only if the f u n c t i o n d satisfying (1.24.1) is unique. Proof. Suppose the endomorphisms ( P ~ ..., , Q

... =

~

of- L~ are not distinct. Then

k < h. Define d l i = d2i = di for i E {1,...,n - 1) - { d k } , d l d k = k and d z d k = Ic + 1. Then it is easily (pk = (Pk+l

=

'ph

for some

checked that both dl and dz verify (1.24.1). Conversely, suppose (PI, ...,pn-l are distinct and let d' be any function satisfying (1.24.1). Then L is a generalized n-valued LM-(pre-)algebra with negation with respect t o d', hence it i s an n-valued Moisil (pre-)algebra by Theorem 1.25, therefore d'i = n - i ( i = 1, ..., n - 1) by Theorem 1.18. 0

We already know by Theorem 1.16 that certain 8-valued LM-algebras cannot be made into d-valued LM-algebras with negation. For d = n we can say even more: there are n-valued LM-algebras t h a t cannot be made into generalized n-valued Moisil algebras. (Take e.g. L4 and L3 viewed as fourvalued LM-algebras, construct componentwise the fou r-val ued LM-algebra

L4 x L3 and note that its endomorphisms are distinct because so are the endomorphisms of L4.If L 4 x Ls could be made into a four-valued generalized Moisil algebra, this would be a four-valued M-algebra by Theorem 1.25; but L4 x L3 has 12 elements in contradiction with Corollary 6.1.19, which implies that every M4-algebra is a direct product of copies of Lz and/or L4. It is important t o note that in view of Proposition 1.24 a large part of t he theory of LM-algebras with negation remains valid for generalized

Basic concepts

121

LM-algebras with negation because, as can be easily seen, in many proofs the fact that d is a decreasing involution is used only through the weaker

properties (1.48) and (1.49). On the other hand Theorem 1.25 shows t h a t there is not much point in this generalization, because the new algebras are obtained from th e old ones by duplicating certain endomorphisms, which does not seem too exciting. That is why we have decided t o work within the framework of Definitions 1.1, 1.8 and 1.17.

We also mention a generalization introduced by Suchoti [1974b] = [1975a], namely a kind of rn x n-valued algebras in which the endomorphisms form an m x n matrix. The main result is a representation theorem according t o which the elements of such an algbra L can be viewed as ( m - 1) x (n - 1) matrices whose rows and columns are increasing vectors with components in

the Boolean algebra C ( L ) ;this generalizes Theorem 6.1.8.

We conclude this section with a few words about homomorphisms and subalgebras. 1.28. Gefiiition.

If L and L’ are d-valued LM-(pre-)algebras, a bounded-lattice homomorphism f : L -+ L’ is said t o be an LMS-homomorphism provided it satisfies (1.50)

f((Pi4= (P:f(z)

(1.51)

.f(P;z) =

Zf(z)

(Vi E I> ? (Vi E

I) .

We denote by LMd the category of d-valued LM-algebras and LMdhomomorphisms. In particular LMn is th e category of n-valued LM-algebras. 1.29. Remark.

It follows from (1.3) and Remark 1.2.21 that conditions (1.50) and (1.51) are in fact equivalent for a bounded-lattice homomorphism, so that in practice

it suffices t o check only one of them. Note also that every LMd-homomorphism h between two LMS-algebras with negation preserves negation, because for every i E I ,

(1.52)

cp:h(Nz) = h(cp:Nz) = h(N&z) = h(cpbiz) =

122

Elementary properties of Lukasiewicz-Moisil algebras

-

= y&ih(z)= "cp&,h(z) = cpiN'h(z) .

1.30. Notation. We denote by LMNd the full subcategory o f LMd having as objects the d-valued LM-algebras with negation. In particular the category LMNn will be denoted simply by Mn. 1.31. Remark. In view of Remarks 1.2.20 and 1.29, the following three steps suffice t o prove that

f

is an LMN1S-homomorphism:

(a)

f

preserves A or V;

(b)

f

preserves N , 0 and 1;

(c)

f

satisfies (1.50) or (1.51).

It is easy t o see that the concept of LMNd-homomorphism is actually stronger than the previous typesof homomorphisms. Take e.g. defined by

f

:

L4

+ L4

f(0) = f(1) = 0 and f(2) = f(3) = 3. Then f is a De Morgan

homomorphism but f(cp,2) = 0 whereas cplf(2) = 3. However, for n = 3 we have: 1.32. Proposition (Cignoli [1979]).

The D e Morgan homomorphisms between 3-valued M o i s i l algebras c o i n cide with the M3-homomorphisms.

A , B E M 3 and f : A + B be a De Morgan homomorphism; since cplx = Ncp2Nz by (1.24.1) and Theorem 1.18, it Proof of the non-trivial part. Let

suffices t o prove that

f(cp22)

= cpzf(2).

First we make use of (3.8) and (3.11) t o evaluate

using this and (3.9) we get

123

Basic concepts

=

f(cpz4 A cpzf(4

7

hence f(cpzz) 5 cpzf(z). To obtain the converse inequality use (3.11) and note that x E C ( A ) + f(x) E C(B)by Remark 1.2.21; then cpzf(z) 5 (Pzf((P24

=f((P24.

1.33. Definition. An LMd-subalgebra or simply a subalgebra of a 8-valued LM-(pre-)algebra L is a 0-1-sublattice S of L such that

(1.54)

5

E S =+ Cpjx E S

(Vi E I) .

1.34. Definition. An LMNd-subalgebra or simply a subalgebra of a d-valued LM-(pre-)algebra with negation L is a De Morgan subalgebra S of L fulfilling (1.53) and (1.54). 1.35. Remark.

It follows from (1.23) that conditions (1.53) and (1.54) in Definition 1.34 are in fact equivalent for a De Morgan subalgebra of L , so that in practice it suffices to check only one of them. On the other hand, if L is an algebra with negation, an LMhubalgebra of L (more exactly, of the negation-free part of L ) need not be an LMNd-subalgebra of L : take e.g. the subset s = {0,1,3} of L4 = {0,1,2,3}. 1.36. Remark. In the remainder of this book we shall suppose that the set J has both least

element 0 and greatest element 1.

Elementary properties of Lukasiewicz-Moisil algebras

124

$2. Axiomatics of &valued and n-valued algebras In this and th e subsequent section we present alternative systems of axioms that have been suggested for various types of Lukasiewicz-Moisil algebras. The main result of this section is Theorem 2.4: the class of Moisil n-algebras is equational. For various systems of axioms in lattice theory see Rudeanu [1963].

We begin with th e remark that numerous equivalent variants of Definition 1.1of 8-(pre-)algebras are obtained by substituting in turn various explicit systems of axioms of bounded distributive lattices for the condensed axiom

(1.1) and similarly, replacing axiom (1.22) by various explicit definitions of

De Morgan algebras yields equivalent variants of Definition 1.8 of d-(pre)algebras with negation. Another point is that in the case of 9-algebras axiom (1.2) from Definition 1.1can be relaxed by requiring that cp; be just lattice endomorphisms, which suffices t o ensure that cpi0 = 0 and cpil = 1. For we can prove

I x I cplx

cpoz CpiO

as in Remark 1.9, hence

= (picpo0 = yo0 = 0 via (1.4) and

'pi1

900 5 0 i.e. cpoO = 0, therefore = 1 by duality.

A stronger result holds for 9-algebras with negation: 2.1. Proposition (Cignoli [1969a]).

The following system of axioms defines 9-algebras with negation: (1.4)-

(1.6), (1.22)-(1.24)

(2.2)

and for every x , y E L:

C~;Z A C p i ~= 0

(Vi E I )

.

Proof. Necessity is trivial. To prove sufficiency we compute first

Axiomatics of &valued and n-valued algebras

125

so that (pi are lattice endomorphisms, hence (pi0 = 0 and (pi1 = 1 as re-

marked above, i.e. (1.2) holds. To prove (1.3) we use the fact that (2.2) can be written (pix

(pix

A N(pix = 0 by (1.23), hence N y i x V (pix = 1, that is

v cp;x = 1.

0

The next problem is t o prove that the classes of n-valued LM-algebras and M-algebras are equational. We begin with an explicit statement of an informal remark made in the previous section. 2.2. Remark. (a) The class of 6-pre-algebras is equational, because axiom (1.5) can be replaced by the set of identities (pix A (p,y = (pix for every i , j E I with

i<j. (b) The class of 6-pre-algebras with negation corresponding t o a fixed decreasing involution d is equational, because axiom (1.24) reduces t o the identity (1.24.1).

The next two results are due t o Cignoli [1969a] and Boicescu (19841. 2.3. Proposit ion. In every n-pre-algebra L, the determination principle (1.6) is equivalent t o each of the following conditions: f o r every x , y E L,

Proof.

(1.6)

+ (2.3a):

Set

Z

= X AP;X A ( p i + l y . Then

and using (1.5) we see that if

i
We

get

(PkZ

k 5 i &!ien ( p k Z 5

5 Q;+1y 5 Q k v .

Finally use Remark 1.9.3.

Thus

(pkZ

(PkZ (pkx

= ( p k x ACp;X A pix =

5 (pky for

all

A(pi+ly

0, while for

k, hence Z I y .

Elementary properties of Lukasiewicz-Moisil algebras

126

(2.3a) +- (2.3b): In view of distributivity the left-hand side of (2.3b) is a disjunction of terms of the form (2.3*)

z = x A (Pilx A

... A P,,x

A ( p j , ~A

... A pjpy ,

where ( i l ,...,im,jl,...,j,) is a permutation of (1, ...,n - 1). It suffices to prove that each z 5 y. If rn = 0 then {jl,...,j,} = (1, ...,n - 1) hence the right-hand side of (2.3*) contains the factor (ply, therefore z 5 (ply 5 y. Now suppose

rn > 0 and let i = m a x ( i l , ...,im). If i = n - 1 then (2.3') implies as above z 5 x A (Pn-lx 5 y via (1.5). If i < n - 1 then p > 0 and l e t

+

+

j = min {jl, ...,j,}. But i 1 E {jl,.,.,j,}, hence j 5 i 1, therefore 'pjy 5 ~ ; + so ~ ythat ( 2 3 ) implies z 5 E A Pix A ( ~ i + 5 ~ yy via (2.3a).

(2.3b) +- (1.6): Suppose (pix = (piy for all i. Then pis V (piy = 1for all 0 i , hence (2.3b) reduces to x 5 y and similarly we prove y 5 E. 2.4. Theorem.

The classes of n-algebras with and without negation are equational. Proof. Immediate from Remark 2.2 and Proposition 2.3.

0

The last point in this section is to define LM-algebras using fewer operations than in the usual definitions. For instance it is clear that we can use only the lattice operations and the endomorphisms (pi as basic operations, as did Moisil [1941a] when he introduced the "n-valued Lukasiewicz algebras": 2.5. Remark. (1.2), 19-pre-algebras are characterized by axioms (l.l), (1.3')

(pix

E C(A)

(Vi E I) ,

(1.4) and (1.5); in this approach the dual endomorphisms cp; are defined by (1.13). See Moisil [1972], pp. 298-306, for a discussion of certain implications between various axioms and in particular for a definition via (Pi instead of (pi.

127

Axiomatics of d-valued and n-valued algebras

We present below some further possibilities.

2.6. Proposition (Sicoe [1967b], [1968]). Let d be a fixed external involution f o r 19-algebras. An algebra ( L ,A, N , 1,{cpi}iEl) of type (2,1,0, {l}iEl) is equivalent to a 19-algebra with negation

via the transformations

(2.4)

z V y =N(Nz A

(2.5)

0 = N1

,

= NV;X

(1.23)

Ny) ,

(Vi E I ) ,

if and only if L fulfils the identities (1.6) and N ( N z A Ny) = z

,

(2.6)

zA

(2.7)

z A N ( N y A Nt.) = N [ N ( zA z) A N(y A z)]

(2.8)

z A 1 =z

,

A N [((p;NtA (pkNt)A V] = N A N(N9,jit A V) Proof.

,

[(CpjZ

A vjy) A

U]

A

( V i , j , k E I with i 5 k) .

Recall first that De Morgan lattices are characterized by axioms

(2.6)-(2.8) via the transformation (2.4)-(2.5); cf. Proposition 1.2.9. Now we are going t o prove that a De Morgan algebra endowed with the operations { ( p i } i E ~ becomes a d-algebra with negation via the transformation

(1.23) if and only if it fulfils (2.9) and (1.6). The necessity of (2.9) is easily checked. To prove sufficiency we write (2.9) in the equivalent form [piVj(X A

y) A

U]V

( v ~ N zA (~diz)V (ViNt A V k N t A V ) =

= (cpjz A (pjy A u ) V (Nvdit A v )

( V i , j , k E I with i

5 k) ,

128

Elementary properties of Lukasiewicz-Moisil algebras

and take in turn (u, v) := (O,O), (0, l), (1,O):

Taking x = y in (2.12) we obtain (1.4). Taking i = k in (2.11) we get (1.24). Taking t = N x in (2.11) and using (1.24) we infer (1.5). From (2.12) and (1.4) we obtain (2.1). Replacing i by di in (2.10) and using (1.24) we deduce N 9 ; z A cp;z = 0, whence (2.2) follows by (1.23). Thus L 0 is a d-algebra with negation by Proposition 2.1. Sicoe has also proved the independence of all the axioms but (2.8). 2.7. Corollary (Sicoe [1967c]). T h e system of azioms (1.6), (2.6), (2.7), (2.10)-(2.12) characterizes 6 algebras with negation via the transformations (2.4), (2.5) and (1.23). Proof.

Necessity is immediate. Conversely, (2.6) and (2.7) imply that ( L ,A, V) is a distributive lattice and N an involutive dual endomorphism by Proposition 1.2.9. Then from (2.10)-(2.12) we obtain (1.4), (1.24), (1.5), (2.1) and (2.2) with the same proof as for Proposition 1.5. Now we obtain x 5 ' p l x as in Remark 1.7.3 via Remark 1.4, hence (2.10) implies x A O =x

A Q O N XAcpla:

='poNx A V ~ = X 0,

therefore 0 is least element, so that 1(= NO) is greatest element and (2.8) 0 holds. The proof is concluded by Proposition 2.1. The idea of the last proposition of this section is that not only the dual endomorphisms pi can be dispended with in the definition, but also half of the endomorphisms cp;, on account of (1.24.1). The point is that if n = 2k 1

+

then (1.24.1) matches ( P I ,...,( ~ kto ( P k + l , ...,~ n - 1while if n = 2k then (1.24.1) matches (PI, ...,Q k - 1 to ( P k + l , ...,y n and (Pk to itself. To state our result formally we introduce the following

129

Axiomatics of &valued and n-vdued algebras

2.8. Notation. Let n E

IN,n 2 2, I

.I:[

= (1, ...,n - 11, IC =

AISO,

if ( p q ) is a formula

I , let ( p - qk) denote the formula obtained from ( p q ) when I is replaced by (1, ...,k}.

involving quantifiers over

2.9. Proposition. A n algebra ( L ,A, V,N, 0,1, ( v ; } i = ,...,~ k) of t y p e (2,2,1,0,0, { l } i = ~,...,k) is equivalent to a Moisil n-algebra (1.20) via the transformations

+ 1, ...,n - 1) ,

(2.13)

'pjx = Nvn-jNx

(j =k

(2.14)

pit = Nvix

(Vi E I )

,

if and only if L fu&h (1.4k),(1.5'), (1.22) (2.2k)and ~

Proof. In view of Proposition

O every T

x, y E L :

2.1 it suffices t o check (1.4)-(1.6), (1.24),

(2.1) and (2.2). Let j E {k 1, ...,n - 1). Then n - j E (1, ...,k}, hence using (2.23), (2.13) and (2.2k)we get

+

' p jA ~

@ ~ x= p j A~N ' p j ~=

= Ny,-jNx A y , - j N ~= p,,-jN~A y,-jNx = 0

,

so that (2.2) holds. Further (2.1k)implies

therefore

(2.1) holds too. Now we write (2.13) in the equivalent form

130

(2.13')

Elementary properties of Lukasiewicz-Moisil algebras

N v j N x = cpn-jx

and since n-k-1

( n - j = n - k - 1,..., 1)

is k or k-1 according as n = 2k+1 or n = 2k, we see from

(2.13'), (2.18) and (2.13) that (1.24.1) is verified with di = n - i. Further, if (Phx = (Phy ( h = 1, ...,n - l ) , then p i N x = N(P,-~x= Nvn-iy = cpiNy, hence x = y by (2.17), therefore (1.6) holds. Now let i E (1,...,k} and j , m E {k 1, ...,n - 1 ) . If j 5 m then n - m 5 n - j 5 k, hence vnAm5 vn-j by (1.5k), therefore 'pjx = N p n + N x 5 Nyn-,Nx = y m x , that is cpj 5 qm. O n the other hand $a5 (Pk+l by (2.16) and (2.13), hence ( ~ 5i (Pk 5 ( ~ k + l 5 v j , completing the proof o f (1.5). Finally (1.4) follows from (1.4k) and (2.15) via

+

0

Axiomatics of three-valued algebras

131

93. Axiomatics of three-valued algebras T he first Lukasiewicz algebras invented by Moisil were the three-valued and thefour-valued ones. It turns o u t th a t they have many special properties, as was shown in the literature. In particular several systems of axioms for three-valued LM-algebras were given and some of them will be presented in

[1982]. The starting point is Proposition 2.9 for n = 3. In this case 1 = {1,2}, k = 1, axioms (1.5l) and (2.18) are vacuously fulfilled. Axiom (1.22) is independent of n, while (1.4'), (2.2l)and (2.14)-(2.17) become: ply1 = $91, $91" A N y l x = 0 , $9' is a lattice homomorphism, cplNpl = Nvl, $91 L Np1N and $91" = 'ply & p1Nx = cplNy + x = y, respectively, where we have translated (2.2') via (1.23). However, since most papers this section. See also Figallo and Tolosa

are written in terms of

$92

we will adopt this language and perform the

(2.13)i.e. $92 = NylN. Thus a three-valued Moisil algebra is an algebra ( L ,A , V, N,0 , 1,(p2) of type (2,2,1,0,0,1) such t h a t

translation via

(3.0)

( L ,A, V, N,0,l)is a De Morgan algebra ,

(3.1)

$92(x A

(3.3)

$ 9 2 ~A

(3.4)

$92$92x = Cpzx

Y) = $ 9 2 2 A (P2Y

7

N V ~=X 0, 7

Consider also th e following properties:

(3.8)

$922

A NX= x A NX,

(3.9)

~

V 2NX= 1

(3.10)

N ~ ~ VN ( $ 9x2 ~A v ~ N x V )N Y ~=X 1

2

, ,

Elementary properties of Luhiewicz-Moisil algebras

132 (3.11)

x 5 CPZX

(3.12)

x A N x 5 y V Ny .

9

3.1. Lemma. (3.0) & (3.8) & (3.9)

j(3.11).

Proof. x = x A 1 = x A ('p2x V N x ) = (x A

' p 2 x ) V ('p2x A

N x ) 5 'p2z.

0

3.2. Proposition. T h e following systems of axioms are equivalent: SO = {(3.0), ...,(3.7)) , S1 = {(3.0),(3.1),(3.4),(3.5),(3.8),(3.10),(3.11)}

,

S2 = {(3.0),(3.1),(3.8),(3.9)} , S3 = {(3.0),(3.8),(3.9),(3.12)} . Cornment .

S1 is a simplification of a system due to Moisil [1960]. System S2 was suggested by A. Monteiro [1963], while Becchio [1973] obtained S3 by eliminating a redundant axiom from a system due to Cignoli and A. Monteiro [1965]. Note that unlike our presentation, the original systems use only the unit element as a primitive constant operation, while 0 is defined by 0 = N1. Proof.

SO + S3: Using freely the properties of a Moisil algebra we obtain (3.12) by Proposition 1.13, then

' p ~ ( L 3 2A~ N

x ) = C ~ Z XA Q ~ N =X ' p 2 ( ~A N x ) ,

which prove (3.8), while (3.9) follows from

Axiomatics of three-valued algebras

133

S 3 j S2: We suppose S 3 and prove (3.1). Note that (3.11) holds by Lemma 3.1, hence

because if N x A y I x then

hence y2(x A y ) 2 92x by (3.13) and similarly y 2 ( x A y ) (3.14)

92(x

AY)

59

2

A~9

2

5 y z y , therefore

.~

Furthermore (3.8), (3.12) and (3.11) imply

and similarly N y A

S2 obtain

+ S1:

92s

A (p2y 5

x A y , hence

First we get (3.11) by Lemma 3.1. Then we use (3.8) to

which, together with (3.1) and (3.9) imply

134

Elementary properties of Lukasiewicz-Moisil algebras

92N92~ A9

2

=~ 9 2 ( N 9 2 A ~ X) = 920 = 0

92N92~ V

2

=~ 9 2 N 9 2 ~V N N ~ =~1 x,

9

,

hence (3.17)

92N92x = cpza: ,

which implies in turn, via (3.11), (3.18)

9

(3.19)

92s V N 9 2 = ~ N ( N 9 2 x A v ~ x )= NO = 1

2

A ~N 9 2 ~ 59

2

A~9 2 N 9 2 ~ =0

, ,

therefore

(3.20)

N92z =

w.

Now (3.20) implies (3.5) by comparison with (3.17) and also (3.10) because

= 920 = 0 , therefore (3.4) holds via $925 = 9 2 "

V ((p22 A 92922) = 1A

.

( 9 2 V~ 92922) = 9 2 9 2 ~

S1 =$- SO: Note that ' p 2 is isotone by (3.1). Therefore x 5 x V y implies 92x 5 92(x V y ) and similarly 9 2 y 5 y 2 ( 2 V y ) , which is the starting point in the proof of (3.2) via (3.11), (3.5) and (3.1): 92s

v 92Y I

92(x

v Y ) I 92(92x v 9 2 Y )

=

=

'pz"N92X

NY2Y) = ' p z N ( 9 2 N w 92N92Y) =

=

92N92(NQ2X

A N92Y) = N'p2(N92X A N P 2 Y ) =

Axiomatics of three-valued algebras

135

From (3.5), (3.4) and (3.10)we infer

N ~ ZV X( ~ 2 = 2 p

2 V ~ ( ( ~ 2 2A

N Y ~ XV )N p 2 ~=

A ~ 2 N v 2 2V) N 9 2 9 2 ~= 1

= N V Z N ~ ZV X (WPZX

which is equivalent to (3.3). Further (3.11) implies in turn N x 5 p 2 N x and

N p 2 N x 5 N N x = x 5 pzx, that is (3.6). The proof of (3.7) follows L. Monteiro [1969]: ( N ~ z NVxN x ) A

~

Z

= X N(p2Nx A N N x ) A

= N ( N x A X ) A 9 2 =~ ( X A

9 2 ~ V)

(NxA

c p 2 ~=

~ Z Z )

=

= XV(XANX)=X,

therefore if p2x = y2y and p2Nx = p2Ny we obtain x = y as follows:

The subsequent systems of axioms make use of the following identities: (3.21)

N ( x A N p 2 ~= ) 1

(3.22)

x A N(Nx A Ny) =x

(3.23)

x A N ( N y A N z ) = N ( N ( z A N ( N x A NO)) A

,

A ~

(3.24)

(A N 9 (NxA NO)))

N ((p2(xA y ) A N ( z A N z ) ) A t ) = = N ( (('Pza: A V2Y) A N(cp2z A N z ) ) A

(3.25)

1 Ax = x

t)

7

Elementary properties of Lukasiewicz-Moisil algebras

136 (3.26)

N(z A N1)= 1

(3.27)

(201

A

~

2 A )(203

, A

204)

= (20: A

20;)

A

24

,

where we have set

3.3. Theorem. An algebra (L, A, N , 1,q 2 )of t y p e (2,1,0,1)is equivalent t o a three-valued Lukasiewicz algebra ( L , A , V , N,0,1,q2)if and only if it verifies any of the following equivalent s y s t e m of axioms: S4 = {(3.1),(3.8),(3.21),(3.22),(3.23)}

,

S5 = {(3.21), (3.22),(3.23),(3.24)) , S6 = {(3.25), (3.26),(3.27)) , and one defines V y =N(NzA Ny)

(3.28)

2

(3.29)

0 = N1

,

.

Corn ment . System S4 is due to A. Monteiro; cf. were given by Petcu [1968].

t. Monteiro [1963], while S5 and S6

Proof.

S2: For (3.22) & (3.23) (3.21) H (3.9) via (3.0).

S4

(j

(3.0) by Proposition 1.2.10, while

137

Axiomatics of three-valued algebras

S4 j S5: For (3.1) & (3.8) + (3.24). S5 j S6: L i s a De Morgan algebra by (3.22) and (3.23). This implies readily (3.25) and (3.26), as well as wl = x1 A(yl Vzl) = w:,then w2 = w: by (3.8), w3 = 1 by (3.9) and w1= z4. The latter identities imply (3.27). S6 + S4: We are going to obtain the axioms of S4 as particular cases of (3.27) via (3.28), (3.25) and (3.26). Thus for z1= y1 = z1 = 1 and t2 = 5 3 = N1 we have w1 = N ( N 1 A

N1) = 1, w2 = 1, w 3 = 1, y1 A z1 = 1, wi = 1, wi = 1, hence (3.27) becomes w4 = x4, that is (3.22). For z4 = y4 = 1 and t 2 = z3= N1 we obtain w2 = 1, w 3 = 1,w4 = 1, wk = 1. introducing these values into (3.27) and taking into account that c A 1 = c by (3.22) with y = 1, we get w1 = w:,that is (3.23). Thus L is a De Morgan algebra, which facilitates the remainder of the proof. Taking z1 = y1 = 5 4 = 1 and t 2 = 0 yields w1 = w2 = w4 = wi = 20’ 2 -1, hence (3.27) reduces to w3 = 1, that is (3.21). For 51 = y1 = x4 = 1 and z3 = 0 we infer w1 = w 3 = w4 = wi = 1, hence (3.27) reduces to w2 = w;. Taking 2 2 = t 2 = 1 in w2 = w: we get Ncp2(22 A y2) = N(cp2x2A cp2y2), that is (3.1). Taking x = 1 in (3.21) we obtain cp21 = 1, therefore taking 2 2 = y2 = 0 t 2 = 1 in w2 = w: we deduce (3.8). Three more systems of axioms make use of the identities (3.30)

z V 1= 1 ,

(3.31)

zA

(3.32)

x A (y V z ) = ( z A x ) V (y A x)

(3.33)

NNx =x

(3.34)

N ( x A y) = Nz V Ny

(x V y) = z ,

, ,

and are based on the following result.

,

Elementary properties of Lukasiewicz-Moisil algebras

138

3.4. Lemma. If N and 92 are unary operations o n the lattice (L, A , V) and 1 E L, t h e n (3.9)& (3.33)j (3.30). Proof. Using (3.33)we can write (3.9)as

vzNxV x

= 1, hence x V 1 =

x V 92N~ V x = 1.

3.5. Corollary. Each of the following systems of axioms characterizes three-valued Moisil algebras:

S2’ = ((3.31)-(3.34),(3.1), (3.8),(3.9)} ,

S3’= ((3.31)-(3.34),(3.8),(3.9),(3.12)}, S4’= ((3.31)-(3.34),(3.1), (3.8),(3.21)) , S5‘ = ((3.31)-(3.34), (3.21),(3.24)) . Com ment. Systems S2’ and S4’ were given by A. Monteiro [1963],while S3’ is due t o Becchio [1973]and S5’ is new. Proof. Clearly SIC

+ SK’ (K = 2,...,5). To prove the converse, note first

L is a distributive lattice by (3.31),(3.32);cf. Proposition 1.2.4.Since SK’ contains also (3.9)or its equivalent (3.21)(4. (3.33) and the lattice properties), Lemma 3.4 ensures t h a t (3.30) is valid. Thus L fulfils (3.30)-(3.34), hence it is a De Morgan algebra by Propositions 1.1.31-1.1.32,therefore SIP + (3.0)and SIP + (3.22)& (3.23),that is 0 SIP =j S K (I< = 2,...,5).

that each SIP implies that

3.6. Remark. A. Monteiro, Petcu and Becchio also proved the independence of the systems S2’, S4’,S6 and S5‘, respectively. Thus e.g. Petcu gives the following models for the independence of S6: th e set (0,l) endowed, in turn, with arbitrary ‘p2and z A y = 1 and Nx = s for (3.25), x A y = min(z,y) and Ns = 0 for (3.26),x A y = y and Nx = x for (3.27).

Axiomatics of three-valued algebras

139

The next axiomatic system is in terms of dual endomorphisms and does not use negation as a primitive operation. 3.7. Theorem (Moisil [1960]).

An algebra ( L ,A, V,O,

of type (2,2,0,0,1,1) as equavdent t o a three-valued M o i s i l algebra ( L ,A, V, N , 0,1, pl,cp2) via t h e t r a n s f o r m a t i ons 1,$1,$~)

and (3.36)

p k x

(3.37)

N X= $

( k = 172)

= +k'$kx 2 ~ V

(X A $ 1 ~ )

7

,

if a n d only if (3.38)

( L ,A, V, 0,1) is a bounded distributive lattice ,

(3.39)

$k

(3.42)

41*2x

= *2$25

(3.43)

$i$ix

V

(3.44)

$kx = $ k y

are dual e n d o m o r p h i s m s (k = 1,2)

($13

,

7

A $ z $ ~ x )V

(k = 1,2)

= 1,

$2.

?!'

=9

-

Proof. According to Proposition 1.5.42 and Definition 1.5.43, the proof will consist in two steps.

I) Take a Moisil 3-algebra L , define

by (3.35) and prove that (3.38)(3.44) hold as well as (3.36)-(3.37). But $h$i = cpi by (1.19) and this makes the proof almost trivial. We only mention that (3.43) follows from $k

140

Elementary properties of Lukasiewicz-Moisil algebras while (3.37) is checked using the determination principle.

II) Suppose L fulfils (3.38)-(3.44), define 9 k by (3.36) and N by (3.37) and prove th a t L is a Moisil 3-algebra and (3.35) holds.

I $ 1 and ~ $ 1 ~ 5 $1$1$1x by From $1$1x I x we deduce $I$I$IX taking x := $Is and by applying $1 via (3.39), respectively. Therefore $I$I$I = $1 and similarly $2$2$2 = $2. B u t $ i $ j = $j$j = cp, by (3.41), (3.42) and (3.36). W e summarize all these results as

and use t hem in the sequel. Thus

(3.46)

91 I 'p2

by (3.40) and then

8z $2 L $1 $2

= $292

<_

$291

= $1. Further

by (3.46) and (3.43), hence

where t he second identity follows fro m the first: (Pk A $k = ~ k ( $ kV (Pk) = $kl

= 0. It f O l b V S th a t

that is

$2N = (pl and similarly $,N = cp2, hence

because (PkN = $ k $ k N

=

$k(P3-k

prove (3.35) and (3.0)-(3.7). First

= q3-k.

NOW

We Use (3.36)-(3.48) to

Axiomatics of three-valued algebras

141

that is (3.35). Further (3.1) and (3.2) follow easily from (3.39), as well as (3.3) from (3.47) via (3.35) and (3.4) from (3.45).

From (3.48) and (3.45) we get pzNpz = h c p z = $2 = Np2, that is (3.5), and N p z N = $zN = p1 I p 2 , that is (3.6). To prove (3.7) suppose p z N x = p 2 N y and pzx = p2y. Then $ 1 ~= $ l y and x = y by (3.44). Finally

$25

= N p 2 x = N p 2 y = $2y hence

hence N x V N y = N ( x A y ) by (3.44) and also

N N x = $ ~ N Vx ( N x A $ l N x ) =

comple ing he proof of (3.0). We still have t o note that p1 given by (3.36)

coincides with p1 given by (2.13),i.e.

= N v z N , which is true because

N p z N = N$1 = 91 = $1$1.

0

The next description of three-valued LM-algebras, in terms of implication, has been used in the axiomatisation of the three-valued Lukasiewicz logic, cf. Ch. 9, $1. 3.8. Theorem (Becchio [1978d]).

An algebra ( L ,+, N , 1) o f type (2,1,0) i3 equivalent t o a three-valued LM-algebra ( L ,A, V, N , 0,1, pz) via the transformations (LW)

+

Y = (cp&

v Y ) A ( Y Z Yv N x )

Elementary properties of Lukasiewicz-Moisil algebras

142 and (WL1)

zvy=(z+y)-+y,

(WL2)

z A y =N(NzV

(WL3)

cp2z

(WL4)

O=N1,

=N z +5

Ny) ,

,

if and o n l y if (Wl)

z -+ (y + z) = 1 ,

(W2)

(.

(W3)

((x + N z ) + z) + 5 = 1 ,

(W4)

(Nz +Ny)

(W5)

l+z=l+z=l,

(W6)

z

--$

3

Y>

+

((Y

+

4

+

+ (y + z)

(.

+

2,)

= 17

= 1,

y = 1 & y -+z = 1 jz = 9 .

Proof.

I) We suppose t h a t ( L , - + , N , l )satisfies (Wl)-(W6) and prove that the algebra ( L ,A, V, N , 0,1, cpz) defined by (WLl)-(WL4) is a three-valued LM-algebra satisfying (LW). To do this we define (3.49)

2

5 y ($2

+y = 1

and keep in mind the translation of axioms (Wl)-(W6) in terms of i.e.

TI)

2

5 y -+ z

5,

etc. We prove in turn several properties.

( L ,5 , l ) is a poset with greatest element .

If z 5 y and y 5 z then (W2) becomes (1 + (1 + (z 3 2))) = 1, whence z + z = 1follows by applying (W5) twice; thus z 5 t. Then (Wl) and (W3) imply z 5 (z -+ Nx) + z 5 z, hence z 5 5. Axiom (W6) expresses the antisymmetry of 5. Finally 1 5 z + 1 by ( W l ) , hence z + 1 = 1 by (W5), i.e. z 5 1.

143

Axiomatics of three-valued algebras T2)

x l y ~ y + z ~ x - + z .

From (W2) via (W5).

.

( x -+ N x ) -+ x = x

T3) x

5 ( x --+ N x )

T4)

--f

5x

x

by ( W l ) and (W3).

Nxlx-ty.

Nx 5 Ny

-+

Nx 5 x

--f

y by ( W l ) and (W4).

Properties T4), (W4) and (W3) imply

NNx that is N N x

5 Nx

5 x.

Nx 5 x

--t

N ( x -+ N x ) 5 ( x -+ N x ) -+ x I x ,

Then N N N x

hence (W4) implies x

Ny

--+

5 N x , that

is N N N x

--f

N x = 1,

5 NNx.

-+

y =NNx

-+

NNy 5 Ny

-+

N x by (W4) and

T5).

x I y + Ny 5 N x

T7)

.

By T6).

-+

x

5 (x

N x ) -+ x = x x --+ N x 5 1, then T3) and (Wl). 1

--+

5 1

-+

x by T2) applied t o

Elementary properties of Lukasiewicz-Moisil algebras

144

x = 1 -i x 5 (x + y)

+ (1 -+ y) =

(x -+

y)

-i

y by T8) and

(W2). T10)

ylzvy.

From (Wl).

T11)

x l y w z V y = y .

If x 5 y then x v y = y by T8). If x V y = y then T12)

x

-+

(y

-+

z) =y

-+

Apply (W2), then T2) to y

(X -+ Z)

2

I y by 7-9).

.

5 y V z:

while the converse inequality follows by symmetry.

From T12).

It follows from (W2) and T8) that

Using T7) and T5) we get

Nz

I N N y = y, whence T2) and T14)

imply

z

-+

N z 5 11: -+ N z 5 x

-+

y

5z ,

Axiomatics of three-valued algebras

145

therefore (3.49), T2) and T3) yield

1= z +

T16)

2

5 ( z -+ N z ) + z = z .

xlxVy-tyVx.

Using (W2) twice, then T2) applied to the instance x

x)

+ x of

5 ((x

-+ y)

4

(Wl), we obtain

whence T16) follows by T13).

T17)

x-+y5xVy-+yVx.

x V y = (x -+ y) -+ y 5 (x + y) + x V y by T14) applied to y 5 x V y; now use T13).

T18)

NyIxVy+yVx.

x V y = ( x -+ y) -t y I N x -+ y = N y + x by T2) applied t o T4), then T6) via T5); now T13), then T14) applied t o T10) yield Ny

T19)

5 x V y -+ x I x V y --+ y v x

xVy=yVx.

T15)-T18) imply x V y 5 y V x , hence y V x

z +y

5 x -+ y

I x V y.

by T2); then T2), T19) and T8) imply

Elementary properties of Lukasiewicz-Moisil algebras

146

zvy= (z+y)+y<(z-iy)+y=zVy

=

- yVz=(y+z)-+z=l+z=z. T21)

(L,A , V, 0 , l )

is a bounded lattice.

T9), T10) and T20) show t h a t ( L ,V) is a join semilattice. But N is a decreasing involution by T5) and T7), while x A y is defined by (WL2), therefore ( L ,A, V ) is a lattice by Proposition 1.1.30. Also, 1 is greatest element by Tl), hence 0 = N1 5 N N z = z by (WL4), T7) and T5). Properties Tl),

N(N2 V NY)= x A y

5 z + Nz I NXV Ny

= (Nx + Ny) + Ny 3 N x + Ny by T5), T7), T13) and

+

5 Nz+ N y e y +x 5 y

5x

+ (z A y) V (x A

2

+z

T6).

by T22), T13) and T2) applied t o

z

=

(Wl). One proves similarly that

z ) , hence

A (Y V Z) I y V z

I x -+ (z A y) V (z A z ) ,

whence T23) follows by T13).

147

Axiomatics of three-valued algebras T24)

z +z

2 z A (y V z ) + (z A y) V (z A z ) .

Applying T13) to (z + z ) + t

5

(z + z ) + t and T14) to ( W l )

we obtain

((z+z)+t)-+t< ((z+z)-+t)+(v+t).

x + 2 <

Taking t = (z + y) V N z , v = z + y V z and using (WLl), T12),

(WLl), (WL2), T5),T6), T12) and (WLl), we get 5 +z

I ((z +

= ((z

+ 2) + ((z -+ y)

v N z ) ) + ((z

((x

=

+ y)VNz))

+z)

v 2)

+

+ (((z + y) + ~ z + ) ~ z ) p) r

v 2) -+ (((z

+ ((z + y

+Nz)

= (((z + y) + (((z

+y

---t

-+ y) + N z ) + N z ) ) =

+ ((z + 2) + N z ) ) +

y) + N 2 ) + ((z + y

v 2) + N z ) )

=

= (((Ny + Nx)+ N z ) + ( ( N z -+ N z ) -+ N z ) ) + +

3

(("Y

Nz)

+

--t

Nz)

+

((N(y v

2)

+Nz)+Nz)) =

= (NyVNz-tNzVNz)

+

+ (NyVNz+N(yVz)VNz)

= (N(zAy)+ -+

(N(y A x )

N(zAz))

=

---$

+ N((y V z ) A z))

= (zAz)+zAy)+

=

((yVz)Az-+yAz)

=

= ( y ~ z ) ~ z + ( ( z ~ a + z ~ y ) + y= ~ z ) = z A (y V z ) + ((z A z ) V

( 2A

y))

.

148

Elementary properties of Lukasiewicz-Moisil algebras T25)

N z 5 z A (y V z ) -t (z A y) V (x A z )

.

Using T4), (WLl), T6), (W2), (WU), T6) and T14) we get

N z < z+

yI(z+y)Vy=((*+y)+y)+y

= zVy+y=Ny+N(yVz)

<

<

5

- (N(yVz)+Nz)-t(Ny+Nz)

I

((Ny

= (ivy

-t

Nz)

--t

=

N z ) + ((N(y v 2) -+ Nx)+ N z ) =

v N z ) -t ( N ( y v 2) v N z )

=

= N ( y A z) + N ( ( y V z ) A z) =

= (yV z) A z

-t

yA z

< z A (y V z ) + (z A y ) V (z A

2).

Now we check system S3 from Proposition 3.2.

It follows from T23)-T25) via T15) that z A ( y V z ) 5 ( z Ay ) V( z Az ) , hence L is a bounded distributive lattice by T21) and Proposition 1.2.1, while N is an involutive dual endomorphism by (WL2) via Proposition 1.1.31. Thus L is a De Morgan algebra, i.e. (3.0) holds. To prove (3.8) we use (3.0), (WLl), T6), (WL3) and T3):

The proof of (3.9) follows from (WLl), (WL4), T3) and (3.49):

To prove (3.12) we first use T4) and T14) and obtain

Nz 5 z

+y

I z -t

yVNy

,

Axiomatics of three-valued algebras

149

hence by T13) and T2) we get (3.50)

x 5 Nx -+ y V Ny 5 x A Nx -+ y V Ny ,

then we note that T2) and T14) imply (3.51)

x

-+

yI x A Nx + y

5 x ANx -+ y V Ny ,

while from ( W l ) and T14) we obtain (3.52)

Ny 5 X A NX -+ Ny 5 x ANx + y V Ny ;

finally T15) applied to (3.50)-(3.52) yields x A Nx

--t

y V Ny = 1, that

is (3.12).

We have thus proved

and it remains to demonstrate (LW).

We have y 5 x + y and Nx 5 x + y by ( W l ) and T4),respectively, while (WL3), T2) and T14) imply

From T14), T6), (W2), (WL3) and (WL1) we obtain

Elementary properties of Lukasiewicz-Moisil algebras

150 hence z + y

Ny

-+

Nx

I Nz V (pzy, which

implies also, via T6), that x + y =

5 y V Q ~ N xtherefore , T28)

holds.

II) We suppose that (L, A, V, N, 0, 1 , ~is ~ a three-valued ) LM-algebra, we define -+ by (LW) and prove (Wl)-(W6) and (WLl)-(WL4). Properties (WL2) and (WL4) are known, while (WL3) follows a t once: Nx + x = 972s

V

x = z. Then we prove (3.49): x + y = l w

'p1(z+y)=1 H

Now (W5) and (W6) express known properties, (LW) implies y

x

+

5

y, which is (Wl), while (W3) and (W4) can be proved in the

stronger form of equalities (using also (WL3)):

=

' p 2 ' p ~ X V X=

x

,

Finally (WL1) and (W2) will be proved with the aid of the determination principle. We use properties cps-iNa = N p i a = tpia (i = 1 , Z ) and compute in turn

Axiomatics of three-valued algebras

151

Elementary properties of Lukasiewicz-Moisil algebras

152

The next two lemmas will be used in Ch. 9, $1. 3.9. Lemma.

The following properties hold: (3.55)

(x +

(. (Y 4 ) ) ((. +

+

+

+

(.

-+

Y))

+

+ (z + (z + 2))) = 1 , z = 1,

(3.56)

N ( z + y)

(3.57)

N(z

(3.58)

x + (Ny -+ N ( z + y)) = 1 .

--t

-+

y) + Ny = 1

,

Proof. Applying T7) to T4) we obtain N ( z + y) 5 z, that is (3.56). Also, (3.57) follows from (Wl)by T6). From T6) and T12) we infer 2 +

(IVY -+ N ( x + y)) =

5

+ ((z + y) + y) =

= (z-+yY)t(x+y)=l. Now we use the determination principle to prove (3.55) in the equivalent form 2 +

(x + (y

-+ 2 ) )

5

(z + (z + y))

using (3.53), (3.54) and T4) we compute in turn Cp*((z -+

(.

+

Y))

= p z ( ( N ( a : + (z

+

-+

.(

Y))

+

(.

A (z

+

+

4))=

(X + 2 ) ) )

V

---$

(.

+ (z + 2 ) ) ;

Axiomatics of three-valued algebras

153

Elementary properties of Lukasiewicz-Moisil algebras

154

3.10. Lemma. The function- defined b y (3.59)

k = x + N X [= N N x + N X = 'pzNx]

s at isfies Es

(3.60)

x-+ x = 1 ,

(3.61)

(x -+ Z)

(3.62)

k

-+

(&

4

Proof. Property Es

X=

while

= 1, (X

-+

9)) = 1 .

(3.60)follows from

V ~ N ~ =~ 9 N2 9 X1 = ~ 91" 5 x ,

(3.61)is equivalent x

while

+2

-+

to

x

-+

k 5k

and in fact

x + k = 2 because

k = 9 2 ( N ~A k ) V N X V Z = 9 2 N ~V 5 = 5 ,

(3.62)follows from (LW):

Ki

+

(x

+

Y> = (P2Y

+

(x

+

Y) L N92Y v (x

L N P ~ VY ( P Z ( N AX Y) = $ 5 2 ~V ( ~ 2 N Ax 9

+

2 ~ )=

Y) 2

Irredundant n-valued Moisil algebras

155

$4. lrredundant n-valued Moisil algebras

4.1. Definition. An n-valued Moisil algebra L is irredundant, if for every subset To

c

(1, ...,n - l}, there exist x , y E L , such t h a t 'pix = (piy, for any i E To, but X#Y.

The question o f characterizing the irredundant Moisil algebras was rained by Rudeanu. The aim o f this section is t o give an answer t o it; cf. Boicescu

[1988]. 4.2. Proposition. Let L E Mn. T h e following are equivalent:

(i) L is irredundant. (ii) FOTa n y j E {1, ...,n - l}, there exist x , y E L, such that (pix = (po;y, f o r i E ( I , ...,12 - l}\{j}, but x # y. 4.3. Proposition. Let L, be the canonical structure of Moisil algebra defined o n the chain (0 = co < c1 < ... < c , - ~ = 1). T h e following conditions are equivalent for a n y j E (1, ..., n - 1):

(ii) x = y

OT

z # y and x, y E {cn-j,

Proof.

(i) + (ii): Suppose x

# y and 'pix = 'pip, for any i E (1, ..., n - l}\{j}. By the determination principle, 'pjx # (pjy and suppose for instance 'pjx = 0 and 'pjy = 1. It follows that x

2 cn-j-l

and y

2 en-j. If x < cn-j-l,

then

= 0 and 'pjtly = 1, a contradiction. If y > cn+, then (pj-ly = 1 and cpj-12 = 0, a contradiction. Hence x , y E {Cn-j,cn-j-l}.

(ii) + (i): Now suppose x # y and x , y E {cn-j,cn-j-l}. We have (pjc,-j = 1 and ~pjc,,-j-~= 0. If i < j, (p;cn-j = = 0 and if i > j,

Elementary properties of Lukasiewicz-Moisil algebras

156

Qicn-j = cpic,-j-i = 1. Therefore 9;x = (piy, Vi E (1, ...,n - l}\{j}. 0

4.4. Corollary.

L, as irredundant, but its proper subalgebras are not irredundant. Proof. To prove the first affirmation, put

2

= cn-j and y = ~ , - j - ~for , any

j E (1, ...,n - 1) and use Propositions 4.2 and 4.3. If LL is a proper subal-

( j E {1,...,n - 2}), and if z,y E Lk, cp;x = cpiy, Vi E (1, ...,n - l}\{j}implies 2 = y , since cn-j-l = Ncj @ LL. gebra of

L,, there exists cj E L,\Lk

D

Remark.

If L E Mn is irredundant, then the endomorphisms cpi are distinct. The converse is false; t o prove this we need a lemma. 4.5. Lemma. If L is a subalgebra of a direct product L f , then the following are equi-

valent: (i)

The endomorphisms cp; are distinct.

(ii) CardIL = 1. ( I L = { d I Q;NZ= Ncpdiz}). (iii) For every z E L,, there exists y E L and j E M , such that pj(y) = x (pj are the canonical projections, j E M ) . Proof.

(i) H (ii): Immediate from Proposition 1.27. (i) + (iii):Suppose there exists x = c;, for some i E {1,2, ...,n - 21, such that for any y E L and j E M , pj(y) # ci. This means

and

Irredundant n-valued Moisil algebras

157

0

7

pj(vn-i-lY) = vn-i-lPj(y) = 19

Vj E M , Vy E L. It follows that

PAY)

< ci

Pj(Y) > cc

= ~ , + i - ~ a, contradiction. (iii) +-(i): Suppose there exist y E L and j E M , such that pj(y) = c,+i, i E (2, ...,n - 1). This implies ‘pn-i

pj(viY) = ‘pi ( ~ j ( y ) )= Vicn-i = 1 ;

Remark.

It follows from the proof of Lemma 4.5 that ‘pi # Cpi+l, i E (1, ..., n - 2) if and only if there exist z E L and j E M , such t h a t pj(x) = c , - ; - ~ . Consider now the following

Cou nt erexa mple. Let

L 5

L5”

=

L5”.

be the canonical five-valued Moisil algebra and L: = {co, c1, c3,cq), { C O , C ~ , C ~ ) its

four- and three-element subalgebras. Consider L: x

The endomorphisms of this five-valued Moisil algebra are distinct

(Lemma 4.5), but disregarding ‘p2, we have P ~ ( X I , X Z )= ( P ~ ( Y I , Y Z ) , i

(pix2

= ‘piy2, a = 1,3,4

(Proposition 4.3) Thus

Lk x L5”is not

= 1,394

* pix1 = Fiyl

+ x1 = y1

and

and x 2 = y2

.

irredundant.

Remark.

As any n-valued Moisil algebra can be embedded in a direct product LF (see Corollary 6.1.9),it means t h a t Lemma 4.5 refers t o t h e whole class of these structures and t o simplify we shall consider further only algebras of this form.

Elementary properties of Lukasiewicz-Moisil algebras

158 4.6. Theorem.

If L

E Mn, then the following are equivalent:

(i) L is irredundant.

(iii) For any element ci E L,, i = 1, ...,n - 1, there ezist z # y E L , such that p j ( z ) # pj(y) + pj(z) = c; and p j ( y ) = ci-1, Vj E M .

if ci E p j ( L ) and ci-1 E p j , ( L ) , i = 1,..., n - 1 and j,j’ E M , there ezist z # y E L, such that p k ( z ) # pk(y) +

(iv) cp; are distinct and p k ( z ) = c;

and pk(y) = c;-~.

Proof.

(i)

+ (ii): Let i

be fixed in t h e set (1, ...,n - l}, otherwise arbitrary.

As t h e n-valued Moisil algebra L is irredundant, it follows t h a t there exist

# y E L , such that cpk(z) = cpk(z), for any k # n - i. Then p j ( c p k z ) = pj(cpky), Vj E M ,k # n - i. Therefore cpkpj(x) = cpkpj(y), Vk # n - i. By Proposition 4.3, it follows t h a t p j ( z ) # pj(y) + { p j ( z ) , p j ( y ) }= { c i , c i - l } . z

(ii) + (iii): If we consider z,y E L t o satisfy t h e conditions of (ii), we put u = z V y and v = z A y and these elements satisfy the conditions of (iii). (iii) + (iv): This follows immediately from Lemma 4.5.

+ (i): Let j E ( 1 ,...,n - 1) and z = c,-j. There exist u # v E L , such that p k ( u ) # pk(v) + p k ( u ) = c,-j and pk(v) = c , - + ~ . We must (iv)

show t h a t cpiu = cpiv,

Vi # j.

160

Elementary properties of Lukasiewicz-Moisil algebras

(i) L i s exactly n-valued. (ii) cpi are distinct and f o r every i = 1, ..., n- 1, if one eliminates cpi and cpn-i, the determination principle fails. The proof is left t o the reader.

It follows from Proposition 4.10 that the above algebra Lk x L5” is a five-valued Moisil algebra, which is exactly 5-valued, but is not irredundant, as was seen before.

4.11.Theorem. If L E M 2 n , n 2 2, t h e n the following are equivalent: (i) L is irredundant.

(ii) L is exactly 2n-valued. Proof.

(i) + (ii): Trivial. (ii) + (i): L has 2 n - 1 endomorphisms and from Proposition 4.10,it follows that for every j E (1, ...,2 n - l}, there exist x # y E L such that vix = viy, for each i E (1, ...,2 n - l}\{j,2n - j } . We shall show that condition (ii) of Theorem 4.6 is satisfied. Indeed, consider first the case j = n. We have 2 n - j = n, hence there

#

# n. Then pk(cpix) = Pk(Cpiy), Vk E M , i # n or vipk(Z) = pipk(y), i # n, Vk E M . From Proposition 4.3,we obtain pk(x) # pk(y) + {pk(x),pk(y)} = {cn,cn-1}. If j 5 n - 1, then 2 n - j 2 n + 1 and we have the following possibilities: exist x

y E

L,such

that 9;s = piy, for any i

Irredundant n-valued Moisil algebras

C)

161

# Qjy and Q z n - j X # Vzn-jy. Let k E M such t h a t p k ( ~ #) p k ( y ) . Suppose V j p k ( 3 ) # ' p j p k ( y ) . Consider for instance p j p k ( z ) = 1 and VjX

= 0. Then V z n - j p k ( z ) = 1. If V z n - j p k ( y ) = 0, then Vnpk(z) = 1, but c p n p k ( y ) = 0, a contradiction. Therefore cpipk(z) = ~ i p k ( y )for

VjPk(y)

any

i # j. Hence

=

{pk(x),pk(y)}

Analogously if

{CZn-j,CZn-j-l}.

V Z ~ - ~ P ~# ( Z ~)z , - j p k ( y ) , it follows that { p k ( z ) , p k ( y ) } = {cj, cj-l}. Consider now the elements u = z A N z and v = y A Ny. Obviously

#

# Pk(v)

and pk(')

If j 2 n

* {Pk(u),Pk(v)} = {cj,

+ 1, then 2n - j

5 n - 1 and

Cj-1).

in accordance with the previ-

# v, such that p k ( u ) # pk(v) =$ Then N u # N V and p k ( N u ) # p k ( N v )

ous case, there exist u

(pk(u),pk(v)} =

{CZn-jTCZn-j-1)-

=+

{ P ~ ( N ~ ) , P ~ (=N{vN)p}k ( u ) ,N p k ( v ) } = {Nczn-j,

NcZn-j-1)

= 0

{Cj-l,cj}*

4.12. Proposition.

If L E M2n + 1, n 2 2, t h e n L i s exactly (2n + l)-valued if and only if the following conditions are satisfied:

FOT every element ci E Lzn+l, i = 1, ...,n - 1 (and also f o r i = n + 2, ...,2n), there exist z # y E L, such that pj(z) # pj(y)

(i)

*

p j ( z ) = ci and pity) = q - 1 .

(iii) Vn

# Vn+l.

Proof.

(+) (i) is proved as in the case j _< n - 1 from the previous theorem. Further on by applying Proposition 4.10, for j = n, it follows that there exist

# y E L , such that cpix = y i y , for any i E (1, ..., 2n}\{n,n a) Qnx

# VnY

and

Vn+1x

{ P ~ ( x )P~(Y)} , = {cn+1, b) Vnz = VnY and Vn+1z follows that

= V n + l Y . By Proposition 4.3, p k ( z ) # pk(y)

*

Cnl.

# vn+1y.

By applying again Proposition 4.3, it

# P ~ ( Y* ) {pk(z),~k(y))= { c n , C n - l } *

P ~ Z )

+ 1). Thus,

Elementary properties of Lukasiewicz-Moisil algebras

162

( e )By

Lemma 4.5 and th e remark that follows it, we conclude that

the endomorphisms cp; are distinct. Using (i) it follows t h a t for any i =

#y

+

L, such that cpjz = cpjy, V j # 2 n 1 - i (the proof o f Theorem 4.6 (iv) + (i)); thus for every i = 1, ..., n - 1, if we 1, ...,n - 1, there exist x drop

'pi

E

and cpn-i the determination principle fails.

Let now z,y E

(1, ...,2 n } \ { n , n

L,

verifying (ii). We shall prove that cpiz = (piy, Vi E

+ 1). Let k E M , such that p k ( z ) # pk(y). We have

Analogously we obt ai n

Thus pk(cpiz) = pk(cpiy), for every

kEM

such that p k ( z ) # pk(y). As for

all the other indices k the equality is obvious, it follows that the determination principle fails if we eliminate {yn,cpntl}. Now Proposition 4.10 implies that

L is exactly ( 2 n + 1)-valued.

4.13. Corollary. There is L E M 2 n not irredundant.

+ 1, n 2 2,

such that L is exactly 2 n

0

+ 1-valued, but

Irredundant n-valued Moisil algebras

In accordance with Proposition 4.12,

L is

163

exactly (2n

+ 1)-valued, but in

view of Corollary 4.7 it is not irredundant.

0

4.14. Proposition.

Let L E Mn, n = 3,4. T h e n the following are equivalent:

L is irredundant.

(i)

(ii) L is ezactly 3(4)-valued. (iii) L has distinct endomorphisms. Proof.

(i) + (ii) + (iii): Trivial.

+ (i):

(iii)

Pj(4 =

If L E M3, there exist z E L and j E M , such t h a t

If Pk(4 E { C O , C Z ) , then Pk(Y) = ( P I P k ( Z ) = p k ( z ) . But if p k ( z ) = c1, it follows that pk(y) = qlpk(z) = q,. Hence p k ( 2 ) # pk(y) + {Pk(Z),Pk(Y)} = {cl,cO}. In the same way, if we take z = ( P Z X , it follows t h a t p k ( z ) # p k ( z ) + {pk(z),pk(z)) = {c2,c1} and the condition (ii) of Theorem 4.6 is satisfied. If L E M4, there exist z E L and j E M , such t h a t p j ( z ) = c1 and we take y = 2 A N z . Obviously p k ( y ) E {q,,cl}, V k E M and

#

c1.

Let Y = (Pl”.

+ pk(y)

= c1. Hence (0,y) satisfies the condition of irredundancy for c1. Analogously (1,Ny) satisfies the condition of irredundancy for cg. Consider then th e elements y and z = N ( P ~V (N~z ) A (z V N z ) and

pky

co

it follows t h a t p k ( z ) E {q,,cg}

+ P&)

+ p k ( z ) = N y l c 3 A cg = co = pk(y) and

= N ~ i c zA cz = cz satisfies the condition of irredundancy for c2.

P ~ ( z )E

{CI,CZ)

#

4.15. Proposition.

Let L E M5. T h e n the following are equivalent: (i)

L is esactly 5-valued.

ci

= pk(y). Thus ( y , ~ ) 0

Elementary properties of Lukasiewicz-Moisil algebras

164

(ii) L has distinct endomorphisms. Proof.

(i) + (ii): Immediately. (ii) + (i): By Proposition 4.12, we must show that: a) There exists

#

x

y E

L,such

that pk(x)

#

pk(y)

j

p k ( z ) = c1 and

pk(y) = CO. Indeed by hypothesis, there exist z E L and j E M , such that pj(x) = c1. Consider then y = N93(z A N z ) A x A N z . We have

*

*

pk(Y) = N93cOAcO = COP Pk(z) E {Clr c3} pk(y) = = c l r and pk(z) = c2 + pk(y) = Nv3~2A c2 = Q . Hence Vk E M , pk(y) E {co,cl} and the elements y and 0 satisfy the required

Pk(5) E {Q,c4} N$oJC1

A

c1

conditions.

b) There exist z # y E L , such that pk(z) # pk(y) + {pk(z),pk(y)} c { c I , c z , c ~ } Consider . again x E L,such that pj(z) = c1, for some j E M and put y = x A

Ns

and z = Nvl(z V

Nx)A (x V Nx). It follows

P k W E {co,c4} + Pk(Y) = co = Pk(Z), Pk(2) = c2 =+ Pk(Y) = P&) = ~ 2 but , pk(x) E {Cl,c3} + pk(y) = c1 and p k ( z ) = c3. Hence that

pk(y) # pk(z)

+ pk(y) = c1 and pk(z) = c3 and obviously y # z.

If we consider algebras without negation, the two notions o f irredundancy are equivalent and th e above characterization of irredundant algebras remains true.

0

165

CHAPTER 4 CONNECTIONS WITH OTHER CLASSES OF LATTICES

Post algebras and Heyting algebras are t w o other important classes of lattices studied i n algebraic logic. In this chapter we investigate i n some detail the fact th a t the following classes of lattices f o r m an increasing chain: Post algebras, axled Lukasiewicz-Moisil algebras, Lukasiewicz-Moisil algebras, Heyting algebras

($51-3); i n subsequent chapters various properties of

LM-algebras will be used i n order to obtain significant properties of Post algebras. In 54 we study Lukasiewicz-Moisil algebras as pseudocomplemented lattices, while the last two sections deal w i th LM-algebras t h a t are also complete lattices a nd m-cornplete lattices, respectively.

$1. Post algebras In this section Post algebras and 8-valued Post algebras are defined and shown to coincide w i th centred n-valued and centred &valued LM-algebras, respectively. See also Wade

(19451.

There are several equivalent definitions of Post algebras, e.g.:

1.1. Definition (Epstein [1960],Traczyk [1963]). A n n-valued Post algebra or simply a Post algebra is an algebra ( L ,A, V, O,l,cl, ...,c,-z) of type (2,2,0,0, {O}i=l, ...,n - ~ )such that:

(Pl)

( L ,A, V, 0 , l ) E DO1 ;

(P3)

every

(1.1)

2

2

E L can be uniquely represented i n t h e form

= (b1 A CI) V ( b z A CZ) V

where bi E

... V (b,-i

A

,

~ ~ - 1 )

C(L)(i = 1,...,n - 1) and b1 1 bz 2 ... 2 b,-l .

166

Connections with o t h e r classes of lattices

T h e elements cl, stants of

...,cn-l are referred to as t h e ascending chain of con-

L,while (1.1) is called the monotone representation of.

Here are t w o basic examples. a) T he n-valued Moisil algebra B[q of all increasing functions from I =

(1, ...,n-1} t o the Boolean algebra B (cf. Examples 3.1.3and 3.1.10.B), where we define

(1.2)

Ci(lc)

=

{

1,

ifkLn-i,

0,

ifk
i, k E {1, ...,n - 1). Then (Pl) and (P2) are verified. To check (P3) take f E B[q and set bi = pn-if. Then all b; E C(B[q), bl 2 ... >_ b,-l and and ~ ( k=) 0, for

(pn-if A c i ) ( k ) = f(n - i) A c i ( k ) = f(n-i)

=

i.e.

(0,

if k > n - i ifk
(1.1) holds.

For uniqueness note th a t

bi(k) ,

=

(0,

if

n-j

2 n-i ,

if n - j < n - i

,

b) T h e n-valued Moisil algebra D ( B ) o f all decreasing functions from I = (1, ...,n - l} to th e Boolean algebra B , where di = n - i (cf. Example

3.1.10.A and Theorem 3.1.18), where we define

Post algebras

(1.2')

167

1,

ifksi,

0,

ifk>i,

=

Ci(k)

and ~ ( k =) 0, for all

i, k E (1, ...,n - 1). The proof is similar t o the

above one. Note t h a t each of the above two examples can be translated into a kind

of "dual" language if one replaces the functions c; by di = c,-~. This remark and Proposition 1.2 below provide the starting point in obtaining a suitable more general concept of &valued Post algebra, t o be related t o d-valued LM-algebras.

1.2. Proposition. T h e following properties hold in B[q:

(i) B

C(B[q)= {f E B'

1 f is constant}.

(iii) Ifa; E C ( B [ q )and i I j

f= V

(ai A

+ ai

di) E B[q and

~i

I a, ( V i , j E I ) then there mists = cpif (Vi E I ) .

$I

Proof.

(i)

The isomorphism B 2 {f E B' I f is constant} is proved by x where x E B and fz(i) = x (Vi E I),while for every f E B'

f

E C ( B " I )H

*f H

f

E B"1 &

f

E B"I &

fi = fj

is constant

.

(ii) For every i,j, k E I we have

cpif

=f

(Vi E I)

(Vj E I ) (Vi E I) H

($

H fit

168

Connections with other classes of lattices

hence cpj(cp; f A d ; ) ( k ) 5 f j = ('pjf)(k). It follows that

'pif

(Vi E I). If g E B[q fulfils 'pif A d; 5 g (Vi E I ) then f

A d;

5f

5 g follows

from

(iii) Define f i = ai (Vi E I ) . Then f E B[q and ((pif)(j) = aj (Vi,j E I ) . Finally, apply (ii). 0 Note t h a t for I = (1, ...,n - 1) we have o = 1, while 1is n - 1 and the algebras B[q dealt with in Proposition 1.2 become n-valued Post algebras if we take c, = 0 (i.e. co(i) = 0 (Vi E I))and ci = d,-i (i = 1,...,n - 1). In view of this remark we see that each of t h e next two definitions generalizes the concept of n-valued Post algebra. 1.3. Definition (Traczyk [1967]). A7 v, 0,1, { V i } i ~ Z { ,d i } i ~ I of ) type (2,2,0,0, { 1 } i c I , { o } i e I ) is An algebra (L, called a &valued generalized Post-algebra with descending c h a i n of con-

stants

( d i ) i E provided ~

it fulfils

,

(Tl)

(L,A, V,O,

(T2)

d, = 1and i 5 j

(T3)

via: E

(T5)

if ai E C(L)and i 5 j =+ a; 5

1) E DO1

+ d, 5 di (Vi,j E I) ,

C(L)and i Ij =+ 'pi 5 'p, (Vx E L ) ( V i , j E I) ,

aj

(Vi, j E I),

Post algebras

169

then there exists z =

V

(u; A d i ) E L and a; = cpiz

(Vi E I) .

icI

1.4. Definition (Georgescu [1971d]).

A &valued LM-algebra L is called a &valued Post algebra provided L E

(C(L ) ) The main result of this section is the equivalence between Definitions 1.3, 1.4 and Definition 1.5 below (cf. Theorem 1.7). 1.5. Definition. A &valued LM-algebra L is said to be centred if there is a family of elements

d; E L (Vi E I),called centres, such that:

(Cl)

(C2)

cpidj

=

I

1,

i f j s i

( V i , j E I) ; 0,

ifj>i

if ui E C(L)and i

Ij=+ ui 5 uj

then there exists z =

V

(ui

( V i , j E I)

A d i ) E L and t h e family

{ui};€~

$I

with these properties is unique. Note again that the Now for every

(1.3)

Pr, : L

di's in (C2) satisfy i 5 j =+ d j 5 di.

L E LMIJ define +

(C(L))", P~(z)(i) =cpg

(Vz E L ) (Vi E I) .

1.6. Proposition (Georgescu and Vraciu [1970], Georgescu [1971d]). For every L E LM6 the following conditions are equivalent: (i)

L is a 6-valued Post algebra;

(ii) PL is a n LM-isomorphism;

Connections with other classes of lattices

170

(iii) f o r every f E

[rl (c(L)) there

is z E

L such that 'pix = fi ( ~ Ei I);

(iv) for every A E L M 9 and every Boolean h o m o m o r p h i s m h : C ( A ) t

C ( L ) there is a n L M 9 - h o m o m o r p h i s m g

:

A

t

L such that

g I C ( A ) = h. Proof.

+ (iii):

C(L). Take an isomorphism a : L t B[q. From f i E C ( L ) and i 5 j j f i 5 fj ( V i , j E I) it follows that a f i E C(B[q)and i 5 j + a f i 5 a f j ( V i , j E I), hence Proposition (i)

Let f E B[q where B =

1.2 (iii) implies the existence of g E B[qsuch t h a t a f i = ' p i g (Vi E I ) . Now

fi = a-"p;g = ' p i ( a - ' g ) = PL(a-'g)(i) (Vz E I ) , t h a t is f = PL(a-'g). (ii) H (iii): Pt is always an injective LM-homomorphism, therefore it is an ismorphism if and only if it is surjective.

(iii)

+ (iv):

Let h be as in (iv) and

B = C ( L ) . In view of (iii) and

the previous equivalence, PL is an isomorphism. Therefore, given z E

A, from h'p;x E B we infer PLh'pix E C ( B [ q )hence there exists w = V (PLh'pix A d i ) E B[q and PLh'p;x = 'piw (Vi E I ) , again by PropicI

osition 1.2 (iii). Define g x =

gx = V

(PL)-'(w). Then g

:

A

t

L

and

(h'piz A c ; ) , where ci = (P~)-'(di), by Proposition 1.23. Note

iEI

also t h a t ' p i g s = (P~)-'(viw) = h'pix. Clearly g is a join homomorphism

5 g ( x ) A g ( y ) . If t 5 gx and t 5 g y then ' p i t 5 ' p i g s = hvix and 'pit 5 hviy, hence yit 5 h'pix A h'p;y = h'pi(s A y ) = 'pig(zA y ) for all i E I , therefore t 5 g(x A y). This proves g ( x A y ) = g ( z ) A g ( y ) .Further g'pjx = V (h'pjxAci)= h'pjz preserving 0 and 1. Thus g is increasing, hence g(x A y )

iEI

because c, = (PL)-'(~-) = 1. for all

So g I C ( A ) = h and g ( p g = h'pix = 'pigx

i E I.

+ (i): Apply (iv) for A = B[qwhere B = C ( L ) ,and h

C(B[q)+ B , the isomorphism in Proposition 1.2 (i). The homomorphism g : B[q + L (iv)

:

is injective because gx = g y implies hqiz = g'piz = vigx = ' p i g y = gyiy =

h'piy (Vi E I ) , hence 'pix = (piy (Vi E I ) , i.e. z = y. To prove surjectivity

Post algebras

L.

171

B

E C ( B [ q ) for , all i E I . Since the family a; = h-'p;x is increasing, a; = p;f (Vi E I) for some f E B[A, by Proposition 1.2 (iii). Now 'pix = ha; = h'pif = 'pihf 0 (Vi E I ) , therefore x = h f . take x E

Then 'pix E

hence 'pix = ha* for some a;

1.7. Theorem.

T h e following conditions are equivalent:

(i) L is a 19-valued Post algebra; (ii) L is a #-valued generalized Post algebra;

(iii) L is a centred 19-valued LM-algebra. Proof.

(i) j (ii): By Proposition 1.2. (ii) j (iii): First we prove in turn

(1.7)

pi(X A y ) =

(1.8)

x E C(L)

But cpixAdi

A ( ~ i y (Vi E I )

+ 'pix = x

5x

(Vi E I) .

and QiyAd; 5 y by (T4), hence ('p;xV'p,y)Ad;5 x V y

(Vi E I ) . If (cpix V ' p i y ) A d; 5 t ( V i E I ) , then x 5 t by (T4) and similarly y 5 t , therefore x V y 5 t , completing the proof of (1.4). For (1.5) we use (T3)and apply (T5) t o ai = 'pix V 'piy, yielding a; = 'pi(" V y) by (1.4). Further apply

(T5)t o obtain the element z

=

V ('pix A ' p i y ) A di;

;€I

from (T4) we see that

z 5 x and z 5 y . If t 5 x and t 5 y then

pit 5 p i x and pit 5 'p;y because 'pi is isotone by (1.5) and therefore pit A d; 5 'pix A p ; y A di 5 z ( V i E I ) , hence t 5 z by (T4). Th'IS proves

Connections with other classes of lattices

172

z = x A y , i.e. (1.6). Now we get (1.7) from (1.6) as (1.5) from (1.4). To

obtain (1.8) take x E C(L)and apply (T5) to ai = x (Vi E I ) , which yields

V (x A di) and x = (piy (Vi E I ) ; but (T2) implies y = x

y =

A do = x .

i€I

Now L is a d-pre-algebra by Remark 3.2.5, in view of (Tl), (1.5) and (1.7), (T3), (1.8) applied to ( p j x and again (T3). If cpiz = (piy (Vi E I ) then (1.4) and (T4) imply x = x V y = y , therefore L is a &valued LM-algebra. Further take j E I and set ai = 1 for i 2 j , ai = 0 for i < j . Then ai E C ( L ) (Vi E I ) and dj =

V

(ui A d i ) by (T2), hence (T5) implies

ieI

ai = (pidj (Vi E I ) , which is (Cl). Finally (C2) follows readily from (T5).

+

(iii) (i): We check condition (iii) from Proposition 1.6. Let f E (C(L))['. In view of (C2) applied to ai = f i (Vi E I ) there is x E L such that x = V (fi A d i ) and if we prove x = V ( ( p i x A d i ) it will i€I

follow that

i€I

fi = (pix (Vi E I ) , as desired. But

(pj(cpix

A d ; ) = (pix A (pjdi

5 j and 0 otherwise, by (Cl). Therefore (pj((p;xA d i ) 5 ( p j x ( V j E I ) , hence (pix A di 5 x (Vi E I ) . If (pix A di 5 t (Vi E I ) then (pix = (pix A (pidi 5 (pit (Vi E I ) , hence x 5 t . 0 equals (pix if i

1.8. Proposition. An n-valued LM-algebra L i s centred if and only if there exist ~ 1 ..., , c,-Z,C,-~ = 1 such that

0 , (1.9)

(picj

ifi+j
=

ifi+j>_n, in which case the centres are 1,

(1.10)

di = c,-i

(i = 1, ...,n - 1) .

Proof. Condition (1.9) coincides with (Cl), therefore it remains to show that (1.9) implies (C2), via (1.10). First we prove uniqueness. Suppose n-1 5

=

V

i=l

Then

(hi

A

di) where bi E C(L)( i = 1, ...,n - 1) and bl 2

... 2 bn-l,

Post dgebras

173

n-1

for all j E I . Finally we prove x =

V ('pn-ixAci).

Denote t h e right-hand

;=I

side by y; then q ~ j y= cpjx ( V j E I ) by ( l . l O ) ,

hence y = x.

0

From Theorem 1.7 and Proposition 1.8 we obtain

1.9. Corollary. L is a n n-valued Post algebra if and only if L is a n n-valued LM-algebra containing n - 1 elements cl, ...,c,-,, c,-~ = 1 satisfying (1.9), in which case c l , ...,c,-~ is the unique ascending chain of constants. 1.10. Corollary. L, is a n n-valued Post algebra. 1.11. Remark.

We have seen that in the case 19 = n axiom (C2) in Definition 1.5 follows from (Cl); this does not hold for arbitrary r9. Take e.g. I = the set of rationals from [0,1]. Then ' hL is a centred algebra and l e t L = {di)iEI U (0) c Lbq. Clearly L is an LM-algebra fulfilling (Cl). Take an irrational number p E ( 0 , l ) and define x p E L,[rl as follows: x p ( i ) = 0 or 1 according as i < p or i

> p.

Then

'pix,

is the constant function (cpix,)(j) = x p ( i ) , t h a t is

= 0 or 1 (= do) according as i < p or i > p . Therefore ' p i x , E C ( L ) for all i and moreover, i 5 j ' p i x , 5 'pjx,; however we shall show that 'pix,

+

V

&I

('p;~,

A d;) does not exist in

L. For every j

E I:

Connections with other classes of lattices

174

@

(Vi,k E I )

(i>pandi
therefore the set of upper bounds of { c p i z P A d j } i Ein ~ L is { d j I j

< p } , which

has no least element. W e are also in a position t o generalize Examples a and b.

1.12. Corollary. For e v e r y 29 a n d e v e r y Boolean abgebra B:

(ii) for a n y set X , ( B [ ~ )and x ( D ( B ) ) are ~ d-valued P o s t algebras. Proof.

(i)

For any

f E B[q,

f

E

C(B"I)@ cpif = f

(Vk,i E I ) H f(i) = f ( k ) For

(Vi E I)

(Vi,k E I ) H f is constant .

D ( B ) the proof is similar.

(ii) Note first that

(1.12) for any

* (cpif)(k)= f ( k )

C ( L X ) = (C($

L E LM19 or L E LMNO, because

Post algebras

175

F E Lx. Now we use (1.12) and (I t o check condition (iii) [q in Proposition 1.6. Given 9 E ( ( C ( B [ q ) )) , we must find F E ( B [ q ) xsuch that y ; F = Q ( i ) (Vi E I ) . For any i E I and s E X , 9 ( i ) ( z )E C (B [q)is a constant function, so that we can set F ( z ) ( i )= 9 ( i ) ( z ) ( k )E B (Vlc E I ) . This defines a function F ( s ) : I --$ B (Vs E X ) . If i , j E I and i 5 j then Q ( i )5 Q(j), hence 9 ( i ) ( z )I 9 ( j ) ( s ) ,therefore F ( s ) ( i )5 F ( z ) ( j ) .This proves F ( s ) E B[q (Vx E X ) and we have thus defined F E ( B [ q ) x .Besides y ; F = Q ( i )follows

2

for any

from

The following concluding remarks will be needed in the subsequent chapters.

1.13. Remark. It follows readily from the equational definition tion

(Cl) of centres (cf.

Defini-

1.5) that any LMd-homomorphism between two d-valued Post algebras

preserves the centres, i.e. it is what one would naturally call a Post-algebra

homomorphism.

1.14. Remark. The class Pn of n-valued Post algebras is equational. This follows from Theorem 3.2.4and again the equational definition (Cl) o f centres.

1.15. Remark. Let f E (C(L))[',where L is a d-valued LM-algebra. Then P ~ ( f ( i ) = ) cpif (Vi E I ) , because f(i) E C(L),hence P~(f(i))(j) = cpj(f(i)) = f ( i ) = ( y i f ) ( j )(Vj E I). Therefore (Cl) implies

f = i vE I (PL(f(4)Ad;)

*

176

Connections with other classes of lat tices

The algebra B[q appears in Moisil [1940], [1968] and Traczyk [1967]. The concept of a Post algebra, due t o Rosenbloom [1942], was developed by Epstein [1960] and Traczyk [1963]. The equivalence between n-valued Post algebras and centred n-valued LM-algebras was proved by Cignoli [1969a]. The monotone representation was given by Moisil [1941b] in t h e three-valued case, by Cignoli I19691 in the case of centred n-valued LM-algebras and independently by Georgescu and Vraciu [1970]. In the next section some properties of n-valued Post algebras will be recaptured within the more general framework of axled algebras which, in their turn, are particular cases of Moisil algebras. Other properties of Post algebras will be established in th e subsequent chapters: see also Balbes and Dwinger [1974]. For other generalizations of Post algebras see Katrina’k and Mitschke [1972].

Axled Lukasiewicz-Moisil algebras

177

$2. Axled Lukasiewicz-Moisil algebras

The following concept was introduced by Moisil [1941b] in t h e case of t hree-valued algebras:

2.1. Definition. Let n 2 3. An n-valued LM-algebra L is said t o be azled provided there exist

L , called the azes of L , such that for every i = 1, ...,n - 1 and every j = 1, ...,n - 2:

a1,az, ...,an-2 E

2.2. Example. Every Boolean algebra is an axled n-valued LM-algebra with axes aj = 0

( j = 1,...,n - 2).

2.3. Example. Every n-valued Post algebra is an n-valued axled LM-algebra, the axes being the centres aj = c j ( j = 1, ...,n

- 2);

cf. Corollary 1.9 and Proposition 1.8.

In the first part of this section we establish several properties of axled algebras. In view of Example 2.3 all of these properties are in particular valid for Post algebras. Then we study in more detail the relationship between Post algebras and axled algebras, including a characterization of Post algebras within axled algebras (Theorem 2.17) and a representation theorem for the latter (Theorem 2.26).

Unless otherwise stated, in this section L will stand for a n axled nvalued LM-algebra with axes u ~..., , an-2. 2.4. Proposition.

The following properties are valid (i) cp;aj = Vhak provided i

+j

2 n and h + k 2 n.

Connections with other classes of lattices

178

(ii) al 5 a2 5

... 5 an-z.

(iii) If L is not a Boolean algebra then aj j = 1,...,n - 3. (iv) L is n o t a Boolean algebra

fj?'

if and only if

C ( L ) and aj < aj+l for

aj

< aj+l f o r

some

j E

{ 1,...,n - 3). Proof.

(i)

It follows from (2.1) and (2.2) that for every j, k 5 n izn-j:

-2

and every

In particular 'p;aj 2 'pn-laj, hence 'piaj = 'pn-laj and similarly =

and similarly

for h 2 n-k. But (2.3) implies also 'pn-laj 2 'pn-lak 2 cpn-laj, therefore 'piaj = ' p h a k .

'pn-lak

(ii) As 'piaj = 0 (i = 1, ...,n-j-1) and 'piaj = 'piaj+l (i = n-j, by (i), we infer aj 5 aj+l ( j = 1,...,n - 2).

...,n-1)

(hi) Take x E L - C ( L ) . Then 'pix < 'pn-lx, hence from (2.2) we get

hence aj !$ C ( L ) ,and also

by (2.1) and (i), hence aj

< ajtl

via (ii).

(iv) If L is a Boolean algebra then 'pix = x (Vx E L )

hence aj = 0 ( j = 1,..., n converse holds by (iii).

- 2)

(Vi= 1,...,n - 1)

by (2.1), therefore aj = aj+l. The 0

Axled Lukasiewicz-Moisil algebras

179

2.5. Theorem. T h e axes are uniquely determined. Proof. Let a;, ..., be axes and prove piaj = piaj (Vi) (Vj). But for i = 1, ...,n - j - 1 we have via, = 0 = 'pa'. ' 3 ' while for i = n - j , ..., n - 1 we get from (i), (2.2) and (2.1) 'piaj = 'pn-jaj = cpn-jaj

v 'plug 2 cpn-la>= via',3

and similarly for the converse inequality.

0

Now we introduce a definition which is justified by the theorem following

it. 2.6. Definition.

A sequence (bl, ...,bn-1) E D ( C ( L ) ) (i.e., bi E C ( L ) (i = 1,...,n - 1) and b1 2 bz 2 ... 2 cf. Example 3.1.19) is called a m o n o t o n e representat i o n of the element x E L provided

sometimes we will identify

(bl, ...,bn-l) with (2.4).

2.7. Theorem (Boicescu [1986]), T h e set MR(x) of all m o n o t o n e representations of a n element x E L consists of all chains (bl, ...,bn-l) E D ( C ( L ) ) such that

Proof. We are going to make use of (2.1), (2.4)' Proposition 2.4 (i), (1.2.18')), (1.2.19'), (1.2.20') and (1.1.16"). First we note that (2.2) implies (Pn-lal

(2.6)

V

(PIX -

1 ( ~ ~ - 1 hence 5, (Pn-lal A ( P I X A v n - l x

( ~ ~ - 1 A ~&x 1

A cpix = 0

,

= 0, therefore

(i = 1,...,n - 1) .

Now we take a chain (bl, ..., bn-1) E D ( C ( L ) ) and write (2.4) in several equivalent forms:

Connections with other classes of lattices

180

(2.4) H

Q ~ X= (b1

A

vial)

V

... V (bn-2

A Q;u,-~)V bn-l

(i = 1,...,n - 1) 913

= bn-1

Q ~ X= (bn-j

A

9jan-j)

V

... V (bn-2

A

~ j a n - 2 )V bn-1

( j = 2, ...,n - 1) bn-1

= QIX

(j=2,. 'pjx = ~

bn-1

~ A (bn+ - V~... V bn-2) a V ~ (plz

( j = 2, ..., n - 1)

=Q ~ X

A v1x = 0 QjX

..,n- 1)

= (bn-j A

(j=2, ...,n-1)

Qn-lal)

V

912

( j = 2, ...,n - 1)

( j = 2, ...,n - 1)

Axled Lukasiewicz-Moisil algebras

&-I= bj

A

181

~ P I X

Vn-jX

= b j A Cpn-jx A yn-lal = 0

( j = 1,...,n - 2)

and the latter statement is equivalent to (2.5).

0

2.8. Corollary. Every x E L is represented an the f o r m

and if L is a Post algebra then (2.7) is the unique monotone representation of x . Proof. ( ( ~ ~ - 1. . .5, c p, 1 x) E D ( C ( L ) ) and fulfils (2.5). If L is a Post algebra = 0 by then (2.5) reduces to bj = cpn-jx ( j = 1, ...,n - 1) because Corollary 1.9. 0 2.9. Proposition. Let S be an LM-subalgebra of L . Then: (i)

If aj E S ( j - 1,...,n - 2 ) then Qx E L ,

(ii) S = L if and only if C(S) = C ( L ) and S is axled. Proof. (i)

Follows from the representation (2.7).

are axes of S then a$ = aj for all j , by the proof of (ii) If a:, Theorem 2.5 applied to a, and a:, then to a: and 'pn-laj. Now S = L follows from (i). 0

182

Connections with other classes of lattices

Proposition 2.9 was proved for three-valued Moisil algebras by L. Monteiro [1974], who also remarked that the converse of Proposition 2.9 (i) does not hold in general, although if is valid for n-valued Post algebras. The next properties involve negation. 2.10. Theorem.

Every axled n-valued LM-algebra is a

Moisil n-algebra.

Comment.

In particular every n-valued Post algebra is a Moisil n-algebra. Proof. Define N : L -+ L by

Noticing that (2.6) for

i = n - 1 yields

we see that (2.9) implies

for each j = 2,

...,n - 1.

Since on the other hand (2.9) implies readily

cplNx = (Pn-12, we have obtained (2.10)

cpiN~ = &,-ix

(i

= 1, ...,n - 1) .

From (2.10) it follows in turn

Axled Lukasiewicz-Moisif algebras for all

i, hence N N x = x. T h e

183

proof of N x V N y = N ( x A y) is easy.

0

Now we wish to describe all the representations similar to (2.7). First let us specify what we mean.

2.11. Definition. Let L E LMn. For each positive integer rn, let (2.11)

L["l = ((2%

...,xm) E L"

1x1

5 ... 5 2,)

L" consisting of all ascending chains and let AB denote the set o f ascending chains ( e l ,..., en-2) E L[n-21such t h a t for every x E L, denote the sublattice o f

(2.12)

x = ('pn-1x A e l ) V

... V ( ' p 2 ~A en-2) V 9 1 "

;

call AB t he set of ascending bases of L . 2.12. Lemma.

If ( e l , ...,en-2) is an ascending basis of an algebra L E M n , then so is (Nen-2, ...,N e l ) . Proof. Note that Nen-2

5 ... 5 N e l .

Then apply (2.12) to N x :

hence

x = ('p1x V N e l ) A

... A (cpn-2x V Nen-2) A 'pn-lx

=

V ( N e l A (92sV Ne2) A ... A ( ~ p n - z xV Nen-2) A Vn-1x) = = cpix V ( N e l A 'p2x) V ( N e 2A ('p3x V N e 3 ) A

... A ( ( ~ ~ -V2 2Nen-2) A ( P ~ - ~ X = )

...

Connections with other classes of lattices

184

... A ( q n - 2 x V Nen-2)A ( P ~ - ~ X=) ...

... = ( P I X V (Nelx A p2x) V ... V (Nen-4 A pn-35) V V (Nen-3 A (vn-2~V Nen-2) A

( ~ ~ - 1 2= )

2.13. Corollary

If (al, ...,an-2) are azes of L, t h e n f o r every x E L:

Proof. From Theorem 2.7 and Lemma 2.12. The next theorem generalizes a result of Abad and

0

L. Monteiro

[1984]

for n = 3.

(19861). AB is a n interval of L"+2], namely 2.14. Theorem (Boicescu

Proof. (a1,...,an-2)

E AB and (Na,-n, ...,Nul) E AB by Theorem 2.7

and Corollary 2.13, respectively. Now take an element (el, ...,en-2)

E AB and use the representations

(2.13) of aj and Nun-j-l; it follows that

Axled Lukasiewicz-Moisil algebras therefore

aj

185

5 ej 5 Nan-jwl ( j = 1, ...,n - 2 ) .

Conversely, take

(el, ...,en-2) E L"+2] satisfying the above inequalities. Then

vn-jx A aj 2 vn-jx A ej 2

vn-jx

A

Nan-j-1

for j = 1, ...,n - 2 therefore (2.7) and (2.13) imply

n-2

hence x = (PIX V

V

('pn-jx A

e j ) , proving that (el, ...,en-2) E AB.

0

j=1

2.15. Proposition.

The axes form the unique ascending basis (el, ..., en-2) E AB such that 'pie, = 0 whenever i j < n.

+

Proof. If i

+j

< n then 'piaj = 0 = cpiej, while for i + j 2 n we

Viaj = pn-jaj V

therefore

aj

2 ej for all j .

9iej

have

2 pn-lej 2 piej ,

The converse inequality holds by Theorem 2.14.0

W e now characterize axled algebras within LM-algebras and Post algebras within Moisil algebras.

2.16. Thexem. The following conditions are equivalent for an n-valued LM-algebra L and n - 2 elements al, ...,an-2 E L;

(i) L is an axled algebra with axes al, ..., an-2; (ii) every x E L is represented in the form (2.7) and there is a E C ( L ) such that

Connections with other classes of lat tices

186

(2.15)

piaj =

{

a ,

ifi+jzn,

0,

ifi+j
Proof.

(i) + (ii): By Proposition 2.4 (i) and Theorem 2.8. (ii) + (i): From (2.15) we get (2.1), while (2.7) and (2.15) imply

that is (2.2). 2.17. Theorem. T h e following conditions are equivalent f o r a n n-valued M O i d algebra L and a n ascending chain a l , ...,an-2 E L"+21:

(i) L is a Post algebra with ascending chain of constants cj = aj ( j = 1, ...,12 - 2 ) , c,-1 = 1; (ii) L is a n asked algebra with azes al, ...,czn-2 1, ...,n - 2 ) ;

( j = 1, ...,n - 2 ) and y;aj = 0 whenever i

(iii) a, = Proof. (i)

and aj = Na,-j_l ( j =

+ (ii):

+j

< n.

First use Example 2.3. Then note that cpiNan-j-l = piaj

because

#

n - i + n - j - 1< n H i

+j

2: n HV'piaj = 1 .

(ii) + (iii): Trivial. (iii) + (i): In view of Corollary 1.9 and Proposition 1.8 it remains to prove that Qiaj = 1for i j 2 n, which is true because

+

Axled Lukasiewicz-Moisil algebras

187

2.18. Corollary. A n n-valued axled algebra L is a Post algebra if and only if it has a unique ascending basis, in which case the latter is the unique chain of constants of L. (j= 1,...,n-2)

Proof. If L has a unique ascending basis then a, = by Theorem 2.14, therefore

L is

a Post algebra by Theorem 2.17. Conver-

sely, Theorems 2.17 and 2.14 imply that in a Post algebra L the chain of constants coincides with the axes and is the unique ascending basis of

L.0

2.19. Corollary (Moisil [1940]). A n element c of a three-vahed Moi3il algebra L is the centre of L if and only if c = N c . Proof. Necessity follows from Theorem 2.17. Conversely, if c = N c then

cp2c = cp2cVc = cp2cVNc = 1by (3.3.9), hence 'plc = cplNc = N p z c = 0 . 0

2.20. Remark. The converse of Proposition 2.15 is not valid for n

>

4. Take e.g. the

subalgebra LL = L, - { c ~ , c , - ~ }of the canonical algebra

L, (cf. Example 3.1.20 and Ncz = ~ ~ - 3 )In. view of Corollaries 1.10 and 1.9 Ln is a Post algebra with ascending basis (q,..., c,-~). Take an arbitrary x E LL. Note that ( ~ n - 2=~ 9 n - 3 ~and 'p3x = cpzx, hence cpn-2x A c2 I (P,-~xA c3 and cp3x A cn-3 I p2x A C n - 2 , therefore the representation (2.12) of x in L, reduces t o (2.16)

2

= V {Vn-jX A C j I j E (1, ...,n - 1) - ( 2 , n - 3))

Further set ci = c l ,

C L - ~= c,-~ and c$ = cj otherwise.

Then

. ( P ~ - ~A X

ci =

'pn-3xAc1 I 'pn-3xAc3 L x and similarly ' ~ ~ x A Ic x~, so - ~that from (2.16) n-1

We

get

2

=

v

j=1

(Yn-jX A <), proving that ( c i , ...,c L - ~ ) is an ascending

LL. Moreover, since ci 5 cj for all j , it follows t h a t 'pic; 5 'picj = 0 whenever i + j < n. To prove uniqueness, suppose ( e l , ...,en-2) is an ascen'piej = 0. Then we see as above ding basis of L; such t h a t i + j < n basis of

188

Connections with other classes of lattices

that every 2 E Lk has a representation of the form (2.16) with respect to ej and since LL is an (n - 2)-valued Post algebra, it follows that ej = c j = ci V j E (1, ...,n -2) - (2, n -3}. Further i < 3 implies = 0 = vicn-4, then v3en-3 = v2en-3 = 0 = ( ~ 3 ~ n - 4while , (~ien-3= 1 = (p;cn-4 for i 2 4; it follows that en-3 = cn-4 = c ; - ~and similarly e2 = ck. Thus ej = ci for all j and so (c;, ...,c : - ~ ) is the unique ascending basis such that pic; = 0 whenever i j < n. However, Lk is not an axled algebra, otherwise ci, ...,C I , - ~ should be the axes of L; by Proposition 2.15 and this contradicts Proposition 2.4 (iii).

+

However, in the cases n = 3,4 we can characterize axled algebras and Post algebras in terms of ascending bases; cf. Theorems 2.21 and 2.22 below. 2.21. Theorem. Let L E M 3 .

(i) L is axled if and only if there exists b E L such that ‘plb = 0 and

in which case b is the axis of L. (ii) (Abad, L. Monteiro [1984]). L is a Post algebra if and only if there is a unique b E L such that (2.17) holds, in which case b is the centre of L.

Proof.

(i) This is Theorem 2.16 for n = 3, since (2.15) reduces to v2al = a & cplal = 0, which is equivalent to p1al = 0 because v2al E C(L) for any at. (ii) Necessity and the fact that b is the centre are the content of Corollary 2.18 for n = 3. Conversely, if (2.17) holds for a unique b, then b = N b , by Lemma 2.12 and by Corollary 2.19 b is the centre of L. 0

Axled Lukasiewicz-Moisil algebras

189

2.22. Theorem (Boicescu [1986]).

Let L E M4. Then L is azled (a Post algebra) if and only if it has an ascending basis (a unique ascending basis). Comment. According t o Definition 2.11, ( b l , bz) is an ascending basis provided bl and for every x E

5 b2

L:

a) We prove the statement relative t o axled algebras.

al) Necessity follows from Theorem 2.14 or Proposition 2.15.

all) Now we suppose ( b l ,b,) is an ascending basis and prove t h a t

(2.19)

a1 = bl A Nbz

,

a2 =

(az A N b l ) A v3(blV N b z )

are axes, using Theorem 2.16.

all.1) First we note that cpiaj = 0 for i

vzbz = 0 , hence cplal = 0 and aiso

+j

< 4,

because yzal = vzblA

a11.2) Next we prove that

which needs a few preliminaries. Set

bz A Nbl = a and note that from

N v l a = v 3 N a = y3NbZ V v3bl we obtain a2 = a A N v l a . Then Lemma 2.12 implies

and from (2.18) and (2.21) we infer

Connections with other classes of lattices

190

therefore we obtain in turn

(2.23)

$922

I

$912 VP1a

,

so that denoting the right side of (2.20) by y and taking into account the

previous results we deduce (ply = (pix,

so that y = 2 , that is (2.20) holds.

a11.3) It remains to show that piuj is a constant for i + j 2 4. T h e representation (2.20) of a2 yields a2 = al V ( v 2 a 2A a 2 ) , hence

Axled Lukasiewicz-MoisiJ algebras

191

while (2.21) implies also (2.25)

I 91" v 92a2

92"

(VX E L)

by (2.25), therefore (2.26)

93al

5 92a2

holds by Remark 1.2.15. Now we obtain (2.27)

93a2

= 92a2

from (2.24) and (2.26). Then it follows from (2.20) (2.27) and (2.25) that

therefore v3x

59

3 ~V 1

p2x, which implies further

hence 93a2 = 9 3 ~ 1 . Using also (2.27) we obtain p3al = 93a2 = y2a2, as desired. b) The proof of the statement relative to Post algebras follows from Corollary

2.18, Theorem 1.22(a) and Theorem 2.17.

0

192

Connections with other classes of lattices

A similar result for the three-valued axled algebras was obtained by Abad and L. Monteiro [1984]. The definition of axled algebras being equational, it follows that every direct product A x B,where A is an n-valued Post algebra and B a Boolean algebra, is an n-axled algebra with axes ( d j , 0) ( j = 1, ...,n - 2) (cf. Example 2.2). We are going t o show that every axled n-valued algebra has the above form. This is a fundamental result proved by Moisil [1941b] in the three-valued case, using methods of ring theory. A lattice-theoretical proof was given by L. Monteiro [1974] and his idea will be used in the sequel. Our proof needs a few preliminaries. 2.23. Lemma.

LMS ( L E LMNS) and ( p ) [a,b] is a Jegment of L closed with respect to the endomorphism ( p i , then [a,b] becomes a 8-algebra

(a) If (a)L E

(8- algebra with negation) b y taking &'"'z = ('pix V a ) A b

(2.28)

(VZ E [a,b ] )

( b y taking (2.28) and

N["*b]z = ( N z V a) A b

(2.29)

(Vz E [ a ,b ] ) ) .

(b) In particular ( p ) holds if a, b E C ( L ) and a 5 b. Proof.

L E LMS one applies Example 1.2.25. If L E LMNS the properties of N[OvbI are also checked easily.

(a) If

(b) For a = Q;U

5 Q ; X 5 cp;b = b (Vz E [ a ,b ])(Vi E I ) .

I emma. If L E LMS (L E LMNS) and a E C ( L ) then 2.24.

(2.30)

f

:

L + (a]x [ u ) ,

f(z)= (z A a , x V a )

17

Axied Lukasiewicz-Moisil algebras

193

is a n isomorphism in L M 6 (in L M N 6 ) . Proof. It is easy t o check (and well known) that

f is an isomorphism in D01.

Besides,

f(cpix) = ('pi" A a, 'pix V a ) =

('pi(3

A a ) ,'pi(x V a ) ) = c p i f ( 2 )

~ ( N z= ) (NzA U , N XV a ) = N(z A a , x V a ) = N f ( z ) .

,

0

2.25. Lemma.

If L is a n azled n-valued algebra t h e n [cpn-lal)

Proof.

(cpn-lal]

is a P o s t algebra and

a Boozean algebra.

L E Mn by Theorem 2.10, hence Lemma 2.23 implies t h a t

('pn-lal]

and [vn-lal) are also Moisil algebras and their endomorphisms are the restrictions of

'pi.

5 ( ~ n - l ~ n -= 2 'pn-1~1,hence aj E ('po,-lal]

Proposition 2.4 yields an-2

for all j , and also via, = 'pn-lal whenever y;aj = 0 for

i

+j

< n.

Therefore

i+j 2

n. On the other hand

is a Post algebra by Corollary

1.9.

If 2 2 yn-lal then 'pix 2

Vn-lal

and since yn-lal V

'p1x

2

'pn-lzby

(2.2) it follows that 'plx 2 'pn-lx,therefore 'plx = 'pn-lx,which implies 'pix = vn-12 for all

quently

i. Thus

5

[yn-lal) = C(['pn-1al))

E C(['p,-lal))

by Proposition 3.1.5, conse-

is a Boolean algebra.

0

2.26. Theorem. Every axled n-valued LM-algebra L is of the f o r m L Z A x B, where A 6s a n n-ualzted Post algebra and B a Boolean algebra. Moreouer, this representation is unique u p t o a n isomorphism. Proof. The representation follows from Lemmas 2.24 and 2.25. Further let A

be an n-valued Post algebra and B a Boolean algebra, satisfying L 2 A x B

L (of A ) ; then (cj,O) are the axes of A x B; cf. Example 2.2. Now Corollary 1.9 implies 'pn-l(cl,O) = ( l , O ) , therefore A ((1,0)] = ('pn-lal]and similarly but otherwise arbitrary. Let a, (let c j ) denote the axes of

=

B

2 [(1,0)) = [qn-lal).

0

Connections with other classes of lattices

194

We are now in a position t o say more about the lattices MR(z) and AB from Theorems 2.7 and 2.14, respectively. For each poset L , let us introduce the notation (2.31)

LIP[= {(zl,...,z p )E LP I z1 2 x 2 2

... 2 z,} .

2.27. Proposition.

(i) For each z E L, MR(z) is a n ( n - 1)-ualued Post algebra, namely

MR(z) 2 [z V ( ~ ~ - l a l ) ] ".- ~ [

(2.32)

(ii) AB is a n ( n - 1)-valued Post algebra isomorphic to [ ( ~ , - ~ a ~ ) [ " - ~ ] . Proof. xVcpn-lal 2 ( ~ , + ~ and a l by Lemma 2.25, it follows that xVqn-lalE

c ([(Pn-lull) . By Lemma 2.23 we obtain cp;z V(Pn-lal= z v ( ~ ~ (i- =~1 , a...,~n - 1 ) . Further in view of Theorem 2.26 and Lemma 2.24 we identify L with ( ( ~ ~ - 1 ~ 1x1 [qn-lal)and each element z E L with (z A qn-lal,z V qn-lal). In particular each aj is identified with ( a j , qn-lal). Taking also into account Lemma 2.23 we obtain ~ j ( a n - j ,(~n-lal) =(

[0,V,-lall ~ j

an-j?

= (Pjan-j A (Pn-lal,PjcPn-lal

-"pn-lal,l1 vj (Pn-lal)

v (Pn-1a1) =

=

Axled Lukasiewicz-Moisil algebras

195

while t he greatest element is th e j o i n of t h e least element and t h e chain

consequently the greatest element is

Thus t he elements o f

where bl

[x V of

2 ... 2

vn-1al)In-2[.

[vn-1al),

MR(2) are precisely t h e chains of t h e form

bn-2

2x

V

vn-lul,

i.e. where (bl,bz ,...,bn-2) E

This proves (2.32). B u t [z V ( ~ ~ - is~ aasegment ~ )

hence a Boolean algebra by Example 1.2.25. Finally note

that if B is a Boolean algebra then

is a p v a l u e d Post algebra:

BlP-'[

the easy proof is similar t o th a t for B[q at the beginning of $1; cf. Example 3.1.19.

(ii) We apply Theorem 2.14 and th e above preliminaries. Th e least element of AB is the chain ( ( U ~ , ~ ~ - ~ ..., U ~( a )n -, 2 , p n - 1 a l ) ) while Lemma 2.23 implies

hence t he greatest element of AB is t h e chain

((Q, l),...,(an-2,1)).

Therefore th e elements of AB are precisely t h e chains of the form

((al,bl),..., (an -2,

bn-2)),

where

5

( ~ ~ - 1 ~ bl1

1. ... 5 bn-2,

i.e. where

196

Connections with other classes of lattices

(bi,

...,bn-2)

E ['pn-l~l)[n-21.This proves

AB

E ['p,-lal)[n-21.

But

Boolean algebra by Lemma 2.25, therefore AB is an ( n - 1)-valued Post algebra as explained a t the beginning of $1. 0 [cpn-lal) is a

2.28. Corollary (Abad and L. Monteiro [1984]). If rn = 3 t h e n AB = [.,Nu] i s a Boolean algebra, and a: = N x . Proof. AB E [ ~ , , - ~isaa) Boolean algebra; cf. Proposition 2.27 and Lemma 2.25. But AB is a Kleene algebra, and it follows that a: = Nx (Cignoli [19651). 0 2.29. Definition. Given a De Morgan algebra L , l e t I ( L ) denote the set of all intervals of L of the form [ x , N x ] , x 5 Nx. Such an interval is said to be Boolean provided it is a Boolean algebra and a: = Nx.The concept of Boolean interval, suggested by Corollary 2.28, is also due to Abad and L. Monteiro [1984]. 2.30. Proposition. Let L be a Kleene algebra ( i e . L E Mg and (3.1.36) holds). T h e n :

(i) ( I ( L ) C) , is a distributive lattice with greatest element [0,11, in which (2.33')

[x,N 4 A [Y,NYl = [. v Y, Na: A NYl

,

(2.33" )

[x,NxI v [Y,NYl = [xA Y, Ns v NYI

*

(ii) An interval [x,Nx]is Boolean if and only if it is the leas- element of

( W9 ,

Proof.

(ii) Let [ x , N s ] be a Boolean interval. If [y,Ny] C [ x , N s ] then x 5 y 5 Ny 5 Nx and

Axled Lukasiewicz-Moisil algebras

197

showing that y = y A N y = x. Thus [ x , N x ] is a minimal element of

the lattice I ( L ) , therefore it is the least element. Conversely, suppose [ x , N x ] is least element in I ( L ) . For every y E

[x,Nx] it follows that x 5 y 5 Nx,hence x 5 Ny 5 Nx,therefore [y A Ny, y V Ny] E [x,N z ] , consequently [y A Ny, y V Ny] = [x,Nx]. This shows that Ny = $ z ~ N " ] , i.e. [ x , N z ] is a Boolean algebra. 0 2.31. Corollary.

In a Kleene algebra there is a t most one Boolean interval. 2.32. Corollary.

L as a complete Kleene algebra interval.

+

I ( L ) is complete

there

as

a Boolean

Proof. The first implication follows from an obvious generalization of formulas (2.33), while the second implication is a consequence of Proposition

2.30 (ii). 2.33. Proposition.

Every axled algebra has

Proof. Suppose L aj

5 Na,,-j-l

a

Boolean interval.

E Mn is axled. Note that for each j

E (1, ..., n

- 2},

by Theorem 2.14.

If n is odd then for j = = n - j - 1we obtain aj 5 Naj and prove that I, = [uj,Naj] is a Boolean interval. For every x E Ij, we have also Nx E Ij, hence aj 5 x A Nx. Further for each k 5 j , from k + j < n we get (Pk(2 A N Z )

while from

'pl(x

A

5 'pj(S A Nx)= 'pjx A P j X

= O = 'pkaj

,

N z ) = y l x A (Pn-lx = 0 we see that if k > j then

Proposition 2.4 (i) and (2.2) imply

'pkaj = 'pkaj v therefore

aj

2 x A Nx

A

Nx)2 'pn-1("

A

Nx)2 'pk(Z

A Nz)

by the determination principle. It follows that aj =

x A Nx, hence N a j = Nx V x, so that Nx = %?I.

198

Connections with other classes of lattices

If n is even then for j = is completed as above.

we obtain aj

2.34. Theorem (Boicescu

If L E M3

OT

5 Nu? 5 Naj

and t h e proof 0

[1986]).

L E M4 the following conditions are equivalent:

(i) L 4s azled; (ii) L has a Boolean interval. When this is the case the Boolean interval is [al,Nul]. Proof.

(i) =+ (ii): By Proposition 2.33. (ii) + (i): If L E M3 and [ u , N a ] is its Boolean interval we are going to prove t h a t a is the axis of L. From cpla 5 y2a 5 p2Na = &a we obtain cpla = 0, that is (2.1) and it remains t o prove (2.2). Take x E L. Then y = (x V a ) A N u E [a, Nu], hence = N y by (2.34), that is y A Ny = a , therefore y["iN"]

If L E M4 and [al,Nul] is i t s Boolean interval, we define a2 = NalAp3al and prove t h a t al, a2 are th e axes of L. It is shown as above t h a t ~ 2 a = l 0 and cp3al V and

cp1x

2 cp3x

for every x E L. Further

cpla2

=

A p3al = 0

AxJed Lukasiewicz-MoisiJ algebras

199

2.35. Remark. Theorem 2.34 cannot be extended to n > 4. Thus e.g. it was shown in Remark 2.20 that the Moisil algebra Lk is not axled, however, LL has a Boolean interval, namely {c+l,cg = N q - 1 ) if n is even and { c =N ~ c y } if n is odd. 2.36. Corollary (A. Monteiro [1980], Boicescu [1986]). T h e following conditions are equivalent f o r L E M3 o r L E M4;

(i)

L is complete;

(ii) L is complete and azled; (iii) L is of t h e f o r m L r2 A x B, where A i s a complete n-valued P o s t algebra a n d B is a complete Boolean algebra. Comment . In particular every finite algebra L E M3 or L E M4 is axled (Moisil [1940]). Proof.

(ii): L has a Boolean interval by Corollary 2.32, therefore L is (i) axled by Theorem 2.34. (ii) + (iii): Follows easily from Theorem 2.26. (iii) + (i): Immediate. 0 2.37. Proposition. If L E Mn is azled t h e n [an-2) E C ( L ) . Proof. Define f : [an-2) ---f C ( L ) by f(x) = ylx. Clearly f is a surjective morphism in D01. To prove injectivity suppose 'plx = plx' for x,x' E [an-2) and note that for each i E (2, ...,n - 1)

Connections with other classes of lat tices

200

which implies x 2 x' by Remark 3.1.4; similarly x' 2 x, therefore x = 2'. Thus f is an isomorphism in DO1 and Remark 1.2.21 applied to f - l shows 0 that the isomorphism holds in B too. 2.38. Proposition. Suppose L E M3 OT L € M4. If L is axled t h e n [ N u l )

I(L).

Proof. Note that

and define f : [ N u l ) I ( L ) by f(z) = [ N x , z ] ,which makes sense in view of (2.35). For every [ z , N z ] E I ( L ) we have z 5 a1 5 Nu1 5 N z by Theorem 2.34, therefore N z E [ N u l ) and [ z ,N z ] = f ( N z ) ;this proves that f is surjective. It is readily seen from (2.35) that z 5 y u f(x ) 5 f(y) for z , y E [ N u l ) . Therefore Remark 1.1.25 and Proposition 1.1.23 imply that f is a lattice isomorphism; besides f ( N u 1 ) = [al, Null and f(1) = [0,1], i.e. I7 f is an isomorphism in D01. .--)

2.39. Theorem (Abad and L. Monteiro [1984]). Suppose L E M3 OT L E M4. If L is axled t h e n I ( L ) a's a B o o l e a n algebra. Proof. Inasmuch as a,-2 5 N u l , Proposition 2.38 and 2.37 show that I ( L ) is isomorphic to an interval of the Boolean algebra C ( L ) ,therefore I ( L ) is I7 a Boolean algebra itself by Example 1.2.25.

Heyting dgebras

201

$3. Heyting algebras Heyting algebras constitute one of the fundamental structures generated by mathematical logic. Therefore the problem of investigating the relationships between Lukasiewicz-Moisil algebras and Heyting algebras is a natural one. The fact that every Moisil algebra is a Heyting algebra was first proved by Moisil [1942c], [1963a] for three-valued algebras, then generalized t o the n-valued case (Moisil [1965]); see also Cignoli [1969a] and lturrioz [1977]. For every

L E LMn

l e t us introduce the following generalizations of the

residuation considered by Moisil [1965]:

3.1. Lemma.

In every n-valued LM-algebra

Proof. Let

zk

and

tk

denote the left and right side of (3.2) respectively.

Then

which proves (3.2) for

k - 1factors of

tk-1

k = 1. If (3.2)

by tk-l then

holds for

tk-1

=

k - 1 and we denote the first A

Pn-kX

and

Connections with other classes of lattices

202

Recall t hat if

2, y

are elements of a lattice L, the relative pseudocom-

pl rnent of x wi th respect to y, provided it exists, is the greatest element z such t hat zAx

5 y. A lattice L is said to be relatively pseudocomplemented

provided t he relative pseudocomplement for every x , y E L; then

=$

5

+ y of x w i t h respect to y exists

can be viewed as a binary operation on L. A

Heyting algebra is a relatively pseudocomplemented lattice w i t h 0.

Hey t ing algebras

203

The following properties are well known and easy t o prove. The definition of + reads as follows: z A x 5 y if and only if z 5 x + y. A first consequence is y I: 2 + y, which implies x A (x +- y) = x A y. Another consequence is that every Heyting algebra is a distributive lattice

(x A ( y V~ y2) = (x A y ~ V) (x A y2) because x A (y1 V y2) is an upper bound of x A y1 and x A y2 and if z is another upper bound then yh 5 x + z ( h = 1,2), hence y1 V y2 5 2 + z therefore x A (yl V y2) 5 z ) with 0 and 1 = x + t for any x. The basic definition also implies t h a t the inequality x 5 y is equivalent t o x + y = 1, therefore 0 + x = x 3 2 = x + 1 = 1. Note also that 1 + x = x. If L is totally ordered then y < x implies x + y = y. Cf. e.g. Balbes and Dwinger [1974]. 3.3. Theorem (Cignoli [1975]). If L E LMn then ( L ,+) defined by (3.1) (or (3.3)) is a Heyting algebra. Proof. First we use (3.2) t o compute (3.4)

qi(x =

+ Y)

((PiY V

= ( P ~ Yv Flx)A

*.*

v v i y ) A ... A (Fn-12 v pn-iy)) =

((~11

A ((Pig V Pix) A (Cpi+lz V vi+ly) A

It follows that

for

i

= 1, ..., n - 1, therefore

x A (x + y) 5 y. Finally if x A

z

5y

then

cpixAp;z 5 'p;y hence 'pixA(~izA@iy= 0 or equivalently 'piz 5 CpixVp;y for i = 1, ...,n-1, therefore using also (3.4) we obtain cpiz = VizA ...Acpn-lz 5 vi(x

* y) (i = 1,...,n - I), consequently z 5 x + y.

0

In particular every Post algebra is a Heyting algebra. This was first shown by Rousseau [1970] using a direct proof; cf. Corollary 3.8 (ii) below.

Connections with other classes of lattices

204

3.4. Corollary (Moisil [1965], Cignoli [1969a], lturrioz [1977]). Every n-valued Moisil algebra is a Heyting algebra in which

3.5. Corollary (Moisil [1963a], A. Monteiro (19801). Every three-valued Moisil algebra is a Heyting algebra in which

Proof. Apply (3.5) for n = 3 and use distributivity. Now we note that x cation 2 V y.

0

+ y is a good generalization of the Boolean impli-

3.6. Proposition.

If L E L M n then:

+ y = 5 V y; if y E C ( L ) then x + y = (Pn-lx V y.

(i) i f x E C(L)then x (ii)

Proof. If 2 E C(L)then (3.2) reduces to

x =+ y = y V ((3 V ‘ply) A ... A (2 V cp,-ly))

=

= yV2Vcp1y=yVa:,

while for y E C(L)we get x

+ y = y V (((PIX

V 9) A

3.7. Corollary. In every n-valued LM-algebra

... A (Cpn-lx V y ) ) = (pn-i2 V y .

Heyting algebras

205

Proof. This is a rewriting of (3.4) via the fact that in a Boolean algebra z

=zvt.

J t

0

3.8.Corollary.

(i) In every axled n-va2ued LM-algebra with azes al, ...,un-2, an-l = 1:

n-1

V

(ii) In every n-valued P o s t algebra, if x =

(xi A ci) and y =

i=l

n- 1

V (yi A ci) are m o n o t o n e representations t h e n

i= 1

is the m o n o t o n e representation of x ( j = 1, ...)n - 1).

+y

and x j

j

yj = Z j V yj

Proof.

(i) From Corollaries 2.8 and 3.7. (ii) From (i),Theorem 2.17, Corollary 2.8 and Proposition 3.6 (i).

3.9. ProDosition. Suppose L i s a n n-valued LM-pre-algebra and bra. T h e n L E L M n i f and only if (3.3) holds.

0

(L,=+) i s a A e y t i n g alge-

Proof. Necessity follows from Theorem 3.3 and the uniqueness of the relative pseudocomplement. Conversely, suppose (3.3) holds and take x , E~L

Connections with other classes of lattices

206 such that 'pix = 'piy

(i = 1, ...,n - 1).

x = x A 1 5 y and similarly y

Then x 3 y = y V 1 = 1 hence

5 x , therefore x

= y.

0

A similar result was obtained by Varlet [1969] for three-valued Moisil algebras.

3.10. Proposition. Let I = ( 1 , ...,n - 1). An algebra ( L ,A, V, 0, +,{ c p i } i E ~ , { & } i E ~ ) o f type (2,2,0,2, { l } i E 1 ,{ l } i E ~ as ) equivalent t o a n n-valued LM-algebra ( L , A , V , O , l , { ' p i } i ~ ~ , { ~ i } via i € ~ the ) transformations 1 = 0 +- 0 and (3.3) if and only i f the following condition and identities hold f o T i = 1, ...,n - 1: (3.10)

( L ,A, V, 0 , a )is a Heyting algebra ,

(3.11)

' p i ( x A y) = p i x A 'piy and ' p i ( x V y) = pix v 'piy

(3.12)

'plx V p l x = x

Proof.

+x

and y l x A

p1X =

,

o,

Necessity follows from Theorem 3.3, the fact t h a t x 3 x = Cplx V x =

j

x = 1,

1 by Proposition 3.6 (i) and by Corollary 3.7. Moreover, (3.3) holds by Proposition 3.9. Conversely, suppose (3.7) and (3.10)-(3.15) hold. Then (3.10) implies 1 =0 0 and L E D01, i.e. axiom (3.1.1) holds. Now (3.15) reads ' p l x + x = 1 i.e. ' p l x 5 x . In particular 'p10 = 0 therefore (3.14) implies that 'pi0 = ' p i ' p I O = 'p10 = 0 for each i, while 'pi1 = q i ( x =+ x ) = 1 by (3.7). Taking also into account (3.11) we see that axiom (3.1.2) holds as well. Further from (3.14) (3.13) and (3.12) we deduce then

(PIX

Heyting algebras

207

and similarly cpiz A Fix = 0, i.e. (3.1.3) is verified. Axiom (3.1.4) coincides with (3.14). It follows from (3.7) that

therefore (piz 5 'pjx whenever i 5 j , i.e. (3.1.5) is fulfilled. Finally suppose (pix = cpiy for all i. Then (3.7) implies (pl(z + y ) = 1 and since q l ( z + y) 5 z =+ y we have in fact x y = 1, that is z 5 y , similarly 0 y 5 z, therefore z = y. This proves (3.1.6). A similar result is due t o lturrioz [1977], who has shown that a Moisil n-valued algebra can be characterized as a s y m m e t r i c H e y t i n g algebra (i.e., a Heyting algebra that is also a De Morgan algebra) endowed with n - 1 unary operations 'pi satisfying (3.7), (3.14) and

(3.18)

c p i N ~= N(p,-ia:

,

V Npis = 1 .

(3.19)

A natural problem is that of characterizing LM-algebra within Heyting algebras exclusively in terms of the latter structure. 3.11. Def inition. A Heyting algebra satisfying the identity (3.20)

(z

+ y) V (y + z) = 1

is called a linear Heyting ulgebru (A. Monteiro; cf. L. Monteiro [1970a] or an L-algebra (Horn [1969]).

Moisil [1942c] proved that if L E M3 then L is a linear Heyting algebra. The logic corresponding to the l a t t e r algebras was studied by Moisil [1942c],

Connections with other classes of lattices

208

Dummet [1959] and Horn f19691. 3.12. Proposition. If L E LMn then L i s a linear Heyting algebra. Proof. From (3.1) we obtain

The fact that every LM-algebra is an L-algebra has important consequences.

3.13. Proposition.

In every n-valued LM-algebra:

(3.22)

+ (y V Z) = (x + y) V (Z + z ) , (Z A y) + z = (z + z ) V (y + z ) ,

(3.23)

x V y = ((z

(3.21)

x

=k- y)

+ y)

A ((y

=+

z)

+ Z) .

Proof. The above identities hold in every L-algebra, 6. L. Monteiro [1970a]. 0

3.14. Definition (A. Monteiro [1974]). A bounded distributive lattice L is fulZy normal provided for every Z, y E L

209

Heyting algebras there exist u, v E L such that

x A u = y A v = x A y and

u V v = 1.

3.15. Proposition.

If L E LMn t h e n L i s fully normal. Proof. Take u = z =+ y and

=y

=k-

x.

Proposition 3.15 can be generalized as follows.

3.16. Proposition. T h e following conditions are equivalent f o r a Heyting algebra L:

(i)

L is a h e a r Heyting algebra;

(ii) L is fully normal; (iii) A proper filter of L is p r i m e if and only if it includes a prime filter. Proof.

(i) =$ (ii): Same proof as for Proposition 3.15.

(ii) =+ (i): Take z,y E L. Then take u,w E L as in Definition 3.14. It follows t h a t x A u 5 y hence u 5 z + y and similarly w 5 y + z,therefore

1 5 (z =+ Y) v (Y =. 4. (ii) +$ (iii): Proved by A. Monteiro [1974].

0

3.17. Definition. A P-algebra is a double L-algebra, i.e. an L-algebra such that i t s dual is also an L-algebra. We denote by K the class of all P- algebras. P-algebras were introduced by Epstein and Horn [1974a] as generalizations of Post algebras; cf. Balbes and Dwinger [1974]. We are going t o relate them to LM-algebras.

3.18. Notation. For every L E LMn and z,y E L, set

Connections with other classes of lattices

210

3.19. Remark. In view of Proposition 3.12 and the duality principle, the n-valued LMalgebras are also P-algebras with respect to the above defined

% and g.

3.20. Proposition.

Suppose L E LMn. For every x,y E L, x properties hold:

(ii)

if z E C ( L ) a n d z A x

(iii) (x (iv)

x

S-

g

y) v (y y =y

3

5 y t h e n z _< x % y,

x) = 1,

2,

Proof.

(i)

For each

$ y E C ( L ) a n d the following

i = 1,...,n - 1,

Heyting algebra

211

(ii) The hypotheses imply z 5 x x y.

4

(iii) (x

%

y)

v (y %

+y

hence z =

x) = 9 1 ((x 3 y)

'p1z

v (y 3 x))

5

+

~ 1 ( x y) =

= p l l = 1.

(iv)-(vii) Obvious from (3.3) and (3.24)-(3.26).

0

As a matter of fact the above properties (i)-(vii)

hold in any P-algebra;

cf. Epstein and Horn [1974a].

3.21. Proposition (Moisil (1942~1). If L E Mn then

(i) x

e y =N(Nx +Ny),

(ii) z

y =N ( N ~

3

~ y ) .

Proof. Routine from (3.3) and (3.24)-(3.26) via (3.1.23) and (3.2.13). 3.22. Proposition (Cignoli and De Gallego [1981]).

Suppose L E Mn. (3.27)

NX

=%

If n

i s even then

x =NX

x

='p$x,

while i j n is odd then (3.28)

NX

S

x ='pyx

and

NX

x ='p-x.

Proof. From

N X =$ x

=

n-1

n-1

A

(@NxV C ~ ~ = X) A

i= 1

(cpn-ia:V'pi~)

i= 1

we see that if n is even then

... A ( ~ ~ - 1=2V E X while if n is odd then

CI

212

Connections with other classes of lattices

Finally since N x x = N N x % N x by Proposition 3.20 (iv), identities (3.27) imply that if n is even then N x % x = @iqNx = 'piqx, while 0 if n is odd then N x x = @ + N x = 99". 3.23. Corollarv.

(i) I f L E M5 then ( P I X = 1 $042

=o

%-

x , (p2x = N x

x,

(p32 = N

x =$ x,

5.

(ii) If L E M4 then (plx = 1 (p3x = o 5.

=%

x , 'p2x = N X

(iii) I f L E M3 then ' p l x = 1 N x 3 x.

3

x = NX

x =NX

3

2,

g

X , (p2x = 0

x =

As remarked by Cignoli and De Gallego [1981], Corollary 3.23 shows

that if n E {3,4,5} the structure of n-valued Moisil algebra is completely determined by the structure of De Morgan algebra. We will sharpen this result in Ch. 6, $2.

213

Pseudocomplement ed lattices

54. Pseudocomplemented lattices W e have seen th a t the n-valued LM-algebras can be made i n t o Heyting algebras. Although we cannot extend this property to &valued algebras, the latter turn o u t to be Stone algebras and this fact is also an efficient tool in the study of LM-algebras.

4.1. Definition. Let

L E D01. If x E L and there is x* E L such t h a t for every y E L

(4.1)

xAy=OHy<x*,

x and x is said to be p s e u d o complemented. L is called a pseudocomplemented lattice if every x € L is

then x* is termed th e pseudocomplement o f pseudocomplemented; if, moreover,

(4.2)

x* V x** = 1

for all x E L then

L

is known as a S t o n e algebra. W e also work w i t h the

dual concepts of dual pseudocomplement, denoted by

x+ (i.e., x V y = 1

xt 5 y ) , dually pseudocomplemented lattice and dual S t o n e algebra (i.e., x+ A xtt = 0). B y a double pseudocomplemented lattice (double S t o n e algebra) we mean a pseudocomplemented lattice (Stone algebra) which is also a dually pseudocomplemented lattice (double Stone algebra). A lattice is said to be relatively complemented provided every interval [u,b]is c o m plemented, i.e. if a 5 x b then there is y such t h a t x A y = u and x V y = b.

<

It is well known and easy t o prove th a t * is a decreasing operation which satisfies also (xV y)* = z*A y* and x*** = x*,while ** is a closure operator such that (x Ay)" = x**Ay**; dual properties hold for +. In a Stone algebra the supplementary properties ( x A y)* = x* V y* and (x V y)** = x** V y** also hold. A double Stone algebra satisfies the identity x*+ = x** (because x** = x**Al = x**A(x*+Vx*)= x**Ax*+hence x** x*+, while x*+ 5 x**

<

follows f rom x* V x** = 1). This implies further x*+* = x*++ = x**+ = x*, then x*

5 st (apply *

t o xt+

5

x**) and t h e dual identities. See Gratzer

Connections with other classes of lattices

214

[1971] and Balbes and Dwinger [1974] for the theory of pseudocomplemented lattices.

4.2. Theorem.

Every L E LM19 is a double Stone algebra. Proof. We define (4.3)

x* = CplX

,

x+ = pax

and check e.g. x* V x** = pls v cplx = 1,

4.3. Remark. If L E LMn then x* = x =+ 0 and x+ = x

+

1. If L E Mn then x* = Ncp,-lx and z+ = N'plx therefore x* 5 N x 5 x+; two out of the three dualities coincide if and only if L is a Boolean algebra. The three-valued algebras are characterized by a kind of determination principle for double Stone algebras. 4.4. Theorem (Moisil [1960], Varlet "681). T h e following conditions are equivalent for L E D01:

(i)

L E LM3;

(ii) L is a double Stone algebra satisfying (4.4)

vxvy (x* = y* & x+ = y+

* x = y) ;

(iii) L E M3; (iv) L i s a generalized three-valued Moisil algebra (cf. Definition 3.1.23).

Proof. (i) =+ (ii): By (4.3) and the determination principle. (ii) + (iii): We set

215

Pseudocomplemented lattices (4.5)

y2z = x**

and

Nx = (x A xt) V x*

and check the system S2 in Proposition 1.3.2. We compute in turn

therefore N N s = x. Further

( N z V Ny)* = (Nx)* A (Ny)'= xtt A y++ = (X A y)++ = = (N(XAY))*

and similarly

(NxV Ny)+ = ( N ( x A y))

+ , therefore Nx V N g = N ( x A y).

Since L E DO1 it follows that L is a De Morgan algebra, i.e. axiom (3.5.0) holds. The fact that 9 2 is a meet homomorphism is precisely axiom (5.3.1). The last two axioms are (5.3.8) and (5.3.9): cpzx A

Nx

= c**A ((x A

3')

V x*) = x** A x A xt = x A x+ =

= (x A x A xt) V (x A x*) = x A N z

(iii) =+ (iv)

=+ (i): Trivial.

,

0

The equivalence (ii) e (iii) was first proved by Varlet using the definition of three-valued algebras in terms of x* and z+, given by Moisil [1960]. Varlet's paper was in f a c t the first one establishing a connection between LM-algebras and double Stone algebras. Formula Nx = (xAx+) V x * is due t o Moisil [1942c], [1960]. See also Theorem 4.9 below.

Connections with other classes of lattices

216

The theory of pseudocomplemented lattices points out t h e role of the so-called regular and dense elements. Let us relate them t o LM-algebras. 4.5. Definition.

If L E DO1 is pseudocomplemented, th e elements of the sets

Rg(L) = {x E L I x** = z} ,

(4.6)

D s ( L ) = {x E L

I Z* = 0)

and said t o be regular and dense, respectively. If D s ( L ) = L - (0) we say that L is a dense lattice. Similarly, if L is a dually pseudocomplemented lattice we set

RS(L')= {x E L I$++

(4.7)

= x}

,

DSL = {x E L Ix+

= 1)

,

Many of the properties established in the sequel are in fact valid under more genera I hypotheses. 4.6. Proposition.

I f L E LMt9 then: (i) R g ( L ) = RS(L)= C ( L ) ;

(iii)

'Ds(LJJ = {x I v o x = 0} = {y

I

A cpoy y E L } and it is an ideal;

(iv) The meet subsemilattice of L generated b y Rg(L) U D s ( L ) is L . Proof.

(i)

x E R g ( L ) H x = 'plx H x E C(L).

(ii) and (iii): If x E D s ( L ) then cplx = 0 hence x = x V p l x ; conversely, y V Cply E D s ( L ) because (y V 'ply)* = ply A 'ply = 0. The rest is obvious. (iv) From x = cplx A

(x V pix) via (i) and (ii).

Pseudocomplemented lattices

217

4.7. Proposition. The following conditions are equivalent for L E LM19:

(i) L is dense; (ii) L is a subalgebra of Liq; Comment. Condition (i) says that the filter D s ( L ) is maximal; in Theorem 5.2.22 we will prove that the minimality of

D s ( L ) (= (1)) is equivalent t o L being a

Boolean algebra.

L is dense then for every z E C(L)- (0) we get z = y1z = 1, hence C ( L ) = L2 and (1.3) yields the embedding PL : L + Li4. Conversely, if L is a subalgebra of Li4 then for every f E L - (0) it follows that 0 (plf)(i) = f(i) = 1for all i E I , hence cplf = 1 i.e. f E D s ( L ) . Proof. If

One of the basic results in the theory of pseudocomplemented lattices and in particular in the theory of Heyting algebras is the Glivenko theorem.

We give below a variant of it for LM-algebras. 4.8. Theorem.

Let L E LM8 and for each i E I regard ‘pi as a map Then:

pi

:

L

4

C(L).

(i) p1 and cpo are morphisms in the categories of pseudocomplemented lattices and of dually pseudocomplemented lattices, respectively, and C(L ) % L / D s (L ) s L/Ds(L). (ii) If L E LMn then ( P ~ - and ~ p1 are morphisms an the categories of Heyting algebras and of dual Heyting algebras, respectively. Proof of the * and

(i)

+ parts.

surjective morphism in DO1 and cp~(z*)= ~ ~ ( ‘ p l=z ‘pl(cplz) ) = (cplz)*.Then C ( L )% L/kercpl by the homomorphism theorem 1.5.16

91 is a

and in view of Corollary 1.3.22 it remains t o prove that modDs(L) =

Connections with other classes of lattices

218

Ker'pl. If ( x , y ) E modDs(L) then z A t = y A t for some t satisfying 'plt = 1, therefore 'p1x = 'p1(z A t ) = p l ( y A t ) = ' p l y . Conversely, if ( z , y ) E Ker'pl then the element t = (x A y ) V Cplz = (z A y ) V p l y satisfies 'pit = 'pix V cplx = 1 and x A t = z A y = y A t.

(ii) By Corollary 3.7.

0

The set of dense element occurs also in the next theorem, which is a generalization of Theorem 4.4. 4.9. Theorem (Varlet [1968], [1969], [1972], Katrinik [1973]).

T h e following conditions are equivalent in a double pseudocomplemented distributive lattice L:

(i)

L satisfies the determination principle (4.4);

(iii)

every chain of p r i m e ideals contains at m o s t t w o elements;

(iv)

every chain of p r i m e filters contains a t m o s t two elements;

(v)

e

(vi)

x A x+ 5 y V y* f o r every z , y E L ;

(vii)

the distributive lattice D s ( L ) is relatively complemented;

5 d f o r every e E Di(TJ and d E D S ( L ) ;

(viii) the distributive lattice

is relatively complemented.

Proof.

(i) + (ii): If x** 2 y 2 z 2 y++ then z* = x*** 5 y* 5 z* hence z* = y* and similarly x+ = y+, therefore z = y . (ii) + (iii): Suppose Pl c Pz c P3 are prime ideals and suppose also, without loss of generality, that Pl is minimal and P3 is maximal. Then Theorem 8.2.2 in Balbes and Dwinger [1974] and its dual imply t h e existence of 2 E D s ( L ) n Pz and z E 'Ds(t)n ( L - Pz). Then y = x V z satisfies x < y (otherwise x 2 z E Pz)and this contradicts (ii) because

Pseudocomplement ed lattices

219

(iii) @ (iv): By the duality principle. (iii) + (i): Suppose there are x , y such that x* = y* and x+ = y+ although x # y, say x 2 y. Let P be a prime ideal such that x E P and y $! P. Then L-P is a prime filter and x** = y** 2 y hence x** E L-P and similarly y++ E P. Applying again Theorem 8.2.2 in Balbes and Dwinger [1974] and its dual it follows that L - P is not a maximal filter and P is not a maximal ideal. Thus the prime ideal P is neither minimal nor maximal, which contradicts (iii). (ii)+ (v): If d E D s ( L ) and e E Ds(LJ then (dve)' = d*Ae* = 0 = d* and (dVe)++ = d++Ve++ = d++ hence ( d v e ) + = d+. Therefore d V e = d i.e. e 5 d. (v) + (vi): Because x A x+ E Ds(L) and y V y* f Ds(L). (vi + (vii): Suppose a , x , b E D s ( L ) and a 5 x 5 b. Then y = (x+ V a ) A b satisfies y** = (x+** V a**)A b** = 1, i.e. y E D s ( L ) . Further x V y = x V (x+ A b ) V a = x V b V a = b, while x A y = x A (x+ V a) = (x A x+) V a = a because x A x+ 5 a V a* = a . (vii) 3 (i): Suppose x* = y* and z+ = y+. Note first that

v .*++)+ = (y++ v y*++)+ = (y v y*)+++ = ( Y v Y*)+

(x v x*)+ = (x v x*)+++ =

(,++

1

Further set a = (xVz*)A(yVy*). Then xVx*, yVy*, a E

D s ( L ) hence

[a, 1) C Ds(L). Thus [a,1) is complemented and for every z E [a, 1)we have 2&"') = z + [ ~ * '= ) z+ V a by the duals of Theorems 8.4.5 and 8.4.1 in Balbes and Dwinger [1974], respectively. Therefore the equality (z V

x*)+ V a =

(y V y*)+ V a reads FVZ[all) = yVy'[""), implying z V x* = y V y*. Finally

(vi)

+ (viii) + (i): By duality.

0

Connections with other classes of lattices

220 4.10. Corollary (Varlet [1968]).

A double pseudocomplemented distributive lattice L i s a three-valued Moisil algebra i f and only if the sets Spec(L) and PFl(L), ordered by inclusion, are cardinal s u m of chains of cardinality at m o s t 2. Proof. Apply in turn Theorem 4.4, Theorem 4.9 and the fact that a double pseudocomplemented lattice is a double Stone algebra if and only if each prime filter (prime ideal) is included in a unique maximal filter (maximal ideal) and includes a unique minimal prime filter (minimal prime ideal); cf. Varlet [1966].

0

Corollary 4.10 and Theorem 4.4 characterize t h e three-valued Moisil algebras within double pseudocomplemented distributive lattices and within double Stone algebras, respectively. In view of Corollary 3.5 and its dual, every three-valued Moisil algebra is also a Heyting-Brouwer algebra as defined below, so that it is natural t o look for a characterization in terms of the latter algebras as well. 4.11. Definition. By a Heyting-Brouwer algebra we mean a Heyting algebra such that its dual is also a Heyting algebra. 4.12. Theorem (Iturrioz [1976a], Becchio [1978a]).

T h e following conditions are equivalent f o r a (distributive) lattice L:

(i)

L is a dually pseudocomplemented Heyting algebra satisfying the id entity

(4.8)

(z

* y) v (y *

5+*)=

1;

(ii) L E M 3 ; (iii) L is a Heyting-Brouwer aZgebra satisfying (4.8).

Pseudocomplement ed lattices

221

Proof.

(i) =+ (ii): In view of Theorem 4.4 it suffices t o prove that L is a double Stone algebra satisfying (4.4). The first step is t o show that L is an Lalgebra; cf. Definition 3.11. We establish in turn several identities. The identity z V y 5 (z + y) =+ y, valid in all Heyting algebras, yields (z + z+)

+ z+ = 1, i.e.

z

=$ Z+

general, we get the identity

x

5

z+; since the converse inequality holds in

+ z+ = z+. We use it and 0 5 z+ t o obtain

+ 0 5 x + z+ = z+, hence x* 5 z+. Taking x+ in the role of z we obtain zt* 5 z++ 5 z, hence y + z+* 5 y + z. From (4.8) and the previous inequality we obtain (z+ y) V (y + z) = 1, i.e. L is an L-algebra. x* = z

Therefore L is a Stone algebra by Theorem 9.2.10 in Balbes and Dwinger

[1974]. The next step is t o prove that L is a dual Stone algebra, i.e. t o check the identity z+ A z++ = 0. From

z + z* = 5 =3 (x we obtain the identity z+ identity z

+ 0) = (z =3 z) + (z + 0) = 1 + z* = z*

+ z+* = z+*. Using it, the previously established

+ z+ = z+ and (4.8), it follows that 1 = z+ V z+*.

Therefore

x++ 5 x+*, hence xt Axc++5 x+ A x t * = 0. To prove (4.4) we use the inequality at*

5 a established above and

5 b* u Q A b A c = 0 u b A c 5 a * ,

aAc which implies a

=+ b* = a* + b. Therefore if z = z* and y = y* then we

get in turn

+ y+* = z * + y+ = y * + z+ = y + Z+* , z + y = (z + y) v (z =+ y) 2 (z + y) v (y + z+*) = 1 5 =$

y2z

by (4.8), hence x

5y

and similarly y

5 z, thus z = y.

(ii)=+ (iii): L is a Heyting-Brouwer algebra by Corollary 3.5 and i t s dual. Then (4.3) yields zt* = (p2(plx= ylz,hence from (3.5) we obtain

(. + Y) v (Y + z+*) 1 (N91zA P2Y)v CPlZ v N Y z Y =

222

Connections with other classes of lattices

(iii) + (i): Obvious.

0

4.13. Theorem (Varlet [1969], Beazer [1975], A. Monteiro [1980]). T h e following conditions are equivalent f o r a double Stone algebra and a n element a E L:

L is a n axled three-valued Moisil algebra with axis a ;

(i)

(ii) L E M3 and D s ( L ) = ( a ] ; = ( a ] and D s ( L ) = [Nu);

(iii) L E M3 and (iv)

Ds(L) = ( a ] and D s ( L ) = [b) f o r s o m e b 2 a .

Proof. Note that if L E

M3 then (4.3) yields Ds(L) = {x E L 1

~ 2 = 2

1)

and = {x E L I p l x = 0). If, moreover, L is axled then Definition 2.1 reduces t o p l a = 0 and p2a V p l x 2 p2x.

(i) =F- (ii): x E Ds(LJ x

* p1x = o * p l x = o =

'pla

& y2x I p2a H

5 a ++ x E (a]. (ii) + (iii): x E D s ( L ) H 'p2x = 1 e p l N x = Np2x = 0 u N x E NX < a + + x 1 N ~ . (iii) + (iv): By Theorem 4.9.

(iv) + (i): L satisfies (4.4) by Theorem 4.9, therefore L E M3 by Theorem 4.4. Then p l a = 0 because a E Ds(L). Besides p l ( x A p l x ) = 0

hence 2 A Cplx E Ds(LJ, i.e. x A Cplx

5 a , therefore

= p1x V (pzx A (Ax) I 'PIX V

p2a

.

0

4.14. Corollary (Varlet [1969]).

A) T h e following conditions are equivalent f o r a double Stone algebra L: (i) L i s a three-valued Post algebra; (ii) L E M3 and D s ( L ) n Ds(L) # 0; (iii) D S ( L )n

- m = { c } f o r s o m e c E L.

223

Pseudocomplemented lattices B) W h e n this is the c m e the element c in (iii) is the center of L . Proof.

(i) =+ (ii):L E M3 by Theorem 1.7and t h e center of L satisfies c = N c by Corollary 2.19,therefore c E D s ( L ) n Ds(L) by Theorem 4.13. (ii) + (iii): It follows fro m Theorem 4.9 t h a t if x , y E D s ( L ) n Ds(L) then x 5 y and y 5 x , therefore z = y. (iii) =$ (i) and B): First we note th a t c E D s ( L ) implies [c ) C D s ( L ) . Conversely if x E D s ( L ) then (ZA c)* = ( X A c)*** = (x** A c**)*=

1*= 0

5 c E Ds(LJ, therefore E [ c ) . Thus D s ( L ) = [c) and similarly Ds(L) = ( c ] . Taking

i.e. x A c E D s ( L ) ; but x A c E I)s(L) because x A c

X A C= c, i.e. x

4.13 it follows th a t L E M3 and c = N c , therefore c 0 is t he center of L by Corollary 2.19. int o account Theorem

T he last problem in this section is about t h e structure of t h e segments of an LM-algebra.

4.15. Definition. Let KO be a class of bounded lattices. B y a relative KO-algebra we mean a bounded lattice L such th a t all of i ts segments, endowed w i t h t h e lattice structure inherited fro m L , belong t o KO. Clearly every relative KO-algebra belongs to

KO.

In particular every

L E LMn is a relative L-algebra by Proposition 3.12, hence a relative Stone algebra (Varlet [1966],Balbes and Dwinger [1974],Theorem 9.2.10), therefore a relative double Stone algebra (using dual arguments). This is a

[1969],which says t h a t every three-valued algebra is relative double Stone. Varlet I19691 has proved t h a t generalization of a theorem due to KatrinAk

every three-valued Moisil algebra is a relative three-valued Moisil algebra. W hat about the class Mn for n

> 3 ? It is

easily seen t h a t t h e segment

{ c ~ , c ~ ,ofc ~L4} E M4 cannot be made i n t o a four-valued Moisil algebra and a similar remark holds for any class of t h e form M 2 k where k >_ 2; for

Connections with other classes of lattices

224

the classes M(2k + 1) see Proposition 6.2.3). Yet the above result of Varlet can be generalized to n-valued algebras without negation: 4.16. Theorem (Boicescu [1984]). Every algebra L E LMn i s relative LMn. Proof. It suffices to prove that the principal ideals [0, a] E LMn because this will imply by duality that the segments [a, 11 E LMn, which will imply that the principal ideals [a,b] of [a,11 belong to LMn. As [O,a]E D01,axiom (3.1.1) holds. Define

and (pP'"]x = 'pp'"]x. Then (3.1.3) holds too. Axiom (3.1.4) is obvious. If x , y E [O,a] satisfy 'pp"]x = 'pP'"]y (Vi), then by applying 'pi we obtain 'p;a A 'pix = v i a A ' p i y (Vi), i.e. ' p i x = (piy (Vi), therefore z = y ; thus (3.1.6) holds. To prove (3.1.4) and (3.1.2) we compute in turn

225

Pseudocomplemented lattices

cppal

and it remains to check that preserves joins. W e set 'pY'"](zV y ) = A , 'pi[O '4 x V cpp'"y = B and prove that 'pjA = cpjB for all j . If j 5 i then

(4.10) yields:

which implies immediately cpjA = cpjB. If i

<j

we obtain in turn

226

Connections with other classes of lattices

Pseudocomplemented lattices

227

4.17. Corollary .

If L E LMn then D s ( L ) is relative LM(n - 1 ) .

< b.

Clearly [ a ,b] C D s ( L ) . As [a,b] = [0, b] n [ a ,11, the duality principle implies that it suffices to obtain [ a ,11 E LM(n-1). T h e dual of Theorem 4.16 yields [a,11 E LMn with Proof. Take a , b E D s ( L ) , a

but

(Pn-la

= 0, therefore cpnLlx [a 11 = p n [a711 m 2 x ,as desired.

4.18. Corollary (Varlet [1968]). If L E LM3 t h e n D s ( L ) is a relatively complemented distributive lattice.

Connections with other classes of lattices

228

$5. Completeness properties and atomicity B y completeness properties we mean results of the following type: if certain (possibly infinite) meets or joins exist, then other meets or joins exist and are equal to the former. W e are also interested i n t h e existence of atoms and t he representation of elements as joins of atoms. W e establish such properties for LM-algebras and Post algebras and relate these results to the corresponding properties of t h e associated Boolean algebras of chrysippian elements. In this section, whenever we state that the existence of a certain meet

o r j o i n implies a n equality involving that m e e t o r j o i n and another meet or join, we m e a n that the latter exists too and the equality holds. Th u s e.g. in Lemma 5.1. below by “if I\ a, exists then cpo A a, = A yoas” SES

SES

A

we mean “if

a, exists then

cpoa, exists and cpo

sES

sES

A yoas”. Also,

A

SES

we say th a t an algebra

A

a, =

SES

L E LMt9 is m-complete ( c o m -

SES

plete) if the lattice L is m-complete (complete). 5.1. Lemma.

Let (a,),Es G L E LMt9. If

Proof. Let a =

A

A

,ES

a,. Then a

a , ezists, t h e n cpo

A

a, =

SES

5 a,

A

Qoas.

SES

Vs E S, hence cpoa

5 yoas Vs

E S.

sES

Take b

5 yoas for

every s E S; then y l b

5

yoas

I a,

Vs E S, hence

plb 5 a , therefore b L cplb = poylb5 you. Th u s cpoa = A

yoas.

0

sES

5.2. Proposition. Let L E LMd. A subset of

C(L)has

a meet (join)

in L if and only i f it

Completeness properties and atomicity

229

has a meet (join) in C(L),an which case the two meets (joins) are equal. Proof. Take (a,),Es Then a

I a,

(Vs E

G C(L). Suppose ( a a ) , Ehas ~ a meet S). If b 5 a, (Vs E S) then plb 5

(Vs E S) and ' p l b E

A

a =

A

a,. Conversely, if a =

A

= a, a . Thus

'plus

C ( L ) , therefore plb 5 a , hence b 5

BES

'poa =

a in C ( L ) .

a, exists, then by Lemma 5.1

BES

poa, =

A

a, = a , therefore a E

C(L)and

this implies

BES

BES

easily that a is the meet of

(U,),~S

in

C(L).

0

5.3. Corollary.

If L

E

LM19 i s m-complete (complete), then C(L)is

However, if

C(L)is complete, L

so.

need not be complete (Cignoli [1969]),

we shall illustrate this fact in Example 5.25.

It seems natural t o ask whether the endomorphisms 'pi commute with infinite meets and joins. A. Monteiro (cf. L. Monteiro [1974]) and Cignoli [1984] answered in the affirmative for three-valued and n-valued Moisil algebras, respectively. We present below the general case. 5.4. Definition (Georgescu [1969], [1971a]).

A &valued Lukasiewiu-Moisil algebra L is said t o be completely chrysippian provided for every ( u , ) , ~ C s L : (C') if A a , exists then ,ES

and ( C ' ) if

V a, SES

exists then

Connections with other classes of lattices

230 5.5. Proposition.

Conditions (C’) and (C”) are equivalent for any L E LMNd. Proof. If

(C’) holds and

V a,

exists then using Proposition 1.2.26 we

SES

obtain

5.6. Lemma.

Let B be a Boolean algebra and ( a , ) , € ~ B[q. Then:

A a,

(5.2‘)

( ?,s

exists

SES

V

(5.2”)

a, exists

aS)(i) =

,,

(Vi E I) ,

a&)

j sES

SES

A

Proof. Suppose

(

A

sES

a, exists and fix an element i E

I.

Note first that

sES

u , ) ( i ) 5 a , ( i ) (Vs E S ) by definition (cf. Example 3.1.3). Then take

z E B such that z 5 a s ( i ) (Vs E S). Define f : I + B by f(j) = 0 if j < i , f(j) = z if j 2 i. Clearly f E B[q and f 5 a, (Vs E S), hence f 5 A a,, therefore z = f ( i ) 5 A a , ) ( ; ) .

(

SES

sES

5.7. Proposition.

Every 6-valued Post atgebra is eompletety chrysippian and fulfils the following converse properties:

(P’) if

A

( Pi a s

exists f o r each a E I then

sES

(P“) if

V sES

A a, exists, and SES

vias exists

for each i E I then

v SES

a , exists.

Completeness properties and atomicity

231

Proof. According t o Definition 1.4 it suffices t o establish the property for an LM-algebra of the form

A

Suppose

B[q where B E B. Take ( U , ) , ~ S E B[q.

a , exists and fix an element i E

I.

Note first t h a t

SES

(Pi

A a, 5

(Pias

(VS E S). Then let b E B[q fulfil b

5

pias (Vs E S ) ;

sES

using Example 3.1.3 (1.11) and Lemma

therefore b

A

2 cpi

5.6,we obtain

in turn

as.

sES

Conversely, suppose bi =

A

pias exists for each i E I .

Then

SES

b; E C ( L ) by Proposition 5.2 and clearly i 5 j =+- bi 2 b j , therefore by Proposition 1.2 (iii) there is an element a such that bi = (P,U (Vi E I). Let us prove that a =

A

a s . But

SES

5 a, (Vs E S ) by th e determination principle. (Vs E S). Then v i b 5 q;as(Vs E S ) (Vi E I), hence

therefore a

therefore b

5 a.

Now take b

5 a,

0

5.8. Corollary. Let L be a d-valued Post algebra OT a n axled n-valued LM-algebra. T h e n L is complete if and only if C ( L ) is complete. Proof. In view of Corollary 5.3 it remains to prove t h a t

C ( L )complete j L

Connections with other classes of lattices

232

complete. If L is a Post algebra this follows from Proposition

5.7. In the case of an axled algebra L E LMn the result follows from Theorem 2.26 and 0

the first half of the present corollary.

5.7 and the first half of Corollary 5.8 were proved by Epstein (19601 in the case 19 = 7 t , while the second half of Corollary 5.8 was established by L. Monteiro [1974]for 3-valued Moisil algebras. Proposition

5.9. Corollary. Let L be a 3(4)-valued Post algebra. T h e n L is cumplete if and only if I ( L ) is complete. Proof. In view of Corollary 2.32 it remains t o prove that I ( L ) complete

+L

complete. In a Post algebra N u l = an-2 and by Propositions

2.37 and 2.38 it follows that I ( L ) 2 C ( L ) . The result follows from Corollary 5.8.

5.10. Proposition (Georgescu [1969],(1971al). The following conditions are equivalent f o r a n algebra L E LM8:

(i) L is completely chrysippian; (ii) the m a p Pr, : L + (C(L))[4 defined by (1.3) is a complete m o r phism. Proof. We suppose

A a, RES

exists and use Lemma

5.6.

233

Completeness properties and atomicity

The infinite distributivity which holds in Boolean algebras (cf. Corollary 1.2.28) can be generalized as follows. 5.11. Proposition. In every completely chrysippian fl-valued Lukasiewicz-Moisil algebra

V a, exists +

(5.3')

A

(5.3")

v

aA

3ES

a, =

3ES

a, exists

+

A a,

aV

=

3ES

3ES

V ( a A a,) , 3ES

A

(ava,)

.

sES

Proof. Using Proposition 5.2 and Corollary 1.2.28 we evaluate

NOWa A

V a, 1 a A as

SES

(Vi E I ) , therefore b 2 a A V

(Vs

E S). Take b 2 a A a , (Vs E S). Then

a,.

CI

S€S

5.12. Theorem (Cignoli [1984]). Every n-valued Lukasiewicz-Moisal algebra is completely chrysippian. Proof. Suppose a = A

a, exists. First we prove that for i = 1, ...,n - 2:

aES

Let j E I and s , t E S. We obtain in turn

234

Connections with other classes of lattices

hence a, A q i a , as A (Pias

I at

(Vs,t E S). It follows that for each s E S we have

5 a , hence

vi+las A pias

5 vi+la, that is

(Pias

5 vi+la,

or

equivalently via, V ~ ; + = ~ 1. a Finally (5.4') follows from

Now we prove via =

A v i a s by induction

on

i. For i = 1 this holds

SES

by Lemma 5.1, so that we perform the inductive step from i to i

+ 1. Note

a , E S ) and prove that 'pi+la i vi+lag (Vs E S). Then take b 5 ( ~ i + ~ (Vs that b 5 ~ i + ~ a . We introduce the elements a', = cp,-lb V ~ i + V~ a,a and prove they fulfil the hypotheses of (5.4'). First cpi+la', = (Pndlb V ( ~ i + ~=a 1 , because

b 5 vi+las implies cpn-lb 5 v;+las.Also, it follows from Proposition 1.2.27 that a' = A a', exists a' = p,-lb V + ~ i + V ~ aa . This implies 'pia' = SES

V ~ ; + ~while a , from via',= pn-lb V 'p;+la V pias and the inductive

hypothesis we see that

A

.QES

Now we apply (5.4') and obtain 1 = ~ ; + ~=a pn-lb ' V~

b 5 cpn-lb L cpi+la, as desired.

i + ~therefore a ,

0

5.13. Remark. It follows from Proposition 5.11 and Theorem 5.12 that properties (5.3) hold in every L E LMn. A direct proof of this f a c t for Monteiro [1974].

TZ

= 3 was given by L.

Completeness properties and atomicity

235

5.14. Lemma.

If L E L M n is infinite then card L = cardC(L). Proof. Taking into acount that C(L)

L and the mapping Pr, : L

--t

[C(L)][q defined by (1.3) is obviously injective, we have card C(L) 5 card L 5 card (( C(L)) [n-ll 5 (card C(L)) n-l hence cardC(L) the proof.

2 Xo, therefore (cardC(L))n-l

,

= cardC(L), completing 0

5.15. Proposition.

A necessary and s u f i c i e n t condition in order that a n infinite cardinal m be the power of a complete n-valued LM-algebra is mxo = m. Comment. In the case n = 2 of Boolean algebras this was proved by Pierce in 1958 (cf. Sikorski [1964], 25.4). Proof. If m = card L where L E L M n is complete, then C ( L ) is complete by Corollary 5.3 and cardC(L) = m by Lemma 5.14, therefore mxo = m by the Pierce theorem. Conversely, if mXo= m then the same theorem implies

the existence of a complete Boolean algebra B such that cardB = m. But L = B"+l] is an n-vahed Post algebra as was seen in $1, B E C(L)by Proposition 1.2 (i), hence card L = m by Lemma 5.14, while Corollary 5.8 shows that L is complete.

0

5.16. Definition. Let L be a poset with first element 0; in particular L may be a d-valued LM-algebra with or without negation. An element a E L is called an a t o m provided a

# 0 and there is no element I E L with 0 c II: < a.

by At(L) the set of all atoms of L. If for every a

a

I

We denote

E L - (0) there is an atom

5 I then L is said t o be atomic. The condition for being an atom is often verified in t h e form 0 5 I < z = 0 or in the form 0 < I 5 a + I = a. In particular if a and b are

236

Connections with other classes of lattices

atoms and b

I a then b = a .

5.17. Lemma. Let L E LMn and a (5.5)

E

At(L). Then there is a n indez io such that

pioa = (pio+la=

... = (pn-1a E A t ( C ( L ) )

and i f i o > 1 then

(5.6)

9

1

=~ 9

2

=~ ... = (pio-1a = 0

.

# 0 that {i I (pia # 0) # 0.

io = min {i 1 via # 0); then (5.6) holds. Now take an arbitrary but fixed i >_ io. Then via 2 yioa # 0 hence v i a E C ( L ) - (0). If 0 < z I via then (pi@ A U ) = Proof. It follows from a

Let

# 0, hence 0 < z A a I a , therefore z A a = a , t h a t is a 5 z , which implies v i a I cpiz = z, whence z = via. Thus (pia E A t ( C ( L ) ) for i = io, io + 1, ...,n - 1; since z A (pia = z

0

Fioa

I Vio+la I 5 vn-la ,

this is possible only if

*.*

(5.5)holds.

0

5.18. Lemma. Let L E LMn and c E A t ( C ( L ) ) . Then there ezists a unique a E A t ( L ) such that vn-1a = C .

A = {z E L I (pn-lx = c ) . Then A # 0 because c E A. Further A is finite because if x E A then for each i = l, ..., n - l we have (pix I (pn--lx = c hence (pix E {O,c}, therefore there is a finite number of Proof. Let

possibilities for the vector (cplz,...,cpn-lz) which determines completely z.

A and i h = min {i I (PiXh # 0) (h = 1,2), we have e.g. il I i 2 , therefore Lemma 5.17 shows that i < il + (pix1 = 0 I ( p i x 2 while i 2 il j (pixl = ( ~ , , - ~ z= ~ c = (pn-1z2= ( p i 2 x 2 , so that pjxl I (pix2 (Vi), whcne 51 5 2 2 . It follows from the above properties that A has least element a. Then Pn-la = c and it remains t o prove that a E At(L). If 0 < z 5 a , then 0 < (Pn-lZ I (pn-la = c , hence cpn-lz = c, therefore z E A , which implies aI x whence x = a. Finally A is totally ordered because if zl, z2 E

Completeness properties and atomicity If a' E At(L) fulfils ~ , , - ~ a=' c then a' E A , hence 0 a = a!.

237

< a 5 a', therefore

5.19, Corollary. If L E LMn t h e n card A t ( L ) = card At (C(L ) ). Proof. Define f : At(L) -+ At(C(L)) by f(a) = ( ~ ~ -Vu~ EaA,t ( L ) . The map f exists by Lemma 5.17 and is a bijection by Lemma 5.18. 0 5.20. Corollary. If L E LMn t h e n L is atomless if and only if C ( L ) is atomless. 5.21. Theorem (Georgescu [1969], [1971a], Boicescu [1984]). If L E LMn then L is atomic if and only if C ( L ) is atomic. Proof. Let L be atomic and s E C ( L ) - ( 0 ) . Then x E L - ( 0 } , hence there exists a E At(L) such that a 5 s; it follows that ( P , , - ~ U E A t ( C ( L ) ) by Lemma 5.17 and ( P , , - ~ U 5 ~ ~ = x. ~ s Conversely, let C ( L ) be atomic and x E L - ( 0 ) . Then ( P ~ - ~ XE C ( L ) - { 0 } , hence there exists c E A t ( C ( L ) ) such that c 5 ( P ~ - ~ X .In view of Lemma 5.18 we find a E At(L) such that ( ~ ~ = - ~c. a Then pn-l(aAs) = cA\,,-lx = c # 0, hence 0 < a A x 5 a , therefore a A x = a , that is a 5 x. 5.22. Proposition. If L E LM29 is atomic t h e n C ( L ) is atomic. Proof. If a E At(L) then la # 0. Let z E C ( L ) , 0 < z 5 'pla. We have cpl(z A a ) = z A 'pla = z # 0. Thus z A a # 0. This implies z 2 a . Since z E C ( L ) it follows z 2 via. Therefore z = qla. It follows that ~ l EaAt(C(L)). The result follows as in the first part of Theorem 5.21.0 The converse fails.

Connections with other classes of lattices

238

5.23. Definition (cf. Day [1965] in the case of Boolean algebras). An algebra

L E L M S will be called superatomic if every LMS-subalgebra

of

L is atomic. 5.24. Corollary.

If L E LMn t h e n L

is superatomic

if and only if C ( L ) is superatomic.

Proof. From Theorem 5.21 and the easy remark that t h e subalgebras of

C(L)coincide with the sets of the form C(S) for

some subalgebra S of L .

0

5.25. Example.

L = {z E LF I 3n,n > n, =+ zn E {0,2}} is a 3-valued algebra for which C ( L ) = L r is an atomic and complete Boolean algebra. However, l e t us prove that L is not complete. Let xi E L be defined by xi = 1 and xi = 0 for n # i. Let z E L be an upper bound of (zi)iEN. Then z, E {1,2} for n 5 n, and x n = 2 for n > n,. Now choose no > n, and define y E LF by yno = 1 and Yn = zn for n # no. It follows that y E L, y < z and y is also . this set has no infimum. an upper bound of ( z i ) i E ~Therefore

Note also that th e lattice L of Example 5.25 is completely chryssipian by Theorem 5.12 but does not satisfy conditions (P) from Proposition 5.7. 5.26. Definition.

A poset L with first element 0 (and in particular a S-valued LM-algebra L ) is said t o be strongly atomic provided each element of L is a join of atoms. 5.27. Remark. In every poset with 0: a) 0 is the join of the empty set of atoms, and

b) an element z is a join of atoms if and only if included in it.

2

is the join of all atoms

Completeness properties and atomicity

239

In the literature one works usually with a single kind of atomic lattices, introduced either by Definition 5.16 or by Definition 5.26. This is probably explained by the fact that th e two definitions are equivalent in t h e case of Boolean algebras; cf. Remark 5.30 below. However, Definition 5.26 is actually stronger than Definition 5.16 outside the class of Boolean algebras, as shown e.g. by the n-valued Moisil algebra

L,, which is atomic but for

n > 2 is not strongly atomic. This example shows also t h a t Theorem 5.21 does not extend t o strong atomicity, since C(L,) = L2 is of course strongly atomic. 5.28. Lemma.

Let L E LMn. If an element of L can be expressed as a join of atoms, this representation is unique. Proof. We take Al,A2

2 At(L)such t h a t V A1 = V A2 and prove A1 = Az.

Let a E Al. Using Remark 5.13 we get

u = u A V A ~ = U A V A ~ V=

(5.7)

(uAx);

zEA2

but a A x

5a

hence a A x = 0 or a A x = a for each x E A2 and it follows

3x E A2 such implies a = 3: E A2. Thus Al C A2 and

from (5.7) that the elements a A x cannot be all 0. Therefore that a A x = a , hence a similarly A2 C Al.

5z

which

0

5.29. Theorem (Boicescu [1984]).

The following conditions are equivalent for an algebra L E LMn:

(i)

L

is

strongly atomic;

(ii) & is an atomic Boolean algebra;

(iii) 1 = V At(L). Proof.

(i) + (iii): Trivial. (iii) =$ (ii): First we prove that L is strongly atomic. It follows from Remark 5.13 that for each

xEL

Connections with other classes of lat tices

240

because for each a E

At(&)either

5A

a = 0 or x A a = a , i.e. a

5 x.

The next step is t o prove t h a t A t ( L ) = At(C(L)). We introduce the auxiliary set

A = { a E At(L) 1 via and note that

# 0) = { a E At(L) 1 via = a }

A 5 At(L) while Lemma 5.17 implies A

s At(C(L));

there-

fore it will suffice t o prove the converse inclusion. It follows from Theorem

5.12 that

1 = ~ p 1 1= (PI

V At(L) = V { ~ p l aI u E At(L)}

=

= V { c p 1 a I a E A } = V A I V A ~ ( C ( ~ ) 7) hence

V A = V At(C(L)),

therefore A = At(C(L)) by Lemma 5.28 ap-

C(L).Now suppose there is b E At(L) - A . it follows that a E C(L), while Lemma 5.17 implies a plied t o

#

Setting a = cpn-lb E

At(C(L)) = A.

b and ( ~ , , - ~=aa = yn-lb, which contradicts Lemma 5.18. This proves that At(L) A hence At(L) = A = At(C(L)). Finally we summarize the previous results: each element x is a join of therefore x E C(L)by Proposition atoms and the atoms belong t o C(L), 5.2. Thus L = C ( L )is a strongly atomic Boolean algebra. (ii) + (i): We take x E L - { 0 } , set A = { a E At(L) 1 a 5 x} and prove that x = V A. The condition a 5 z (Va E A ) holds trivially. Now we take y E L such that a 5 y ('da E A ) and must prove x 5 y. If this is not true, it follows that x A jj # 0 and since L is atomic there is an atom a 5 x A jj. Then a 5 y and a 5 z, hence a E A , therefore a 5 y, which 0 implies a 5 y A y = 0, a contradiction. Therefore a

E At(L), a

s

5.30. Remark.

The above implication (ii) =+ (i), due t o Halmos [1963],is worth being stated separately: a Boolean algebra is strongly atomic if and only if it is atomic.

241

m-algebras 56. rn-algebras

One of t he current trends in lattice theory consists in extending the properties of various classes of lattices t o the corresponding classes of mcomplete lattices. As will be seen in Chapters 5-6, t h e appropriate tool for such a generalization in the case of LM-algebras is the concept of m-algebra, which we introduce in this section. The MacNeille completiom L s of the important algebra

Le

is an example of m-algebra and it is isomorphic t o

D({O,l}) (Theorem 6.12). Both the concept of rn-algebra and Theorem 6.12 are due t o lorgulescu

[1984~1. 6.1. Definition. Let rn be an infinite cardinal number. A d-valued LM-algebra (possibly with negation) is said t o be an m-dgebra provided the following conditions hold:

(i) the lattice L is m-complete (cf. Definition 1.1.11); (ii) L is m-completely chrysippian, i.e.

(6.1')

Cpi

A

XA

=

XEA

A

C~;XX

(Vi E I)

XEA

for every subset { X X } X ~ Aof

L of cardinality 5

m.

Open problem.

Is it true that (i) implies (ii)? 6.2. Definition.

L

is called a fully complete algebra if L is an m-algebra for every m. In other words, a fully complete algebra L is a complete lattice L and a

completely-chrysippian LM-algebra

L.

Connections with other classes of lattices

242

Every finite d-algebra is fully complete.

Non-trivial examples of m-

complete and fully complete algebras are given below.

6.3. Example. Every m-complete Boolean algebra is an m-algebra (20 = 2). 6.4. Remark.

If L is an m-algebra (for arbitrary S) then C ( L ) is an m-complete Boolean algebra.

6.5. Proposition. If B is a n m-complete Boolean algebra t h e n B[q and D(B)are m-algebras (cf. Esamples 1.1.3, 1.1.10and 1.1.19). Proof. Formulas

(6.2‘)

A f x ) ( i )= A f x ( i ) ,

( XGA

cardA I m

,

&A

yield m-completeness both for B[q and

D(B),while

e.g. (6.1’) for

D(B)

follows from

(W Ax f x ) ( j ) =

( Ax

fx)(di)= A x fx(di) =

6.6. Corollary. If B is a complete Boolean algebra t h e n B[q and D ( B ) are f u l l y complete algebras. 6.7. Definition.

The set I is said t o be regular provided it has least element 0 and greatest

m-algebras

243

element 1, it satisfies th e equivalent conditions i) and ii) in Remark 3.1.11 and is endowed with an antitone involution d : I algebra (an rn-algebra, a fully complete algebra)

--f

I.

We say that an

L E LM9 or L E LMN9

is

regular provided the corresponding set I is regular. Suppose L is regular. Then the algebra Lo in Examples 3.1.2.b, 3.1.12 and 3.1.20 is regular and

C(&) = L2.This

algebra is not fully complete

but we are going t o prove that the MacNeille completion made into a fully complete algebra isomorphic t o L, [rl .

L, of Lg can

be

6.8. Remark.

If the set I is regular then for every two subsets A and B of I that satisfy conditions (i) and (ii) of Definition 1.1.37, conditions (iii) and (iv) of that definition are equivalent. For if a = supA in I then a+ = inf B in I and dually, if b = inf B in I then b- = supA i n I . 6.9. Remark. For every f E

(6.3) then

Lh',

set

U f = {i E I I f(di) = 1) ;

Ufand I - Uffulfil conditions (i) d (ii) in

(6.4)

i <j

(6.5)

V f U ( I - U f )= I ,

because if j

Definition 1.1.37, i.

(Vi E V f ) ( V j E I - U j ) ,

< i for

some

i E U f and j E I - U f we would contradict the

fact that f is increasing. In the remainder of this section we denote by sup t h e g.1.b. in

6.10. Remark. It follows immediately from Remark 6.9 that i f f E Li', then (6.6)

Uj = {i E I I i 5 supUf} ,

(6.7)

I - Uf= { j E I I SUP U f < j

}

.

J.

Connections with other classes of lattices

244

6.11. Lemma. The m a p s 0 : Lh4 -+

g and

(6.8)

0(f)= s u p U f

(6.9)

Q(z)(k)=

+ LIZ'] defined b y

\E :

(Vf E Li')

i

,

d k i x ;

1,

(Vk E I ) (Vx E

J) ,

dk>x;

0 ,

respectively, establish a bounded-lattice isomorphism. Proof. For every

x E J and every f E LL',

(G o 9 ) ( ~=)supUq(,) =

{k E I I Q ( x ) ( d k ) = 1)

=

sup{kEIIkIx}=z,

(Q o i P ) ( f ) ( k ) = 1

++

= SUP

* Q ( @ ( f ) ) ( k )= 1 H dk I@(f)

dkIsupUf~$kEUf*ff(lc)=l.

We have thus proved that the functions

G

and

is! are inverse to each

0 and 1, therefore in view of Proposition 1.1.23 (ii) it remains to show that G and Q are isotone. But f I g implies Uj C U, hence G(f) I G ( g ) . Also, if x 5 y then other. Clearly Q preserves

therefore Q ( x )

I \E(y).

0

6.12. Theorem. Let I be a regular set and

7 the MacNeille completion of J . Then:

(i) The bounded distrabutive lattice ( i E I ) and N defined by

(6.10)

(6.11)

(pix

=

I

1,

x>di

0 ,

x
3

endowed with the operations

(Vx E

J) (Vi E I) ,

N s = s u p ( k ~ I I d k > x } (VZEJ),

'pi

m-alge bras

245

ES isomorphic to Lkq.

is a fully complete algebra (ii)

LS i s a subalgebra of ZS.

Proof.

(i) L;'

6.6. In view of Lemma 6.11 it suffices t o transfer the operations yi and ' of Lifl (6. notation (3.1.11) (3.1.30))t o 7, i.e. t o define cp;(O(f)) = @(yif) and N ( @ ( f ) )= @(f,). We obtain is a fully complete algebra by Corollary

pix = v ~ ( Q ( Q ( ~=)@ ) )( y i ( Q ( X ) ) ) = = sup {k E

I I T ; ( Q j ( z ) ) ( d k )= I} =

= sup {k E I1 * ( z ) ( i )= 1) = supI,

di < s

1,

z2di

sup0,

di > z

0,

x
?

N z = N ( O ( Q ( z ) ) )= @((*(z))')

=

= sup{k E 11 ( Q ( z ) ) ' ( d k ) = I} = = sup {k E 11 * ( z ) ( k ) = 0) = sup {Ic E I I dk

> z} .

(ii) If z E J formula (6.10) coincides with (3.1.8'),that i s cpjz = (pax. Further from (6.11)we get NO = sup1 = 1, N1 = sup0 = 0 while for z E I - (1) we obtain N z = sup {k E I I k c dx) = ( d z ) - . Thus if z E J formula (6.11)coincides with (3.1.35),that is N z =- z.

6.13. Corollary. ~f I = (1, ...,n - 1) then

L'

Y

L,.

This Page Intentionally Left Blank

247

CHAPT E R 5

FILTERS, IDEALS AND 8-CONGRUENCES

T he aim of this chapter is t o construct a theory of filters, ideals and congruences in LM-algebras generalizing the main results of t h e correspon-

ding theory for distributive lattices (6.

Ch. 1, 93).

T h e starting point is

a convenient definition o f th e concepts of filter, ideal and congruence in LM-algebras, obtained by adding appropriate extra conditions to the corresponding concepts of lattice theory and universal algebra, respectively (cf.

Ch. 1). T he last section answers a problem raised by Cignoli i n connection wit h a property o f prime filters in Post algebras.

$1.d-filters, d-ideals and 8-congruences

A 9-filter (&ideal) of an LM-algebra is quite naturally defined as a filter (an ideal) compatible w i th the endomorphisms cpi. It turns o u t t h a t &filters coincide wit h Stone filters and w i th deductive systems. Since the quotient

L / p of an LM-algebra L by a congruence p (in the sense of universal algebra) need not satisfy th e determination principle, we introduce t h e stronger concept of 9-congruence. T h e “metatheorem” of this section is t h a t 6- filters (&ideals),

d-congruences and quotient algebras behave as one would

expect. See also Hatvany [1984].

1.1. Definition. Let L E LMd. A d-filter of L is a filter F of L such t h a t A d-ideal of L is an ideal H of

x E F jcpox L such th a t x E H + cp1z E H .

E

F.

In the sequel we only deal w i th &filters b u t all the results can be transferred by duality to 1’1-ideals.

Filters, ideals and 6-congruences

248

1.2. Remark. For every &filter

F

The concept of &filter was introduced by Moisil [1940], [1963c] for nalgebras and later [1968] for d-algebras, under the name of strong filter (because there exist filters that are not &filters, e.g.

F

#

(1) of L d : if x E F - (1) then pox = 0

proper d-filter of

Liq is (1)). However,

#

every proper filter

F ; similarly the unique

it turns out t h a t in the case of

LM-algebras 6-filters coincide with Stone filters introduced by A. Monteiro

[1954] (6. A. Monteiro [1974]) and studied by Cignoli [1966a], [1966b] and Cherciu [1971].

1.3. Definition.

A Stone filter of a bounded (distributive) lattice L is a filter F of L such that for every x E F there is y E F n C ( L ) such t h a t y _< z. We set

F* = F n C ( L ) .

(1.1)

1.4. Proposition.

The following conditions are equivalent for a filter F of an algebra L E LM6: (i) F is a 6-filter; (ii) F is

a

Stone filter;

(iii) F is generated by F*; (iv)

F = cp;l(F*).

Proof.

+ (ii): For x E F take y = pas. + (iii): Obvious. + (iv): Because z E F * pox E F H 'pox E F*

(i) (ii) (iii)

&filters, 19-ideals and 19-congruences (iv) =+ (i). Because x E

249

F =+ 'pox E F' G F.

Comment.

It is easily seen that F* is the unique filter of C(L) satisfying (iv).

1.5. Definition. Let L E LM9. For every subset X of L we denote by [ X ) , the &filter generated by X , i.e. t h e least 19-filter including X . Also, we denote by Fl(L) and F119(L) the lattice of all filters of L and t h e lattice of all 19-filters of L, respectively.

1.6. Remark. Let L E LM19. Then [0)4 = (1) while if (1.2)

[XIS

= {Y E L I 'po(zi A

I

0# X

L then

... A xn> I

y;sl,...,2n~~;n~mr-{0}};

C C ( L ) then [ X ) , = [ X ) while if X is a filter of L then [X>S= {Y E L I 'pox I y, 2 E X } .

in particulx if X

1.7. Theorem.

For every L E LM19 the m a p s * : F119(L) + Fl(C(L)) defined by (1.1) and 'pi' : Fl(C(L)) -+ F119(L) are lattice isomorphisms anverse t o each other. Proof. Clearly the maps * and 'pol are isotone and their targets are those indicated above. The composites

'pi'

o * and * o 'pi' are the identity

mappings by Proposition 1.4 and

g,'(P) n C(L) = P which follows from x E

( V P E FlC(L)) ,

C(L)n 'p;'(P) -s x = 'pox E P -s 5 E P .

The congruences in the classes LMQ and LMNQ are introduced according t o the general Definition 1.5.11. However, compatibility with 9; and N follows from the other conditions:

Filters, ideals and 8-congruences

250

1.8. Proposition (Boicescu [1984]). T h e following conditions are equivalent f o r a n equivalence relation p of a n algebra L E L M 8 :

(i) p is a congruence; (ii) p is compatible with A, V and

'p;

(Vi E I ) .

Proof.

(i) j (ii): Trivial. (ii) + (i): We suppose x p y and prove CpixpCpiy. But ' p i x p p i y hence 1p('piy V p i x ) and similarly 1p ('pix V Cpiy). This implies in turn

and similarly piypCp;x A Cpiy, therefore p i x p p i y

(Vi E I ) .

0

The construction of quotient algebra requires a stronger concept (see Remark 1.16). 1.9. Definition (Boicescu [1984], lorgulescu [1984c]).

A 29-congruence of an algebra L E L M 8 or L E LMN29 is a congruence of L E DO1 such that x p y ('pix p (piy (Vi E I ) ) . 1.10. Proposition.

T h e following conditions are equivalent f o r a binary relation p of a n algebra L E LM29;

(i) p is a 29-congruence; (ii) there is a congruence p' of L such that z p y Proof.

(i) + (ii): Take p' = p.

@

(Pix p ' v i y (Vi E I ) ) .

d-filters, 19-ideals and &congruences

251

(ii) + (i): Clearly p is a congruence and if 'pjxpcpjy (Vj) then 0

cpicpjx p'(pj(pjy ( V i , j ) , therefore (pjx p'(pjy (Vj), i.e. z py.

1.11. Definition. Let L E LM29 or

L E LMN29. For every congruence p of L we denote by

p the d-congruence

generated by p , i.e. the least d-congruence including p.

O ( L ) and Oo(L) the lattice of all congruences of L and the lattice of all &congruences of L , respectively. Also, we denote by

1.12. Remark. Let L E LMr9 or L E LMNd and p E z P Y @ (piz ppiy

(1.3)

O ( L ) . Then

(vi E 1))

*

To establish t h e relationship between d-filters and d-congruences we intraduce the maps kerl;, modl; and modl;, or simply ker, mod and mod: (1.4)

ker : Oo(L) --t F119(L) ,

(1.5)

mod : F119(L) --t O ( L ) ,

(1.6)

mod

: Fld(L) + Oo(L)

defined by (1.7) (1.8) (1.9)

ker p = {z E L I z p l }

(Vp E Oo(L)) ,

z m o d F z ' @ 3u E F z A u = z' A u

- mod =

o mod

(VF E Fld(L))

,

(cf. Definition 1.11) ,

respectively.

1.13. Theorem (Boicescu [1984],lorgulescu [1984c]). Let L E LM19 OT L E LMN.9. The maps mod and ker satisfy mod = ker-' and establish an isomorphism F129(L) S Oo(L). Proof. It is easy t o check that ker p E Fld(L) even for p E O(L); e.g. if z E ker p and z

5 y then y = z V y p 1V y = 1i.e.

y E ker p. Let us prove

252

Filters, ideals and 19-congruences

that m o d F E O ( L ) for F E F119(L). Take x m o d F x ' a n d y m o d F y ' i.e. x Au = x'Au a n d y A v = y ' A v for u,v E F. Then (PiXAviu = (PiYAviu where (piu E F, hence 'pix mod F'piy, while from x A u A v = x' A u A v and y A u A v = y' A u A v where u A v E F, we deduce immediately that If L E LMN6 then N x V N u = Nx'VNu. ~ A y p ~ ' A y ' a nxVypx'Vy'. d But from

we get N u A 'pou = 0 hence

NXA YOU = ( N x V N U )A C ~ O U = ( N d V N U ) A

YOU

=

where pou E F, showing that Nxmod F Nx'. Clearly the maps ker, mod and are increasing. To prove ker o m o d = 1 F l q ~ take ) F E Fld(L). Then

-

-

ker(modF) = {x E L zmodFI} = = { X E L vix mod F 1 (Vi E

I)}

=

= {XEL

by Remark 1.2. To prove mod o ker = 1e0(,qnote first that you 5 u and ker p n C(L)= ker(p n (C(L))')and recall Proposition 1.3.26. Then for every p E Oo(L)

zmod

o ker p y H pixmod ker pviy

(Vi E I)

19-filters, 19-ideals and d-congruences

253

1.14. Proposition (Boicescu [1984]). Let L E LMn or L E Mn. T h e n Oo(L)= O ( L ) and the maps mod and ker satisfy mod = ker-' and establish a n isomorphism Fln(L) O ( L ) . Proof. Let p E

-

O ( L ) . If x mod o ker p y then (i = 1,...,n - 1)

cpixmod o ker pcp;y

w (3u; u ; p l & cpix A ui = (piy A ui) therefore taking u = u1 A

... A u,-'

(i = 1, ...,n - 1)

we obtain u p 1 and cpix A u = (piy A u

(i = 1,...,n - 1) hence

(i = 1,...,n - 1) , therefore x A u = y A u, showing that

3 mod

-

o ker p y. Thus mod o ker p

E

mod o ker p therefore

(1.10)

-

mod o ker p = m o d o ker p E Oo(L) .

Taking into account Theorem 1.13 and (1.10) we have

-

xpy+xpy*xmod (j

o kerpy

cpixmod o ker pcp;y

w

(i = 1, ...,n - 1) e

* (3% u; p 1 & pix A u; = cp;y A u;)

(2

= 1,...,n - 1)

+

+- (3% (o;u~p 1 & c p ; ~A ~piui= (P;Y A 9;";)

(i = 1, ...,n - 1)

+ (i = 1,...,n - 1)

+

=$

(pix mod o ker py;y

=$

xmodokerpyw3uupl&xAu=yAu

+- 3~ x = x A I p s A U = Y

A U ~ YA 1 = y

proving that p = mod o ker p, therefore p

O(L)= Oo(L), hence by Theorem 1.13.

-

+. X

+ ~

Y

E Oo(L)by (1.10). This proves

is the identity mapping and the proof is complete 0

254

Filters, ideals and d-congruences

1.15. Corollary.

I f B is a Boolean algebra t h e n Oo(B)= O ( B ) . F12(B) = FI(B) and the m a p s mod and ker satisfy mod = ker-' and establish a n i s o m o r p h i s m Fl(B) O ( B ) . 1.16. Remark. Definition 1.9 ensures that if p is a 29-congruence of an algebra L E LM6 ( L E LMN29) then the determination principle holds in L / p , therefore L / p E LM19 ( L / p E LMNB). In view of Theorem 1.13 every &congruence p is of t he form mod F for some &filter F (= ker p). The quotient 29algebra L / m o d F will be denoted simply by L / F . In particular if L E LMn or L E Mn then every n- congruence is of t h e form m o d F and L / F will stand for L/mod F .

1.17. Remarks. a) In a Moisil n-algebra Proposition 1.8 remains true: If L E Mn we still

have t o prove that x p y implies N x p N y . This follows from

Therefore a l l types o f congruences reduce t o lattice congruences compatible with 'pi.

b) However, this does not hold in general, as shown e.g. by the following example due t o lorgulescu 11984~1.Suppse I is an infinite regular set (6. Definition 4.6.7) and take L = (?(I))['. Let Fo C C ( L ) be the filter of those functions f E C ( L )for which the constant value f ( i ) is a cofinite set (i.e., I - f ( i ) is finite). We denote by F the &filter generated by Fo and prove that mod F is not a &congruence. Let g, h E L be defined by g ( i ) = [O,i-]C I for i # 0, g(0) = 0 and h ( i ) = [O,i]G I,for Vi E I . For each i E I, define f; E C(L)by f i ( j ) = I - {i} V j E I. Then f i E FO C F and ('pig

fi)(j)

= g ( i ) n f i ( j ) = [O, i-]

n ( I - {i})

= [o, i-]

&filters, 19-ideals and d-congruences

255

and similarly ((pjhA f i ) ( j ) = [0,i-1, therefore (pig A fi = (pihA fi, proving that YigmodFcpih for every i E I . It remains to show that ( g m o d F h )

does not hold, i.e. that g A

f # hAf

for every

f E F.

f E F fo 5 f.

But

implies, via Remark 1.6, the existence of fo E Fo such that

This implies further the existence of a finite subset loc I such that 8 # I - I0 = fo(i) 5 f(i) Vi E I. Take i E I - Io; then g A f # h A f follows from

1.18. Remark.

It follows from Remark 1.17 b) and Proposition 1.8 that for certain sets 1 the corresponding classes LMd and LMNd are not equational (because the homomorphic image L / p of L is outside t h e class whenever the congruence p is not a &congruence).

1.19. Proposition. Let L E LMd or L E LMNd. If F, F’ E Fld(L) and F 5 F‘ then = {i I x E F’} is a 29- filter of L / F . If, moreover, F’ is proper (prime) is so. then

F?

then 0 = ( p i x A ui, s E F’,

Proof. The first statement is obvious. If 0 E

u; E F (Vi E I ) , hence 0 =

(pis A

(p;ui, which implies (piu; 2 p i x E F‘

F‘ and since (p;ui E F C_ F’ it follows t h a t 0 E F’. If 3 V E F? then pi(^ V y) A ui = (pi2 A U i , z E F‘, ui E F (Vi E I ) , hence ( p i x A U i ) V ( v i y A U i ) E F’, implying e.g. (pox A uo E F’, in which case x E F‘ because x 2 (pox A ug, therefore i E F? otherwise 6 E F?. O

therefore piui E

1.20. Remark (lorgulescu [1984c]). The concept of d-filter and th e concept of m-filter in Remark 1.3.31 are

I

23 then the filter I of Ld = J = (0) + I is neither a &filter nor an m-filter. Further b) let E be a set of cardinality m and the filter F of P(B)consisting of all cofinite sets. independent of each other. Thus e.g. a) if

Filters, ideals and &congruences

256

Then E - {z} E

F

for every z E

E and

n

(E- {z})

= 8 !$ F . Take

xEE

L = B[q and let F' be the image o f F by the isomorphism B E C ( L ) . Then [F')s= {f E L 139 E F', g 5 f} by Remark 1.6 and this &filter is not an m-filter. Now c) take an m-algebra L and a E L . Then [ a ) is an m-filter; besides, [a) is a &filter or not according as a E C ( L )or not. A filter which is both a &filter and an m-filter will be called an (m, O)-filter. The reader is urged t o transcribe the results o f this section to the case of the (m, d)-filters of an m-algebra. In particular the restriction of the functions in Theorem

1.7 yields an isomorphism between the complete lattice o f all (m, d)-filters of t he m-algebra

L and the m-filters of the Boolean m-algebra C(L).

1.21. Remark (lorgulescu [1984c]). Let us call m-congruence o f an LM-algebra

L , any congruence relation p on

L satisfying the stronger conditions (i) for every

A h€H

L , c a r d H 5 m, if a h p b h (Vh E provided both meets exist, and

( a h ) h € ~ (, b h ) h E ~C

ahp

A

bh

H ) then

h€H

(ii) the dual condition. Suppose L is an m-algebra (cf. Definition 4.6.1) and let F be a &filter of

L. Then using Proposition 1.2.27 and F = i it is easily seen that m o d F is an m-congruence if and only if F is an (m,d)-filter. Let us call (m,d)congruence every 8-congruence which is also an m-congruence. Thus m o d F is an (m, 29)-congruence for every (m, 9)-filter

F

and as a matter of fact the

isomorphism constructed in Theorem 1.13 can be restricted to an isomorphism between the complete lattice of all (m, 29)-filters and the complete lattice of all (m,d)-congruences.

The reader is urged to transcribe other

results of this section t o the case of (m, 8)-congruences. Now we turn to further consequences of the previous basic results. Proposition 1.14 has a kind of converse for 3-valued Moisil algebras. Let

L

be a double pseudocomplemented lattice; by a ++-closed filter is meant

257

8-filters, 8-ideals and 29-congruences a filter

F of L such that x E F

+ x++

++-closed filter for every congruence p of

E F. It is obvious that ker p is a

L.

1.22. Theorem (Varlet [1972]).

Let L be a double pseudocomplemented lattice. T h e m a p ker is a bijection between O(L)and the set of ++-closed filters of L if and only i f L E M3. Sketch of proof. Sufficiency. If L E M 3 then i t s congruences coincide with

the congruences of the underlying double pseudocomplemented lattice, while the 3-filters are exactly the ++-closed filters. Necessity. In view of Theorem 4.4.4 it suffices t o show that

-

L is a dou-

ble Stone algebra verifying the determination principle. The latter property follows from the fact that the congruence

defined by (z

N

y

+ x* = y*

and z+ = y+) satisfies ker N= (1) = ker

1~ hence -= 1 ~ Further . one proves that if x+ A x++ > 0 then the ++-closed filter [zt)is distinct from ker p for every p E O ( L ) . This contradiction shows that zt A z++ = 0 and 0 one proves similarly that z* V x** = 1. 1.23. Proposition.

If L E LMn t h e n O(L)E O ( C ( L ) ) hence O ( L ) is a Heyting dgebra. Proof. Follows from Theorem 1.7, Proposition 1.14, Corollary 1.15 and e.g.

the dual of Example IX.3.5 in Balbes and Dwinger [1974].

0

The next two propositions were first proved by Beazer [1976] for 3-valued Moisil algebras, using the theory of double palgebras. 1.24. Proposition.

T h e following conditions are equivalent for L E LMn:

(i)

O ( L ) i s a Boolean algebra;

(ii)

C(L)i s finite;

(iii) L i s finite.

Filters, ideals and d-congruences

258 Proof.

(i) ($ O ( C ( L ) ) E B H (ii): By Proposition 1.23 and Berman [1973]. (ii) + (iii): Because L' = (C(L))["-'] is a finite Post algebra and PL : L --f L' is a monomorphism. (iii) + (ii): Trivial. 0 1.25.

ProDosit ion.

T h e following conditions are equivalent foT L E LMn:

(i) @ ( L )is a Stone algebra; (ii) C(L)i s a complete Boolean algebra. Proof. (i) H

O ( C ( L ) ) is a Stone algebra

H (ii) by Proposition 1.22 and

Beazer [1973].

0

1.26. Definition.

A class K: of 7-algebras is said to have the congruence extension property if for every A E K , every subalgebra B of A and every congruence p of B , there is a congruence p' of A such t h a t p'

n B2 = p.

1.27. Proposition (Boicescu (19841).

T h e class LMn has the congruence extension property. Proof. Let A E LMn and B , p as in Definition 1.26. The filter F generated by ker p in A is obviously a d-filter and we shall prove that m o d F n B 2 = p. If ( z , y ) E m o d F f l B2 then x , y E B and there is u E A such that

x A u = y A u and a 5 u for some a E ker p , therefore x A a = y A a, hence ( 2 , ~ )E mod o ker p = p by Proposition 1.14. Thus m o d F n B 2 E p, while 0 the converse is obvious from p = mod o ker p. 1.28. Remark.

It is known (see e.g. Pierce [1968], Lemma 3.1.8) t h a t if A is a 7-algebra and p,u E @ ( A )then p o u E @ ( A ) if and only if p o u = a o p, in which case p o u = p V u in the lattice @ ( A ) .The algebra A is said t o have

259

29-filters,29-ideals and 10-congruences

commuting congruences provided p o a = a o p for every p , a E @ ( A ) . 1.29. Theorem (Boicescu [1984]; cf. Beazer [1972] for M3).

Every L E LMn has commuting congruences. Proof. Let p l , p 2 E @(L).Then p h = modFh ( h = 1,2), where Fl and F .

L. Take (x,y ) E p1 o p2. Then there exist z E L and uh E Fh 1,2) such t h a t x A u1 = z A u1 and z A ug = y A u 2 . This implies

are n-filters of

(h =

x p l x A u 1 , yp2 yAu2 and x A u l A u 2 = y A u 1 A u 2 , therefore x A u l p 2 y A u l

x A uzp1 y A u2. Setting w = (x A u2) V ( y A u l ) and using t h e above

and

relations it follows that

i.e. x p ~ ~ a n d s i m i l a r l y y ptherefore(x,y) ~v, E p20p1. Thusplop,

and

p2

o p1

C p1

p20p1

o p2 by symmetry.

1.30. Remark. It is known that for every filter F of a Boolean algebra B and every x,y E B ,

(1.11)

x mod F y

* (x V jj) A ( 5 V y) E F

(x V jj) A (3 V y ) E F then x A u = x A y = y A u and conversely, if x A u = y A u for some u E F then using (2.18") we get because if u =

x V y V G = ( x A u ) V Y V G = ( y A u ) V j j V i i = 1, hence u (x

v

5 x

V y by (2.19") and similarly u

5

5 V y , therefore u

E F . This property was generalized by Varlet [1972] matter of fact it holds for LMn.

y) A (2 V y)

t o LM3 and as a

1.31. Proposition.

Let L E LMn

(1.12)

OT

5

L E Mn. FOTevery F E Fln(L):

zm o d F y

*

n-1

A (pix V 9 ; ~A ) i=l

V piy) E

F

Filters, ideals and 19-congruences

260

and every congruence p i s obtained in this way: p = mod o ker p. Proof. Using Proposition 1.14 and Remark 1.12 we get

zmodFy @

* z modFy * (i = 1,...,n - 1) ,

cpizmodFcp;y

whence (1.12) follows by Remark 1.20. The equality p = mod o ker p was obtained in the proof of Proposition 1.14.

1.32. Definition. Let us say that: a) An element

z of a .r-algebra A has property R ("regular") provided the

map @ ( A )+ ?(A) defined by p b) An algebra A has property

H

[z],, is injective;

R whenever all of its elements have property

R; c) A class

K: of valgebras has property R if all of its algebras have property

R. Varlet [1972] proved t h a t a double pseudocomplemented lattice has pro-

R if and only if z* = y* & z+ = y + + z = y . In particular L M 3 has property R and this characterizes L M 3 within t h e class of double Stone

perty

algebras. One can prove more: 1.33. Proposition.

The class LMn has property R. Proof. Notice first that the element 1has property R, because if [l]= ,, [1lPz then ker

p1

= ker

p2

whence Proposition 1.30 implies

p1

= p2.

Then take z E L E LMn and p I , p 2 E O ( L ) such that [z],,

= [z],,.

Using the LMn-structures of [0, z]and [z, 1 1 (cf. Theorem 4.4.16) one sees that

pi

= P k fl [ O , Z ] ~

P k fl [z, 11'

(k = 1,2)

(k

= 1,2) are congruences on [O,z] and pk" =

are congruences on [z,11. Furthermore

261

&filters, 8-ideals and 8-congruences

and since x is the unit o f [0, x] it follows that pi = pk and similarly ply’ = p2”.

Now take (u, v) E p1. Then (u A z, v A X ) E pi = p; and ( u V x, v V z) E

p2”, therefore u = u V ( u A X ) pz u V (v A z) = ( u V

proving that ( u , v ) E p2. Thus

p1

v) A

(u V x) p2

C pz and similarly pz

0

pl.

1.34. Remark.

The concept of filter in a Boolean algebra has been given an equivalent definition, very appropriate t o the purposes of symbolic logic: a filter is the same as a deductive system. The latter term designates a subset

D

of the

Boolean algebra B such that

(i) 1E D and (ii) if z E

D

and z + y E

D

then y E

D.

B . Then 1 E F and from x E F and x -+ y =? vyi E F we obtain z ~ = yx A ( % V Y ) E F hence y E F . Conversely, suppose D is a deductive system. Then D # 0 because 1E D. If x E D and x 5 y then x ---f y = 1E D therefore y E D.

To prove the equivalence suppose first that

F is a filter

of

If x , y E D then from

x

-+

x A y = Z V ( x A y ) = f V y L. y

we obtain z + x A y E D by the previous property, therefore x A y E

D.

It turns out that the above properties can be extended t o LM-algebras. The point is a convenient generalization of the Boolean implication, invented by A. Monteiro in 1963 for 3-valued Moisil algebras and called weak implication; cf. Cignoli [1969a]. Then Cignoli [1969a], [1970] defined and studied the weak implication and the deductive systems of Moisil n- algebras. Thus the results presented below are due t o A. Monteiro and Cignoli.

Filters, ideals and d-congruences

262

-

1.35. Definition. The weak implication on an LM-algebra is the operation

x

-

defined

M

y='PoxVy(Vx,yEL).

M

- -

1.36. Proposition.

The weak implication

has the following properties:

M

(1.13)

x

(y

M

(1.14)

(1.15)

x x

z)=l;

M

+ M

-

(y

M

x

y A z = (x

xVy

(x

M

Z = ( X

- -

-

y)

M

(1.19)

1

+ z ;

M

--t

+ M

z)A(y

+ M

2 = 2 ;

M

x=x;

M

(1.20)

z 5 y implies x -+

y =1 ;

M

(1.21)

x 5 y iff (pix

--t

M

(1.22)

2

--t

QoX=1.

M

Proof. Straightforward .

z);

M

(y + z ) = z A y

M

(1.18)

Y)+

M

M

(1.17)

- M

M

(1.16)

; y) + (x

z)=(x

+

(piy= 1

, Vi E I ;

z);

+ M

z);

by

263

6-filters, 6-ideals and 6-congruences

1.37. Definition. If X is a non empty subset of L then we say that x E L is a consequence of m

X if there is a finite subset { x l , ...,xm} of X such that A j=1

xj + M

x = 1.

Let us denote by C s ( X ) th e set of all consequences of X ; we put also CS(0) = (1).

1.38. Proposition.

C s ( X ) = [ X ) , for every subset X & L E LMn. Proof. Cs(0) = (1) = [0)8. Then

m

U

~

A

O j=1

X j S Y

and the proof is completed by Remark 1.6.

1.39. Proposition. Let L E LMn. A subset S

0

L is a deductive s y s t e m if and only if it is

a 6-filter. Proof. Let S be a deductive system. Then S m

x E Cs(S) then

A j=1

xj

M

# 0

because 1 E S. If

x = 1 for some x l , ..., xm E S, therefore

Filters, ideals and &congruences

264

...

z1 + (zZ + M

M

-+ M

...) = 1 by (1.16),

-+ z)

(I,

hence after

M

c

m steps we obtain z E S. Thus Cs(S) C S and since S [S)e = Cs(S) by Proposition 1.38, it follows t h a t S = Cs(S) = [S)s is a &filter. Conversely, let S be a d-filter. Then 1 E S. If z E S and z

yES M

then

(POX

-

E S n C ( L ) = S' and

(Pox

(POY

M

= pox V(P0y = (Po(p0z v y ) = = (Po(z

-

E S' ;

y)

M

9 is obviously a filter of th e Boolean algebra C ( L ) ,hence a deductive system of C ( L ) by Remark 1.34, therefore 'pay E S c S, whence finally y E s. 0 but

1.40. Proposition (the Deduction Theorem).

Let L E LMn, X (1.23)

L

and z, y E

L. Then

y E C s ( X U {z}) H z + y E C s ( X ) . M

Proof. We use Proposition 1.38, Remark 1.6 and the obvious property you

wH

(POU

I (POW.

F

Let

= {y E L I z

--f

y

5

E C s ( X ) } . The equivalences

M

show that F = { y E

L 1 cp0(zl A ... A z,

5 'pay}, whence it is straightX U {z} F and F E F' for any F = [XU (5))s. The above proof i s

F is a $filter, &filter F' such that X F'. Thus also valid for X = 8 if we replace z1 A

forward t o show t h a t

A z)

... A 2,

by 1.

265

Prime filters $2. Prime filters

The aim of this section is the study of prime filters and prime d-filters in a Lukasiewicz-Moisil algebra. A special attention is paid to minimal and maximal prime filters. 2.1. Definition. Let L f LMd. For

L and G

F

(1.1)

F* = F n C ( L ) ,

(2.1)

G;= cp;’(G)

C ( L )we set

(Vi E I ) .

A proper (prime) d-filter is defined as a d-filter which is also a proper (prime) filter. We denote by PFld(L) the set of all prime 8-filters on L. By a masimald-filter is meant a maximal element in the family of proper 8-filters ordered by set inclusion. 2.2. Lemma (Cignoli [1969a]).

Let L E LMd, F a filter on L and G a filter on C(L).Then for every i ,j E I : (i)

G;is a filter on L ;

(ii) i (iii)

<j

+ Gi

Gj;

(G;)*= G;

(iv) F* is a filter on C ( L ) and (F”)o

zF c (F*)l;

(v)

G is prime, or equivalently maximal, in C ( L )($ Gi is prime in L ;

(vi)

if F is prime in L then F* is prime, or equivalently maximal, in C(L).

Proof. Straightforward.

Filters, ideals and 29-congruences

266

2.3. Theorem (Cignoli [1969a], Georgescu and Vraciu [1969a], Georgescu

[197I d ]). Let L E LMd and F a 19-filter o n L. T h e n the following conditions are equivalent:

F is a m a z i m a l O-filter;

(i)

(ii) F* is a n ultrafilter o n C ( L ) ; (iii) F* is a p r i m e filter o n C ( L ) ; (iv) F is a prime 8-filter; (v)

either

(vi)

F is

'piz

E

F

OT t&a:

E F (Vi E I) (Vz E L ) ;

a m i n i m a l p r i m e filter.

Proof.

(i) H (ii): By Theorem 1.7. (ii) e (iii) e (v): By Proposition 1.3.18. (iii) e (iv): Because z V y E F e 'pox V 'poy E F*. (iv) (vi): Take a prime filter F' of L such t h a t F' c F . Then F'* c F* and F'* and F* are prime filters of C ( L ) by Lemma 2.2, hence they are maximal by Proposition 1.3.18, therefore F" = F*. Now it follows by Proposition 1.4 and Lemma 2.2 that F = (F*)o= (F'*), c F', therefore F = F'. (vi)

j

(iv): Trivial.

0

2.4. Remarks.

F is necessarily a &filter, because Lemma 2.2 implies in turn that (F*)o& F , F* is prime in C ( L ) , (F*)o is prime in L , therefore F = (F*)ois a &filter by Theorem 1.7.

a) Any minimal prime filter

b) Prime (or, equivalently, maximal) &filters coincide with minimal prime filters, by Theorem 2.3 and t h e above remark a). c) The map * establishes a bijection between PF16(L) and PFl(C(L)), by

Theorems 1.7 and 2.3.

Prime filters

267

Several characterizations of those lattices

L E DO1 in which maximal

Stone filters coincide with minimal prime filters were obtained by A. Monteiro [1954] and then by Cignoli [1971b], who started from a result of Cherciu

[1971]. 2.5. Proposition.

Suppose L E LMd, F as a 29-filter, 0 # S C L , F n S = 0 and x,y E S + x V y E S. Then there is a prime 19-filter P such that

PnS=Q)andFGP.

F* n (cp0S]c(~) = 0, otherwise x 5 cpos for some x E F* and s E S, which would imply cpos E F , then s E F , a contradiction. According t o Q and Theorem 1.3.10 there is a prime filter Q of C ( L ) such that F* 0 Q n (cpOS]c(~)= 0. Then P = Qo has the desired properties. Proof.

2.6. Remark. Proposition 2.5 yields corollaries similar t o Corollaries 1.3.11-1.3.14 of Theorem 1.3.10. Thus e.g. a) in a d-algebra every proper d-filter is included in a prime &filter and is

an intersection of prime &filters. Also,

b) given z,y E L with x

#

y, a necessary and sufficient condition for the existence of a prime d-filter F such that x E F and y $ F is cpox $ y.

2.7. Proposition.

The intersection of all maximal 9-filters of a n LM19-algebra L is (1). Proof. It follows from Theorem 2.3 that an element x belongs t o all maximal 6-filters

F of L if and only if cpOx belongs to all maximal filters F* of C(L),

and this is equivalent t o pox = 1 by Theorem 1.3.28, therefore t o x = 1 by (3.1.27) and (3.1.2). 2.8. Definition. Let us say t h a t a congruence (a d-congruence) is:

(i) proper if it is distinct from the universal congruence L2,and

0

268

Filters, ideals and 6-congruences

(ii) maximal if it is a maximal element in t h e family of all proper congruences (proper 2')-congruences)ordered by set-inclusion

C.

According t o a general definition of universal algebra, an algebra A i s called semisimple if the intersection of all its maximal congruences is the trivial

congruence A.A. 2.9. Corollarv.

The i n t e r s e c t i o n of a l l maximal 19-congruences of an LMb-algebra L i s

AL. Proof. From Proposition 2.7 and Theorem 1.13.

0

2.10. Corollary. LMn-algebras and Mn-algebras are semisimple. Proof. From Corollary 2.9 and Proposition 1.14.

0

2.11. Remark.

LM6 (LMN1S)-algebras are also semisimple, because maximal 6-congruences are identical with maximal congruences. 2.12. Proposition.

The following conditions are equivalent f o r a prime ideal I of an algebra

L E LM6: (i)

I

(ii)

In Ds(L) = 0;

i s a 2')-ideal;

(iii) for every x E L, e i t h e r x E I o r x* E I. Proof.

(iii): If x E I then 'plx E I hence x* = Cplz # I because I is proper. If x # I then 'plx# I hence the prime ideal I f l C ( L ) of C ( L ) (cf. the dual of Lemma 2.2) contains Cplx (6. t h e dual of Proposition 1.3.18).

(i)

j

(iii)

+ (ii):

contradiction.

If x E I n D s ( L ) then x E I , z* = 0 and x*

# I,

a

Prime filters

269

l(i) =+ l(ii): If z E I and cp1z (i!I then as for (i) + (iii) we get Cplz E I n C ( L ) , hence z V Cplz E I and since cpl(z V plz) = 1 it follows that z V qlzE I n Ds(L).

0

2.13. Remark. In this book there are several uses of t h e symbol Ker which should not be confused, even if they are not distinguished by subscripts. The first meaning is the one used in universal algebra: if A, A' are r-algebras then

(2.2)

kerA,At

: Hom(A,A') + @(A)

is defined as in (1.5.6), i.e. for every f E Hom(A,A'), (2.3)

*

~(ker~ f , )~ ~ ) f(z) = f

( ~ )( V ~ , YE A ) .

The first occurrence of this concept was in formula (1.3.8), in t h e particular case of lattices. As a matter of fact we are interested in t h e more particular case of algebras L , (2.4)

kerLp

L' from LM9 or LMNd. Note that in this case

: Hom(L,

L')

-+

Oo(L)

because if p;z(kerL,Lt f)p,y (Vi E I) then cpif(4

= f ( c p i 4 = f(cpiY) = Vif(Y)

(Vi E I)

therefore f(z)= f(y) i.e. z(kerL,Lt f)y. The next meaning of ker was first used in Theorem 1.13, again for L , L' E LM19 or LMNB: (1.4)

kerL : Oo(L)-+ F16(L)

,

(1.7)

ker p = { z E L I z p l }

(Vp E O O ( L ) )

A third and similar meaning will be introduced now. 2.14. Theorem (Cignoli [1969a], Georgescu [1971d]).

Let C = LM19 or C = LMNB. For every L E C there is a bijection between PFld(L) and Homc(L, L;'). Proof. Set Homc(L,L, UI) = H and KerL = kerL o k e r L , y i.e., for each

f EH,

Filters, ideals and d-congruences

270

K e u f = 1. E L I s(kerLLrJl ' 2 f)ll = f-1([1))

(2.5)

7

l(i) = 1 (Vi E I). As L;' is a chain by Lemma 4.6.11it follows that [l) is a prime d-filter

where 1 E Li',

of Li', hence it is straightforward t o check that f-'([l))is a prime 6-filter of L. Thus KerL :

H + PFlO(L).

Now we prove the existence of KerL', namely

(Keri'F)(x)(i) =

(2.6)

(

1

,

vix E F

0

7

vix $ F

(Vx E L) (Vi E I ) ,

for each F E PF16(L). Let f ( x ) ( i ) stand for the right side of

f(x) E ' iL

(Vx E L).

The next step is t o check that every

(2.6).Clearly

f

E

H . Using (3.1.11) we

see that for

x E L and i,j E I,

* Yjcpix E F H Fix E F =1* (f(x))) (.i> =1

f(vjx)(j) = 1 H

f(

(vi

e

7

proving that f(cpiz) = cpi ( f ( z ) ) .One verifies similarly, using Remark 1.3.5.

1.3.9,that f i s a bounded- lattice homomorphism. In view of Remark 3.1.29 (and of Remark 1.2.29 if C = LMNd) this shows that f E H . Then using Remark 1.2 we obtain a) and Proposition

x E ker f

* f(x) = 1 H f(z)(i) = 1

H cpix~F

satisfies

I) e

(V~EI)HZ€F,

f = F . Finally it remains ker g = F then g = f. This follows from

which proves t h a t ker

(Vi E

t o show that if g E H

271

Prime filters 2.15. Corollary (Georgescu [1971d]). If L E P19 t h e n a) every m o r p h i s m f :

b) L'\

L +L '\

is a surjection, and

has n o proper Post subalgebras.

Proof.

ra

(C(L)) , hence Proposition 4.1.6 yields an element x E L such that 'pix = u ( i ) (Vi E I). Notice that f = ker-' ker f by Theorem 2.14 and f ( u ( i ) ) = 1 or 0

a) Take a E

Lia. But

a can be viewed as a funtion a E

according as u ( i ) = 1or 0. Then f(x) = a because

f(z)(i) = 1 H 'pix f ker f H u(i) E ker f

* f (a(;)) = 1

($

a(;) = 1 ,

b) Apply a) t o the inclusion mapping of a subalgebra L of LiA.

0

2.16. Corollary.

T h e following conditions are equivalent

L E P19 and f E F119(L):

~ O T

(i) F is a maximal 19-filter;

(ii) L / F Proof.

(i) j ( i i ) : The morphism f = Ker-'F : L + is a surjection by Corollary 2.15, therefore L'\ FZ L / ker f by the homomorphism theorem 1.5.16. Thus it remains to prove that ker f = modF. First we apply (2.6) and obtain

Now if z ker f y then for each i E I, (2.7) and Theorem 2.3 imply that either ( p i x , (piy E F in which case we set uj = 'pixA (PigI or (pix, Cpiy E F in which case we set u;= pix A p;y. In both cases

Filters, ideals and 29-congruences

272

that is z m o d F y . Conversely, (2.8) implies that for each i E I,if via: E F then (piy A ui E F hence (pig E F and similarly (p;y E F + pix E F therefore z ker f y by (2.7).

(ii) + (i): Suppose F’ is a proper 29-filter of L such that F E F‘. Then = {2 I z E F’} is a proper &filter of L/F by Proposition 1.19. As 0 remarked after Definition 1.1, this implies F? = {i},that is F‘ = F . 2.17. Proposition. T h e following conditions are equivalent f o r L E LM29 and F E Fl(L):

(i) F is mazimal; (ii) there is a p r i m e filter G of C(L) (necessarily unique: G = F’) such that F = GI. Proof.

(i) + (ii): In view of Lemma 2.2 we obtain in turn that F* is prime in C ( L ) , F C (F*)1 and (F*)l is prime in L , hence proper; therefore F = ( F * ) l . If F = G1then F’ = (Gl)* = G by Lemma 2.2. (ii) =$ (i): If F’ is a proper filter of L such that F F’ then G = F’ (F’)’ C C(L) hence F’ = (F‘)’ therefore F = F’ by Lemma 2.2. 0 2.18. CorolI a ry . If L E LM29 and F E PFl(L) then F’ = (F*)oand F” = ( F * ) l are the unique m i n i m a l p r i m e filter F’ F and the unique m a x i m a l filter F” 2 F , respectively.

s

Proof. In view of Lemma 2.2, F’ is a maximal filter of C(L) and F‘ C F E F”. But F’ is a maximal &filter by Theorem 1.7, therefore a minimal prime filter by Theorem 2.3, while F” is a maximal filter by Proposition 2.17. If P is a maximal filter such that F P then F* & P” and F’ E F”’, therefore P” = F’ = F” hence z E P implies (pox E P’ = F”’ F” then z E F”, which shows that P E F”, therefore P = F”. The uniqueness of F’ is

s

Prime filters

273

proved analogously.

A. Monteiro [1974]has shown that for L E D01, every prime filter is included in a unique maximal filter if and only if L is n o r m a l i.e. for every x,y E L satisfying x Ay = 0, there exist u , v E L such that X A V = y A u = 0 and u V v = 1. Then it follows from Corollary 2.18 t h a t every L E LM9 is normal; this can be also proved directly taking u = 'plyVx and v = cplxVy.

2.19. Definitio n. Let L E LM9. Denote by Rad(L) the intersection of all maximal filters of L. For every proper filter F of L let Rad(F) stand for t h e intersection of all maximal filters of L that include F . 2.20. Proposition. Let L E LM19 and set M = PFl(C(L)). Then

(2.9)

Fl = Ds(L) .

Rad(L) = FEM

Proof. The first equality follows from Proposition

Rad(L) = 9';

( FEM n

F ) = cp~'([l))

2.17,implying further

= Ds(L) .

2.21. Proposition. Let L E LM9 and F a proper filter of L. Then Rad(F) = cpT'(F). Proof. Let

MI be the family

L which include F that M1 = {G1 I G E

of those maximal filters of

M z = { G E PFl(C(L)) I F* G G } ; we claim M z } . If F' E M1 then F' C F", while Proposition 2.17 shows that F' = ( F f * ) land F'* E PFl(C(L)), hence F'* E M2. Conversely, suppose F' = G1 where G f M 2 . Then F' is maximal by Proposition 2.17 and 2 E F implies cplx E F* G G hence x E GI = F', proving that F 5 F', therefore F' E M I . Now using Corollary 1.3.14in C(L) for F* we obtain and

Rad(F) =

n GEM2

cp;l(G)=cp;l(

n GEM2

G)

=

Filters, ideals and 19-congruences

274

The next result connects filters t o the operations Ch. 4, $3.

and

% studied in

2.22. Theorem. The following conditions are equivalent f o r L E LMn: (i)

L is a Boolean algebra;

(ii) Rad(L) = [l);

(iii) every proper filter F is a n intersection of maximal filters, i.e. Rad(F) = F ; (iv)

x

+ y = ( P ~ - v~ y;x

Proof.

(i) + (iii): By Corollary 1.3.14 and Proposition 1.3.18. (iii) + (ii): In particular Rad([l)) = [l); but clearly Rad([l)) = Rad(L ) . (ii) + (i): As x A x* = 0, it remains t o prove that x V x* = 1. But x V x* E D s ( L ) , because (x V x*)* = 0 and D s ( L ) = [l) by Proposition 2.20. (i) + (vi): By Proposition 4.3.6. (vi) + (v): By (4.3.25) and (3.1.27) where 0 is 1. (v)

+ (i): The

(1

+ x) V (x +- 0)

= 1, that is : ( p ; ~ V c p ~= - ~1,xi.e. ( P ~ - 5 ~ X(pix.

hypothesis yields

xVcp,-lx = 1, whence for each i c I Thus (pix = (P,-~X (Vi E I ) , which shows that L (i) + (iv): By Proposition 4.3.6.

= C ( L ) E B.

275

Prime filters (iv)

+ (i): x* V x = qn-1x V I = x =$ x = 1.

0

2.23. Remark. Properties (i)-(iii) in Theorem 2.22 are equivalent even for L E LM6 (same proof).

A. Monteiro (cf. A. Monteiro [1974]) raised the problem of whether a bounded distributive lattice in which every proper filter is an intersection of maximal filters and dually, is a Boolean algebra. The answer is negative, as was shown independently by Balbes [1972] and Adams [1974]. However for

L E LMn the answer is positive; cf. Theorem 2.22. 2.24. Proposition.

The fol2owing conditions are equivalent for L

E LMt9 and

F E PFl(L):

(i) F is maximal; (ii) D s ( L ) Proof.

c F.

*

(i) (ii): By Propositions 2.19 and 2.16. (ii) =$ (i): Let F' be a filter such that F c F'. Take 5 E F' - F. From yl(x V p1x) = 1 we infer I V Flx E D s ( L ) 5 F and since F is prime it 13 follows that $311 E F 5 F', therefore 0 = I h Cplx E F', i.e. F' = L.

In the case of LMn-algebras we can say much more about prime filters. 2.25. Theorem (Boicescu [1984]). T h e following conditions are equivalent f o r L E LMn and a proper filter F of L:

(i)

F is prime;

(ii) there is a prime filter

F' F ;

(iii) ihere is a unique minimal prime filter F'

F;

276

Fiiters, ideals and 6-congruences

(iv) F* is prime in C ( L ) ; (v)

there is a prime filter G o f C ( L ) (necessarily unique: G = F*) and i E (1, ...,n - 1) such that F = G;;

(vi) F is included in a unique maximalfilter. Proof.

(i) j (iii): By Corollary 2.17. (iii) + (ii). Trivial. (ii) +- (iv): F’* E PFl(C(L)) by Lemma 2.2 and F’* = F*.

C_ F*, therefore

c

F (F*)n-l by Lemma 2.2. We prove that F = (F*)jfor j = max {il(F*)iC F } . If j = n-1 then (F*),-lC F hence F = (F*)n-l. If j < n - 1suppose by way of contradiction the existence of x E F - (F*)j. Then ‘pjx @ F’ hence cpp E F*. But (F*)j+l F hence there is y E (F*)j+l such that y @ F . Using Proposition 3.2.3we obtain y 2 x A Cpjx A ( ~ j + E ~ yF therefore y E F , a contradiction. If F = Gi for G E PFl(C(L)) then G = F’ by Lemma 2.2. (v) +- (i): By Lemma 2.2. (i) 3 (vi): By Corollary 2.17. (vi) + (v): F is an intersection of prime filters F’ by Corollary 1.3.14. Each F’ is included in a maximal filter P by Corollary 2.18. But F 5 P therefore P is unique by hypothesis. Then P* and each F” are maximal in C(L)by Lemma 2.2 and also F” P*,therefore F‘* = P* for each F‘. According t o (iv) 3 (v), for each F‘ there i s a j such that F‘ = ( P * ) j ,which implies F = ( P ” ) k ,where k is th e least j. This representation is unique with 0 respect t o P* again by Lemma 2.2. (iv) 3 (v):

(F*),

F“

c

Comment . Theorem

2.25 shows that if 6 = n certain properties of &filters can be

(iii) extended t o filters. Thus the equivalence (i) ($ (iv) generalizes (iv) in Theorem 2.3; (i) ($ (iii) and (i) (vi) generalize Corollary 2.18 and the lattices sharing this property have been characterized by A. Monteiro

[1974]. In that paper it

is also proved that for

L

E

D01,every proper filter

277

Prime filters

which includes a prime filter is also prime if and only if L is fully normal; cf. Definition 4.3.14. The above equivalence (i) e (ii) implies that every LMn-algebra is fully normal; cf. Proposition 4.3.16. The equivalence (i) ($ (v) generalizes a result of Cignoli [1969a] on Moisil n-algebras (see also Cignoli [1975a]). 2.26. Corollary (Cignoli [1969a], [1975a]). If L E LMn then t h e poset PFl(L) is the cardinal s u m of a set of chains of cardinality 5 n - 1 each: (2.10)

{G; I i = 1, ...,n - 1)

, G E PFl(C(L)) .

Proof. Taking into account Theorem 2.25 it only remains to prove that every two elements belonging to distinct chains (2.10) are incomparable. If Gi C Gi for G,G‘E PFl(C(L)) then G = (G;)’ (G>)* = G‘, therefore

G = G‘.

0

2.27. Corollary (Epstein [1960], Traczyk [1963]). I f P E Pn then each chain (2.10) has exactly n - 1 elements. Proof. For each G E PFl(C(L)) and each i = 2, ...,n - 1, from (pic,-i = 1 E G and ( P ; - ~ C , - ~ = 0 @ G we infer c,-i E G; but c,-i @ Gi-1, therefore

Gi-1

c Gi.

0

The converse of the above result will be the object of the next section 2.28. Corollary. If L E LMn, F’,F” E PFl(L) and F’, F” are incomparable t h e n F’VF” = L. Proof. If F’V F” is a proper filter then there is F E PFl(L) such that F’VF” F , which implies F” C F* and F”’ E F’ hence F” = F” = F’”, therefore Theorem 2.25 yields F‘ = (F*)i and F” = (F*)j for some i , j . 0 This contradicts the incomparability of F’and F”.

278

Filters, ideds and d-congruences

2.29. Proposition. If L E LMn is infinite then card L

5 card PFln(L) = card PFl(C(L)) ,

Proof. It follows from Remark 2.4.c and Corollary 2.26 that (2.11)

cardPFl(C(L)) = cardPFln(L) 5 cardPFl(L)

5

5 ( n - 1)cardPFI(C(L)) . But C ( L )is infinite by Proposition 1.25, hence cardC(L) 5 cardPFl(C(L)) (Makinson [1969]), therefore PFl(C(L)) is infinite and (n-l)cardPFl(C(L))

= cardPFl(C(L)) which together with (2.11) yield

cardPFl(L) = cardPFl(C(L)), while cardL Makinson result.

5 cardPFl(L)

again by the 0

Proposition 2.29 cannot be generalized to infinite 6; thus e.g. Liq is infinite while cardPF1d(Lhq) = 1 as noticed after Remark 1.2. 2.30. Remark. The theory presented in this section can be extended in several ways. One of them has been already mentioned: the passage from LM-algebras to bounded distributive lattices. Another idea is followed in Sicoe (19711: the study of the ideals of the form c p i ( K ) or cp;l(IC), where i E I is fixed and I< runs over the ideals of L. lorgulescu [1984c] has obtained a natural generalization of Theorem 2.3 and several related results within the framework of m-concepts; cf. Remarks 1.3.31 and 1.20. In particular it is noted that every maximal 9-filter which is also an m-filter is obviously a maximal (m, d)-filter but it is not known whether these two concepts coincide and their relationship to the concept of maximal 19-filter is also an open problem. lorgulescu [1984c] also proved that in the d-valued algebras (d-infinite) the equivalence (i) % (v) in Theorem 2.25 does not hold.

The Cignoli problem

279

53. The Cignoli problem Epstein [1960] and Traczyk [1963] proved t h a t i n every Post algebra following property

L the

P holds: th e set PFl(L) o f prime filters of L is a cardinal

sum of ( n - 1)-element chains. In his thesis, starting f r o m Moisil algebras, Cignoli gave a new proof o f the above theorem (cf. Corollary 2.26) and asked whether the converse was true. More precisely: is every Moisil n-algebra w i t h

P a Post algebra? In this section we prove t h a t the answer is affirmative even in the more general case when L E LMn; cf. Boicescu [1984]. property

3.1. Lemma.

Let L E LMn, with property P . If F E PFl(C(L)) let Fl

c F2 c ... c

be the chain of prime filters in L determined by F (cf. L e m m a 2.2). FOT each i = 2, ...,n - 1, there exists x b E L, such that:

F,-1

(i) x & E ~ - - - * ( i = 2 ,...,n-I),

(iii) 'pix> =

= ... = ' p n - l x $ .

3.2. Lemma.

Let B be a n n-valued Post algebra and F E PFl(C(B)) and i E ( 2 , ...,n1). T h e n the elements

satisfy conditions (i)-(iii) of L e m m a 3.1. Conversely, every element 2; satisfying conditions (i)-(iii) of L e m m a 3.1 has a unique representation

Filters, ideals and &congruences

280 of the f o r m (3.1), n a m e l y yk = 'pix.

Proof. It follows from (3.1) that 'pjx; = 0 for all j < i and 'pix; = y$ for all j 2 i. Therefore (ii) and (iii) hold. In particular ' p i - 1 ~ ; = 0 hence x; $ F,-1, while 'pixi = yk E F hence 3% E F;. Thus (i) holds and the representation (3.1) is unique. Conversely, if Z> = cn-i

&EB

satisfies conditions (i)-(iii) of Lemma 3.1, then

A 'pix; foIIows from

3.3. Lemma. Under the hypotheses of L e m m a 3.2, there exists a finite subset 'Po C PFl(C(B)) such that

Proof. In view of Proposition 1.3.9, the elements of SpecC(B) are the sets L - F , F E PFl(C(B)). But y> g! L - F hence L - F E s(y$) (cf. Notation 2.1.5) for each F E PFl(C(B)), therefore SpecC(B) = U{s(yk) I F E PFl(C(B))}. Now the compactness of SpecC(B) and Corollary 2.1.6 imply the existence of a finite subset Po S PFl(C(B)) such that

whence (3.2) follows by injectivity of s.

0

3.4. Theorem. If L E LMn and satisfies condition P , t h e n L i s a n n-valued Post afgebra. Considering the canonical monomorphism (4.1.3) F'L : L + C(L)["-'I, it follows that L can be identified with the LMn-subalgebra PL(L) Proof.

The Cignoli problem

281

C(L)["-l]and it is sufficient to prove the theorem for PL(L). As C(L)["-'] is a Post algebra and therefore has property P , it follows that for any F E PFl(C(C(L)"+'])), the chain of prime filters in C(L)"+'], ( ~ ) l ~ i ~ nis- strictly l increasing. As

of the n-valued Post algebra

(3.3)

C(PL(L))= C(C(L)["-']),

P,it follows that the chain of prime filters in PL(L) F1 n PL(L) c FZ n PL(L) C ... C Fn-1 n PL(L).

and PL(L)has property

F

determined by

is

According t o Lemma 3.1, there exists z& E PL(L) satisfying conditions

(i)-(iii). However zk belongs t o the Post algebra C(L)"+'] and therefore by applying Lemma 3.2, it follows that there exists y b E F , such t h a t zb = c,-i A yk, for any i = 2, ...,n - 1 and any F E PFl(C(PL(L))). According t o Lemma 3.3 there exists a finite subset Po c PFl(C(PL(L)))

V

such t h a t

yk = 1. But in

C(L)fn-']

FEPo

V

and because

z& E PL(L),it follows that c,-; E PL(L)i = 2, ...,n-1.

FEPo

Using again (3.3) we see that PL(L) is an n-valued Post algebra.

0

Comment.

The above proof implies in fact (3.5)

4;n-i

= V{z$ I F E PFI(C(PL(L)))}

(i = 2, ...,n - 1)

2 z$ for all F and if z 2 zb (VF) then x 2 c,-i by (3.4). As a matter of fact (3.5) holds in PL(L)because cn-i E PL(L). Conversely, a direct proof of (3.5) i.e., for L and without using Lemma 3.2 and 3.3because c,-i

would yield a direct proof of Theorem 3.4-i.e.

without using Post algebras

or topological arguments. We are able t o give such a proof in the case when

L is complete.

282

Filters, ideals and 19-congruences

3.5. Proposition. Let L E LMn be complete and with property P. Then (3.6)

( i = 2, ...,n - 1) ,

V { x b I F E PFl(C(L))} = cn-;

where x f . are the elements constructed a'n Lemma 3.1 and cn-i satisfy (4.1.9)(cf. C o r o k l ~ y4.1.9). Proof. We adopt (3.6) as a notation and prove (4.1.9), i.e. cpjc,-i = 1or 0 according as j

Let i E (2,

2 i or j < i. It suffices t o prove ( P ; c , - ~= 1and ( ~ i - ~ c = ~ -0.i

...,n - 1).

F E PFl(C(L)), from cn-i 2 x ; and x ; E F; we obtain (Pic,-; 2 (Pix; E F , hence Propositions 2.7 and 1.3.18 imply For every

(Pic,-i

E

n {F I F E PFl(C(L))} = (1) .

Now suppose by way of contradiction that exists F E PFl(C(L)) such that follows t h a t

~ p i - ~ A ~ -cpix; i

we infer x;,

5 pix;

E

(Pi-1Cn-i

which contradicts (3.7).

#

(~i-~c,-i

0. Then there (pix& E F it

F and since

F ,hence

V x ; . Therefore cn-i

turn

E

5 q ; x & V 2% and this

implies in

The Cignoli problem

283

We reminded above t h a t if L E DO1 and L is an n-valued Post algebra, then L has property P . The converse is not true, as was proved by Chang and Horn [1961]. They have considered L = L:, where S = {sili E w @ 1). Obviously L E DO1 and let L1 be the subset of L formed by the elements such t h a t f ( s n ) = c2, for any n E w, n 2 n o , for some no E w and f(sw) = c1 or c2, or f ( s n ) = co, for any n E w, n 2 no, for some noE w and f(sw) = co. One proves that L1is a sublattice with 0 and 1of the lattice L and has property P , but L1 is not a three-valued Post algebra. One can say even more:

3.6. Proposition. L1 E DO1 defined above has no structure of three-valued LukasiewiczMoisil algebra. Proof. As

L1 satisfies condition P

but cannot be a three-valued Post alge-

bra, this follows from Theorem 3.4.

0

This proposition was initially proved by Cignoli [1969b]. Chang and Horn [1961] and then Traczyk [1963] gave necessary and sufficient conditions for a bounded distributive lattice having property P t o be a Post algebra, but in those characterizations appear the constants ci from the definition of a Post algebra.

3.7. Corollary. If L E DO1 and satisfies property P , then the following are equivalent:

(i) L is a n n-valued Post algebra;

(ii) L has a structure of n-valued Lukasiewicz-Moisil algebra. 3.8. Corollary. If L E DO1 and satisfies property P , then L has a unique structure of n-valued Lukasiewiez-Moisil algebra compatible with the lattice structure. Proof. By Theorem 3.4 it follows that L has a structure of n-valued Post algebra. This structure is unique (cf. Balbes and Dwinger [1974] p. 192

284

Filters, ideals and &congruences

or Exercise X.2.8) and it determines uniquely the structure of LukasiewiczMoisil algebra; cf.

(4.1.11)and Corollary 4.1.9.

0

3.9. Corollarv.

If L is a n n-valued P o s t algebra then L has a unique structure of n-valued Lukasiewicz-Moisil algebra compatible with t h e lattice structure of L.

3.10.Corollary. If L E DO1 and PFl(L) as a cardinal s u m of chains of cardinality 2, t h e n the following are equivalent:

L i s a double pseudocomplemented lattace;

(i)

(ii) L E M3; (iii) L is a three-valued Post algebra; (iv) L is a double Stone algebra; (v)

L i s a double pseudocomplemented lattice and D s ( L )n

# 0.

Proof.

(i)

by Varlet [1968]. (ii) + (iii): By Theorem 3.4. (iii) + (iv): By Theorem 4.4.4. (iv)

(iii)

=$-

(ii): Proved

+ (i): Obvious. + (v): There exists c1 E L , such that (plcl= 0 and cpzcl = 1. By

Proposition (v)

4.4.6,c1 E D s ( L ) n

(i): Obvious.

m.

0

285

CHAPTER 6 REPRESENTATION T H EOR E M S AND DUALITY FOR LM-

ALGEBRAS

Various ideas have been used in the literature i n order to describe the structure of LM-algebras in "more concrete" terms: algebraic representations as subdirect products of simple algebras, representations as algebras o f sets or of fuzzy sets, representations as algebras of continuous functions, dualities between th e categories of LM-algebras w i t h and without negation and certain categories of topological spaces, construction of n-valued algebras i n terms o f three-valued ones.

$1. The representation theorem of Moisil One of t he most important results in the theory of Lukasiewicz-Moisil algebras is t he representation theorem given by Moisil i n 1941 for t h e threevalued algebras, in 1963 for th e n-valued algebras and in 1968 for t h e 9valued algebras. In this section we give a sharpened version of t h e Moisil theorem and study various important particular cases of it.

1.1. Proposition.

Let L E LM6 (or L E LMN6) and F E Fld(L). FOTevery z E L (z E C(L))let [z](let 2,) denote its equivalence class with respect t o F ( t o F* = F n C(L)).T h e n

(1.1)

? = [x]n C(L) (VX E C(L))

and the mapping [z]+ p=

(1.2)

(V[z] E C ( L / F ) )establishes a n isomorphism

C ( L / F )2 C ( L ) / F *.

Proof. Let z E for some u

C(L).If y E [z]f l C ( L )then y E C(L)and z A u = y A u

E F , hence 2 A you = y A you where

'pou E

F* therefore y E Z.

286 Conversely, if y

Representation theorems and duality for LM-algebras

E 2 then y E C(L)and x A u = y A u for some u E F n C ( L ) ,

E C(L)n [XI. If [x]E C ( L / F ) then [pox] = pO[s]= [x],hence (1.1) implies p l = [x]fl C(L), which shows that the mapping [x]H p 5 is well defined. It is clearly a homomorphism and also a surjection, because if x E C ( L ) then [x]E C ( L / F ) and 9% = 2. To prove injectivity suppose p 5 = pyy for [x],[y] E C ( L / F ) ;then poxAu = p o y A u for some u E F n C ( L ) ,therefore hence y

0

1x1 = [ C P O ~ I = [VOYl = [Yl.

1.2. Lemma. Let L E LMd (or L E LMNd) and (Fs)SE~ G Fld(L). Let f : C ( L ) t (C(L)/F,') be the morphism f ( z ) = (i?),Es and is : C(L)/F,*-,

n

JES

C(L/F,) the isomorphisms is(?) = [x], (cf. Proposition 1.1). Then there is a unique morphism f : L + (L/F,) such that f I C(L)=

n

SES

Fig. 1.1. Proof. Set f(x) = ( [ S ] , > , ~ S .

The representation theorem of Moisil

287

The above results enable us t o obtain representations of LukasiewiczMoisil algebras from representations of Boolean algebras, as we are going t o show below. The basic tool is the concept of subdirect decomposition in Definition 1.5.18 and Birkhoff’s Theorem 1.5.19. That theorem is based on a lemma according to which for every algebra A and every set of congruences 19, (s E S) of A such that

n

19, = AA, t h e quotient algebras

sES

A/19, are the factors of a subdirect decomposition of A and conversely, every subdirect decomposition of A can be obtained in this way. Then it is proved that one can choose the congruences 6, so as t o fulfil t h e above condition and besides, each algebra A / 6 , be subdirectly irreducible. Moreover, if X: is an equational class and A E X: then each algebra A / d , E K: therefore every algebra of a n equational class has a subdirect decomposition i n t o subdirecth~irreducible factors, within that class. 1.3. Proposition. Let L E LM19 ( O T L E LMNS). There is a bijection between the subdirect decompositions of L in LM19 (in LMN19) and the subdirect decompositions of the Boolean algebra C ( L ) , a n y two such decompositions being connected by the diagram in Fig. 1.1. Proof using t h e above universal algebraic lemma. The factors of the former decompositions are of the form L/19, E LM19 (LMN19), hence 19, E Oo(L).

Now apply Theorem 5.1.13, Proposition 1.3.26, Theorem 5.1.7 and Lemma 1.2.

0

1.4. Definition. An algebra L E LM19 ( L E LMNB) is &simple provided cardL

>

1 and

Oo(L) = {A,r,,L2}. 1.5. Proposition.

Every algebra L E LMI? (L E LMNd) has a subdirect decomposition i n t o E~ the f a m i l y of all m a the 19-simple factors L/Fs (s E S), where ( F s ) s is ximal 19-filters of L.

288

Representation theorems and duality for LM-algebras

f is injective by Theorem 5.2.3,Prop1.3.13,therefore f is a monomorphism, while

Proof. Apply Lemma 1.2 noticing that osition 1.3.18 and Corollary

n

each L / F, is obviously &simple.

1.6. Theorem (Cignoli [1969a], [1975],Boicescu [1984]). If L E LMd or L E LMN.9 and card L > 1 t h e n the folZowing conditions are equivalent: (i)

L i s simple;

(ii)

L i s d-simple;

(iii) Fld(L) = { L ,[l)}; (iv)

C ( L ) Z L,;

(v)

L is isomorphic t o a subalgebra of L [rl , ;

(vi)

L is a chain;

(vii)

all the filters of L are prime;

(viii)

L i s subdirectly irreducible;

(ix)

L is 9-subdirectly irreducible;

(x)

L is directly irreducible;

(xi)

L is 8-directly irreducible.

Proof. The implications (i) j (ii),(vi)

+ (iv),

(viii)

+ (ix),

(viii)

+ (x) +

(xi) and the equivalence (vi) H (vii) are trivial.

(ii) + (iii): Follows from Theorem 5.1.13. (iii) + (iv): The Boolean algebra C(L)i s simple by Theorem 5.1.7, therefore Proposition 1.3.29implies (iv). (iv) + (v): L S PL(L),where PL is defined by (4.1.3),while P1;(L) is a subalgebra of C(L)[ qS {O,l}[q. (v) + (i): Let p E O ( L ) , p # 01;. Take (x,y) E p. x # y. Take i E I such t h a t x(i) # y(i), e.g. x(i) = 0 and y(i) = 1. Then 'pix = 0 and

The representation theorem of Moisil 'piy = 1, therefore ( 0 , l )

E p hence

289

z = z A 1p z A 0 = 0 for every z, i.e.

p = 1L.

=+ (vi):lf z,y E {O,l}[q and x $ y then ~ ( i=) 1 and y(i) = 0 for some i E I,hence j < i + y(j) = 0 5 ~ ( jwhile ) j > i + z ( j ) = 12 y f j ) , therefore y 5 x. (v)

(i) + (viii): Follows from t h e characterization of subdirectly irreducible algebras, see e.g. Pierce [1968], Lemma 2.1.6. (ix) =+ (ii): By Proposition 1.5. (xi) + (iv): Take a E C(L). Then L S (a] x [a) by Lemma 4.2.24, 0 (a] or L E [a), that is a = 1 or a = 0. therefore L 1.7. Remark. Proposition 1.5 and Theorem 1.6 imply that every L E LM6 ( L E LMN6) is a subdirect product of subdirectly irreducible d-valued algebras (8-valued algebras with negation). This result cannot be obtained using the Birkhoff

theorem because the classes LMd and LMNd need not be equational; cf. Remark 5.1.18. Moisil ([1963b], [1968]) proved that every Mn-algebra (LMS-algebra) can be embedded in a direct product of copies of L, (of

Lbq). These pro-

perties are sharpened in the next two results.

1.8. Theorem. Every algebra L E LM6 (or L E LMN6) is a subdirect product of subalgebras of L2 FI. Proof. Immediate from Proposition 1.5 and Theorem 1.6.

0

1.9. Corollary (Cignoli [1968a], [1975]). Every algebra L E LMn (or L E Mn) is a subdirect product of subalgebras of LnProof. Notice that LF-']

Ln;cf. Corollary 4.6.13.

0

Representation theorems and duality for LM-algebras

290 1.10. Corollary.

E v e r y algebra L E Pz') is a subdirect product of copies of Li'. Proof. Use Corollary 2.5.15.b.

0

1.11. Corollary (Epstein [1960]). E v e r y algebra L E Pn is a subdirect product of copies of L,. Note t h a t t h e converse fails: thus e.g. t h e algebra L in Example 4.5.25 is a subdirect product of copies of

L3 but it is not a 3-valued Post algebra

(e.g. by Corollary 4.1.9). 1.12. Lemma.

L, a n d L2 are the only axled n-algebras t h a t are subalgebras of L, in LMn.

Proof. Suppose t h e axled algebra L'

is a subalgebra of L, in Mn. If 0 is an

axis of L' then L' is a Boolean algebra by (4.2.2) therefore L' = L z . Suppose 0 is not an axis of L'. But (4.2.1) shows that 1 is not an axis either, therefore the axes of L' belong t o L, - L'. Thus L' is not a Boolean algebra, hence it has n - 2 distinct axes by Proposition 4.2.4, hence L' = L,.O

1.13. CorolIa ry . E v e r y axled n-valued algebra L is a subdirect product of algebras L , and

L2, in t h e class of axled n-algebras. Proof. If L is axled th e algebras L / F s in the proof of Proposition 1.5 are also axled because th e class Azn of axled algebras is equational. Also, the involved morphisms preserve the axes, therefore t h e decomposition is va-

lid in Azn. But each L/Fd is a simple Mn-algebra, hence a subalgebra of LP-l1 E L, by Theorem 1.6, therefore it is L , or L2 by Lemma 1.12. 0

Our next concern is t o find conditions under which the subdirect decomposition in Theorem 1.8 becomes a direct decomposition.

The representation theorem of Moisil

291

1.14. Proposition.

Let L E LMd and F a filter of L. T h e n : (i) F i s a principal 29-filter H H F = [x)for some x E C(L).

(ii)

F i s a princapal mazimal 6-filter @ H F = [x)f o . Some x E A~(c(L)).

Proof.

(i)

[{z})g

= [cpoz) by Remark 5.1.6.

(ii) Let x E

At(C(L)). In

view of (i) and Theorem 5.2.3 it suffices t o

prove that [x)is prime. If y V z

2x

therefore cpoy A x = x or cpoz A cpoz

then

x = x, hence x 5 p0y 5

5 z. The converse implication from (ii) follows

y or

x 5

easily.

1.15. Corollarv.

If L E LMd there is a bijection between the set of its principal maximal 4-filters and the set of atoms of its center. 1.16. Lemma. If L E LMn OT L E Mn and i s complete and atomic t h e n L is a direct product of subalgebras of LP-']. Proof. We apply Lemma 1.2 t o the family {[x) I x E

At(C(L))}of all

principal maximal d-filters; cf. Proposition 1.14. In this case the morphism

f

in Lemma 1.2 is injective, because if a , b E

is x E

At(C(L))such

6 = (f(b)) .

that x

5

a and x $

C(L)and

a

#

b then there

b, therefore ( f ( a ) ) = i #

f is a monomorphism and if we prove t h a t f is surjective this will show that f : L + L' = n {L/[x) I x E At(C(L))} It follows that

Representation theorems and duality for LM-algebras

292

is an isomorphism, while each L / [ z )is simple, i.e. a subalgebra of L,. Let ([yz])zEA

E L', where

A = A t ( C ( L ) ) . Set y =

V

(yz A z). Then

xEA

Remark 4.5.13 implies, for each

t E A, that t A y

= v(t A yz A x) = yt A

t,

hence

(y, yt) E m o d ker [ t ) c mod ker [t) 0

which proves that f(y) = ([&!t])tcA. 1.17. Theorem (Boicescu [1984]).

T h e following conditions are equivalent f o r a n algebra L E LMn or L E Mn:

(i) L i s a direct product of subalgebras of LP-'] (E L,,); (ii) L is complete and atomic. Proof.

(i) + (ii): Obvious. (ii) + (i): By Lemma 1.16.

0

1.18. Remark. In a finite Boolean algebra every proper filter

F

is a unique intersection of

maximal filters (namely those generated by the atoms included in the generator of

F).

1.19. Corollary (Moisil (1941a1, Cignoli [1969a], [1975]). Every finite algebra L E LMn (or L E Mn) is a direct product of subalgebras of LP-'] (= L,), the number of factors being c a r d A t ( C ( L ) ) or equivalently the number of maximal n-filters. S u c h a decomposition is unique . Proof. The first statement follows from Corollary 1.15 and the remark that every (n-)filter is principal. The second statement follows from Proposition 1.3 and Remark 1.18 applied t o [l).

0

The representation theorem of Moisil

293

1.20. Corollary.

Every finite axled n-algebra is a direct product of algebras L, and Lz,in the class of axled n-algebras. Proof. From Corollary 1.19 and Lemma 1.12. Example 4.5.25 shows that in Theorem 1.17 one cannot replace, in general, condition (ii) by

C(L) being complete and atomic. However this can

be done for t he class Pn: 1.21. Corollary.

T h e following conditions are equivalent for a n algebra L E Pn:

L is a direct product of copies of the algebra LP-'] (E Ln);

(i)

(ii) L is complete and atomic; (iii) C ( L ) is complete and atomic. Proof. By Theorem 1.17, Lemma 1.12, Corollary 4.5.8 and Theorem 4.5.21. 0

It was remarked in Ch. 5, 94, that the atomic finite algebra L, is not strongly atomic. However Proposition 1.23 below shows that the elements of a complete and atomic LMn-algebra do have a representation similar t o that in the definition of strong atomicity.

1.22. Definition. An element z =5

2

of a lattice L is said t o be join-irreducible if Vy, z E L : y V

+ y = o or z = 2.

1.23. Proposition.

If L

E LMn is complete and atomic t h e n every e2ement of L is uniquely

represented as a j o i n of pairwise incomparable join-irreducible elements. Proof. In view of Theorem 1.16 we may suppose that

L=

n

SES

A, where

294

Representation theorems and duality for LM-algebras

each A, is an LMn-subalgebra of L,. Then t h e required unique representation of an element a = ( u , ) , ~ S E L is a =

V xt,where t€S

xt =

(1.3)

{

a,,

0

,

ifs=t,

ifs#t.

1.24. Corollary.

If L E M3 is complete and atomic then every element x E L - (0) is

a

join of elements from At(L)U A t ( C ( L ) ) . Proof.

In this case each A, is a subalgebra of L3 = (0) U At(L3) U

At(C(L3)),therefore if a to A t ( L )U A t ( C ( L ) ) .

#

0 the element zt defined by (1.3) belongs 0

T he next result is a partial generalization of Theorem 1.16 to an arbitrary

6. 1.25. Theorem.

The following conditions are equivalent for a n algebra L E LM6 ( L E LMN6): (i) C ( L ) is finite;

(ii) L is a direct product of finitely many subalgebras of LLq; (iii) F16(L) is finite; (iv) every proper filter of L is a unique intersection of maxima2 6-filters. Proof.

(i) + (ii): By Lemma 1.16. (ii) +- (iii): cardC(Lhq) = 2, hence if t h e number of factors A, of L is rn then cardC(L) = 2", therefore F16(L) is also finite by Theorem 5.1.7.

(iii)

+ (iv):

Theorem 5.1.7 shows th a t Fl(C(L)) is finite hence C ( L )

is finite, therefore each proper filter of

C(L)is

a unique intersection of

The representation theorem of Moisil

295

maximal filters of C(L)by Remark 1.17. The latter property is transferred t o the &filters of L again by Theorem 5.1.7. (iv)

+ (i): Every proper filter of C ( L )is a unique intersection of maximal

filters by Theorem 5.1.7, therefore

C(L)is finite by the converse of Remark

1.18 (Hashimoto [1952]; cf. Balbes and Dwinger [1974], Exercise 3.6.7).

0

1.26. Remark. The direct decompositions in Theorem 1.17 and 1.25 are provided by Lemma 1.16, i.e. (1.4)

L

l-I { L / ( a )1 a E A t ( C ( L ) ) } .

An equivalent form will be obtained below. 1.27. Proposition. Let L E LM6 ( o r L E L M N 6 ) , and u E C ( L ) . If the ideal (u] is endowed with the structure of 6-algebra (6-algebra with negation) defi-

ned in L e m m a 4.2.23, i.e. (1.5)

'p4a: = 'pis

,

p;a: = @a:

Aa

,

( N " x = Na: A a )

,

(Vz E ( a ] ) , t h e n ( a ] Z L / [ a ) . Proof. The canonical mapA : (a] -t L / [ u )is a morphism in L M 6 ( L M N 6 ) . Further for every a:, y E ( a ] ,

i = w 3c mod [ a )y H

H 9 ; s mod [u) 'pjy (Vi

3 ~E ;[u) ' p i x A ~i =

($ 'pix

A ~i (Vi E

E I) e

I) H

A a = 'piy A a (Vz E I ) e cpix = cpdy (Vi E

I)

x =y

,

therefore-is injective. Surjectivity follows from the f a c t that for every x E L ,

(x A a, a ) E mod [ a ) , therefore a : x u = ?.

0

1.28. Corollary . If L E L M n (Mn) is complete and atomic, or if L E L M 6 ( L M N 6 ) and C(L)is finite, t h e n (1.4) holds, OT equivulently

Representation theorems and duality for LM-algebras

296

Proof. From Remark 1.26 and Proposition 1.27.

0

The above corollary generalizes the representation obtained by Cignoli [1969a], [1971] for finite n-valued algebras with or without negation. In view of the previous results we may wish to count t h e subalgebras of

Liq, with and without negation. 1.29. Lemma. If L1 and L2 are subalgebras of L'\

such that L1

9 L2

then

H o ~ L M B L2) ( L ~=, 0.

E L1- L2. Suppose f E HomLMB(L1,L2). Then y = f(x) # x, hence z(io)# y(i0) for some io E I , say x ( i 0 ) = 1 and y(i0) = 0. This Proof. Take

2

implies the contradiction

1.30. CorolIa ry . T w o subalgebras of L,VI having distinct underlying sets cannot be isomorp-

hic. This corollary shows that counting the subalgebras of L '\

i s the same as

counting their underlying sets. 1.31. Theorem (Cignoli [1969a], [1975], Boicescu [1984]).

Let s ( l , p ) ( s ~ ( l , p s) (, I ) ,S N ( I ) )be the number of p-element LMd-subalgebras (p-element LMNd-subalgebras, LMd-subalgebras, LMNzfl-subalgeb ~ a s )of Liq. T h e n :

(i)

If n is even t h e n each MoEsil subalgebra number of elements, s ~ [n-l],2k) ( and sN( [n- 11) = 2(n-2)/2.

=

of LP-']

L, has a n even

( (n - 2)/2 ) (k = 1, ...,n/2) k-1

The representation theorem of Moisil

(iii) For every n, s ( [ n- l ] , p ) =

( n-2 )

297

(1c = 2,3, ...,n ) and s ( [ n-

P-2

11) = 2n-2. (iv)

If1 is infinite then

and these inequalities cannot be improved.

Proof. (i)

T he first statement follows fro m th e fact t h a t every Moisil subalgebra

L of Ln satisfies ci E L w Nci E L ( i = 0,1,

...,n - 1).

Since

L

it follows t h a t t h e 2k-element algebra L i s determined by th e remaining 21c - 2 elements and t h e latter, in their turn, are determined by the first k - 1 elements i n t h e set (c1, ...,c(,-Z)p}.

contains co and

This implies the first statement and

+

1)-element subalgebra of L , contains c ( ~ - ~and ) / ~by eliminating it one obtains a 2k-element subalgebra; the latter can be

(ii) Every (2k

viewed as a subalgebra of L,-t. Therefore s N ( [ n - 1],21c

(iii) Proof left to the reader.

+ 1) =

SN([72 -

1],21c) = s N ( [ n - 2 1 3 ) =

Representation theorems and duality for LM-algebras

298

(iv) It follows from Theorem 1.6 that Li' and its subalgebras are precisely

the subsets that contain 0 and 1, therefore

s ( I ) = cardF(L\' - {0,1)) = 2"dLy1

(1.8)

.

On the other hand the map which associates with each

i

function xi E Lh' defined by x;(j) = 0 or 1 according as j

E I , the

<

i or

j 2 i , is obviously an injection. Therefore card I

5 card Li' 5 2card1,

which together with (1.8) yields (1.7). Finally we prove t h a t th e inequalities (1.9) cannot be improved. If

I = LV @ 1 the above map i H xi is surjective too, because every x E L,[' , x # 0, fulfils x = xio where io is t h e least i such that x ( i ) = 1. Therefore in this case c a r d 1 = cardL2[' . Now let I = [0,1] n Qand for each irrational number p E [0,1] let x p E Li' be defined by xp(i) = 0 or 1according as i < p or i > p . It follows t h a t card Li' 1 2x0 = 2c"dz, hence card Li' = 2c"d1. 1.32. Remark.

If the set I is regular (cf. Definition 4.6.7) then in the previous theory the algebra Li' can be replaced by i t s isomorphic image (cf. Theorem 4.6.12). Thus e.g. we can prove the following analogue of Theorem 5.2.13: 1.33. Proposit ion.

If the algebra L is regular then the map (1.10)

@ : PFld(L) + Hom(L,ze)

(1.11)

@ ( P ) ( X= ) Sup {i E I

,

I YdiX E P}

(VZ E L ) (VP E PFld(L))

is a bijection, whose inverse Q is defined b y (1.12)

ker h = {z E L I h ( z ) = 1)

(Vh E Hom(L,ze))

.

299

The representation theorem of Moisil

Proof. Let ipl

: PFld(L)

--f

Hom(L,Li') and

ipz

:

LL4

+

LS be

the bijections defined in Theorems 5.2.13 and 4.6.12, respectively. Then

H o m ( L , z s ) defined by ip3(h) = ipz o h is also a bijection, therefore so is ip = GJs o @I. Then for every P E PF129(L), 2 E L , h E Hom(L,LS) and i E I,

ip3

: Hom(L,Li')

--f

@ ( P ) ( x= )

@3(@1(P))(x)

=

(a2 0

@l(p))(x)

= sup{i E I I i p @ ) ( x ) ( d i ) = 1) =

(a o

ker)(h)(x) = ip(ker h ) ( z ) =

1.34. Corollary (lorgulescu [1984a]).

Let L be a regular m-algebra, P F l m d ( L ) the set of aZZ prime ( m ,d)-fiZters of L and Homm(L, the set of alZ m-homomorphisms from L to LB. T h e n the restriction ipm = ip I P F l m 6 ( L ) of the m a p in Proposition 1.33 is a bijection

z,)

(1.13)

ipm :

PFlm.S(L) --t Homm(L,zG) .

Proof. It is easy t o check that the prime d-filter P is an m-filter if and only

if @(P)is an m-homomorphism.

0

lorgulescu [1984c] constructs a parallel theory of m-simple and m-semisimple regular m-algebras, culminating in the following analogue of Theorem

2 cardI, card L > 1and there is a family o f maximal (m,d)-filters of L whose intersection is {l},then L is a subdirect product of m-subalgebras of La. 1.7: if L is a regular m-algebra such that m

Representation theorems and duality for LM-algebras

300

$2, Applications of the representation theorem In this section we illustrate the fact that various problems concerning LM-algebras can be solved with the aid of the representation theorem 1.8. Thus e.g.

consider the problem whether there may exist several 19-

algebras ( L ,A, V, {p;}iEr,{p;};Er,0 , l ) ,having the same underlying boundedlattice structure ( L ,A, V, 0 , l ) . 2.1. Proposition.

(i) If L E DO1 t h e n there as at m o s t one structure of LMS-algebra o n L.

(ii) If card1 > 2 t h e n there exist non-isomorphic LM19-algebras having the s a m e underlying DO1-structure. Proof.

(i)

If L E LM3 then (p2 is a closure operator on L (cf. Definition 1.1.14) therefore p2 is uniquely determined by its Moore family C ( L )by Remark 1.1.15 and Proposition 3.1.5. A dual argument holds for t h e interior operator $ol.

(ii) If card1 > 2 it is easy t o construct two subalgebras L1 and L2 of L a E LMIJ such t h a t L1 and L2 are isomorphic sublattices of the chain Lh‘ E DO1 but the underlying sets L1 and L3 are distinct. Then according t o Corollary 1.30 L1 and L2 are not isomorphic in LM19. 0 2.2. Proposition (Cignoli and De Gallego [1981], Boicescu [1984]).

If L E DO1 and n E {3,4,5} n-algebra on L.

there is at most one structure of Moisil

Proof. For n = 3 this follows from Proposition 2.1. Now l e t n = 4 (the case n = 5 can be treated similarly). Suppose

L‘ and L” are Moisil 4- al-

gebras with the same underlying structure L E D01. But cpi = p1” by the argument in the proof of Proposition 2.1 (i), therefore Fld(L’) = Flr9(L”)

Applications of the representation theorem

301

by Remark 5.1.2. Let us apply Proposition 1.5, which involves the family

(F,),,sof

maximal &filters, common t o

L’ and L”.

For each s E S the

algebras L’/F, and L”/F, being simple, are isomorphic t o subalgebras of L4

LZ or L 4 . On th e other hand L’/Fsand L”/Fs are isomorphic in D01, therefore either L’/Fs 2 L”/Fs 2 LZ or L’/Fs L”/F, 2 L 4 in M4. This implies L‘/Fs S L”/Fs, so that (the isomorphic in M4, i.e. t o

n

n

sES

SES

images of) L‘ and L” are subalgebras of the latter direct product and have the same underlying set L, hence they coincide. Cignoli and De Gallego

0

[1981]have shown that for n 2 6 there exist

subalgebras of L, that are not isomorphic in Mn although their underlying Mg-structures are isomorphic. Let us consider again the problem whether every interval of a Moisil n-

algebra can be endowed with a structure of M o i l n-algebra. We have the comment t o Definition 4.4.15 t h a t if n is even t h e answer is in the negative. However if n is odd we have the following partial positive answer. 2.3. Proposition.

If L

E M(2n

+ l ) , n 2 1 and L is finite then every interval [a,b] 5 L can

be made into a M o i s i l ( 2 n

+ 1)-algebra.

+

Proof. The interval [a,b] belongs t o LM(2n 1) by Theorem 4.4.16 and it is finite, hence [a,b] is a direct product of LM(2n 1)-subalgebras of Lp“] by Corollary 1.19. Therefore [a,b] is isomorphic in DO1 t o a product of chains of cardinalities k f { 2 , 3 , ...,2n 1) and it suffices t o prove

+

+

that such a chain

M(2n

C

can be viewed as a Moisil subalgebra of Lz,+l

E

+ 1 ) . This follows immediately from the following DO1-isomorphism:

C Z {co, ...,c ~ - ~ , c ~..., ~ cZn} - ~ +C ~L2n+l , for k = 2p and C 2 {co, ...,c p - l , c,, C Z , - ~ + ~ , ...,cZn} C Lzn+lfor k = 2p + 1.

0

The class of axled n-algebras is a proper subclass of the class of Moisil n-algebras (cf. Theorem 4.2.10 and e.g. Remark 4.2.20) and is characterized by the representation theorem 4.2.26: every axled n-algebra is of the

Representation theorems and duality for LM-algebras

302 form A x

B with A E Pn and B E B. Moisil [1940] raised t h e problem of

obtaining a simpler theorem in the general case and introduced the concept of strictly chrysippian element as a tool for solving this problem. 2.4. Definition. An element m of a lattice L E DO1 will be called strictly chrysippian (du-

ally strictly chrysippian) if m V x E c ( L ) (rn A x E c ( L ) )for every x E L .

S ( L ) and S'(L) denote the sets of strictly chrysippian elements and dually strictly chrysippian elements of L , respectively. Let

2.5. Proposition.

T h e following conditions are equivalent for a n element m of a n algebra L E LMn:

(i)

m E S(L);

(ii) m E C ( L ) and [m,1] is a Boolean algebra;

(iv) cplm E S ( L ) . Proof.

(i) =+ (ii): Obvious; the complement of y E [m,1) in [m,11 is $jV rn. (ii) + (iii): Let x E L. Take y E L such that (m V x) A (m V y) = m and ( mV x) V ( mV y) = 1. Then x A y 5 m and m V x V y = 1, hence using (1.2.19):

Cpix A 'pig A 6 = 0 and m V pix V (piy = 1 ViY

I m V Cp;x and m V cp;x 5 'p;y

therefore m A Cplx A

= 0.

(iii) + (i): We get in turn

(Vi E I) ; (Vi E I) ,

Applications of the representation theorem

303

2.6. Proposition (Sicoe [1966]). Let L E LMn. Then: (i)

t/m E L(m E S ( L ) H plm E s'(L)).

(ii) I f L E Mn then Vm E L ( m E S ( L ) @ N m E S ' ( L ) ) . (iii) L E B H S ( L ) n S'(L) #

0.

Proof.

(i) Proposition 2.5 and its dual imply

(ii) Similar t o the proof of (i), (iii) If rn E S ( L )n S'(L) then Lemma 4.2.24, Proposition 2.5 and its dual 0 imply L ( m ]x [m)E B. The converse is trivial.

2.7. Proposition. Let L E LMn. Then: (i)

S ( L ) E Fln(L) and it is the greatest element F E Fln(L) such that cpl(F) = F.

(ii) S ( L ) E Fl(C(L)) and it is the greatest element F E Fi(C(L)) szlch that & ( F ) = F .

(iii) s ( L / s ( L ) ) = [i).

Representation theorems and duality for LM-algebras

304 Proof.

(i)

(pl(S(L)) = S(L) by Proposition 2.5. Therefore S ( L ) E Fln(L). If cpr(F) = F E Fln(L) then for every m E F and x E L we get m V x E F E C ( L ) hence m E S(L). Clearly

S(L)is

a filter and

(ii) S(L)E Fl(C(L)) by (i), while cpT1 (S(L))= S(L)by Proposition 2.5. If F E Fl(C(L)) and cp;'(F) = F then for every m E F and x E L we get ' p l ( m V x ) = m V v l x E F hence m v x E cp;'(F) = F G C ( L ) therefore rn E S(L). (iii) Let B E S ( L / S ( L ) ) .Take x E L; then (Pn-1$Ap13ASc= ( ~ , + ~ i A p ~ $ , hence there is m E S(L)such that

therefore s A m E

S ( L ) hence s E S ( L ) by (i),that

In the sequel we work with t h e filter

(2.1)

D s ( L ) = { x E L I (Pn-lx = 1)

(cf. Proposition 4.4.6) and the n-filter generated by it:

(2.2)

D n ( L )= { x E L I 3 y E D s ( L ) , 'ply I x }

2.8. Proposition. Let L E LMn. Then:

(i)

L / D n ( L ) E B.

(ii) D s ( L ) = D n ( L ) w L E B. (iii) S ( L )n D n ( L ) = [I).

*

is

B = i.

0

Applications of the representation theorem

305

Proof.

(i)

The Glivenko theorem 4.4.8 constructs the Heyting-algebra isomorphism L / D s ( L ) 2 C(L).But D s ( L ) Dn(L) hence L/Dn(L) is a quotient algebra of L / D s ( L ) by the second isomorphism theorem (cf. e.g. the literature quoted in the introduction t o Ch. 1, $5) therefore L / D s ( L ) E B.

(ii) Suppose D s ( L ) = Dn(L). If x E D s ( L ) then q1x E D s ( L ) hence q1x = vn-Iv1x = 1, therefore x = 1. Thus D s ( L ) = [l) hence L E B again by Theorem 4.4.8. The converse is trivial.

(iii) If m E S ( L ) n Dn(L) then q l x 5 m for some x E D s ( L ) . But (P,,-~X A plx 5 m hence

2.9. Theorem (Moisil [1940], Boicescu [1986a]).

E v e r y algebra L E LMn is a subdirect product of a n n-valued algebra having exactly o n e strictly chrysippian element and a Boolean algebra. Proof. Construct t h e map

(2.3)

h

:

L

+

(L/s(L))x (L/D~(L))9 h ( ~ )= ([xIS(L), [x]Dn(L))

and use Propositions 2.7 and 2.8. 2.10. Remark.

If L is an axled n-algebra then a)

S ( L )= [vn-lal) because if m E S ( L ) and x E L then

hence

0

Representation theorems and duality for LM-algebras

306

b) L / S ( L ) 2 (‘pn-lal] is a Post algebra by Proposition 1.27 and Lemma 4.2.25. 0 2.11. Corollarv.

If L (i)

E LMn is finite then:

L

(L/s(L))x ( L / D ~ ( L ) ) .

(ii) S(L)= (1) H L has n o direct factor L2 H L has n o direct factor B E B, cardB > 1. (iii) If, moreover, n = 3 t h e n S ( L ) = (1) H L E P3. Proof.

(i)

Let S ( L ) = [s) and D s ( L ) = [t),and z E L. According t o Proposition 4.4.6 we get t 5 x V @+lx, hence q l t 5 ‘plx V (Pn-lx, therefore

I pit.

This proves that (Pit E S ( L ) hence s 5 p l t or s A v l t = 0. On th e other hand s V q l t E S ( L ) n D n ( L ) , hence s V qlt = 1 by Proposition 2.8. Thus 'pit = S and t h e target of t h e map h in (2.3) can be written L‘ = ( L / [ s ) )x ( L / [ s ) ) .For every (il,;) E L’we have (il,;) = h ( z )where z = ( U A S ) V ( V A \ S ) , therefore the embedding h is an isomorphism. (PIX A vn-lx

(ii) Since S ( L 2 )= L2and S(Lk) = (1) for k 1.19 that S ( L ) = { 1) H

> 2, it follows from Corollary

L has no direct factor L2. The representation

B = L;@),p ( B ) E N , for each B E B, completes the proof. (iii) S ( L ) = (1) u L = L; by (ii).

0

The above corollary specifies the structure of finite algebras; cf. Corollary 1.19. In the infinite case the situation is difFerent. Thus e.g. the algebra

L in Example 4.5.25 satisfies S(L)= (1) but L $ P3 (e.g. by Corollary 4.1.9). However Corollary 2.11 (ii) is also valid in the infinite case, as shown by the next result which we quote without proof 2.12. Proposition (Boicescu [1986a]).

T h e following conditions are equivalent f o r L E LMn:

Applications of the representation theorem

307

(i) S ( L )= (1); (ii) Vm E C ( L ) ( m< 1

* 3z E [m,11,

5

# C[m, I]);

(iii) L is a subdirect product of simple algebras Li, Li

# Lz;

(iv) L has no non-trivial Boolean direct factor.

The next result provides a technique for proving identities. The intuitive concept of identity is clear; the precise technical meaning is given in Definition 1.5.26.

2.13. Proposition. The following conditions are equivalent for two LMn-polynomials p , q (i.e., words of the Peano algebra of the same type as LMn): the identity p = q holds in every L E LMn;

(i)

(ii) the identity p = q holds in every chain L E LMn;

(iii) the identity p = q holds in LF-'] (2 L,). Proof.

(i) + (ii) + (iii): Trivial. (iii) + (i): Clearly p = q holds in every subalgebra of LF-'], hence in every L E LMn by Corollary 1.9. Now we refer the reader t o Definition 4.3.17 (together with Definition 4.3.11) for the concept of P-algebra and the class

Iir

of P-algebras. We

have noticed in Remark 4.3.19 that every LMn-algebra can be made into a P-algebra. Since every chain with 0 and 1is clearly a P-algebra (recall that z

5

y if and only if z

+ y = l), while an infinite chain L cannot be an

LMn-algebra (otherwise L should be subdirectly irreducible by Theorem 1.6,

L, by Corollary 1.9 and this is a contradiction). Therefore the inclusion LMn c K is proper. To say more about this topic we shall investigate the equational classes of K. First we need some further therefore a subalgebra of

concepts.

Representation theorems and duality for LM-algebras

308

2.14. Definition (Epstein and Horn [1974a]).

A lattice L E DO1 is called a B-algebra if for every z,y E L there is x =$ y which is t h e least element z E C(L)such that z A x 5 y. If, moreover, y) V (y =$ x) = 1 (Vz,y E L ) then L is termed a BL-algebra. (z See Proposition 4.3.20. Epstein and Horn [1974a] proved t h a t in fact P-algebras coincide with BL-algebras and with those L-algebras whose duals are Stone algebras or,

L which satisfy the following property: for every z E L there is !X which is the greatest element z E C(L) such t h a t z 5 x and !(zVy) =!zV!y. The operation ! satisfies the following properties: equivalently, with those L-algebras

(2.4)

!a: = xt+ = 1

%

x

,

!(x =+y) = x

C + y .

2.15. Definition. Let Cn be the n-element chain viewed as a lattice and K, t h e equational subclass of

K

generated by

C,.

2.16. Proposition. The following conditions are equivalent f o r L E K and n 2 2:

n

(iii)

v

i=l

(z;

S ziti) = 1 (vz1,...,z,+l

E L);

Applications of the representation theorem

309

(viii) L i s a subdirect product of chains of cardinalities (ix)

5:

n;

PFl(L) is a cardinal sum of chains of cardinalities 5 n - 1.

Proof.

(i) + (ii): It suffices t o prove (ii) in Cn, where 2 1 > x 2 > ... > xn+1 is for some i, therefore xi + xi+l = 1. impossible, hence xi 5 (ii) + (iii): Via (2.4) and !1 = 1. (iii) =+ (ii): Because x % y 5 z + y. (ii) + (viii): L E K is a subdirect product o f chains with 0 and 1 (Epstein and Horn [1974a]). It follows that each o f those chains satisfies (ii), therefore it has at most n elements, otherwise x1 > ... > xn+l would yield a contradiction.

+ (i): Because I<, is an equational class. (viii) ++ (ix): Proved by Epstein and Horn [1974a]. (viii) + (iv): It suffices t o prove (iv) for a chain. Let y be the left side o f (iv). If xi 5 x;+l for some i then y = (1 + x1) + x 1 = x1 =+ x1 = 1, while if x1 > x 2 > ... > x, then y = (((Q V ... V s,) + xl)) + x1 = ( x 2 j x1) + 51 = 51 + 21 = 1. (iv) + (v): Because ( V a;) + b = A (u ; + b) in every Heyting (viii)

i

algebra.

i

Representation theorems and duality for LM-algebras

310

(v)

+ (vi):

Let y stand for the left side of (vi). Then

+ b in every Heyting algebra, while y 5 x1 by (v). (vi) + (vii): Trivial.

b5a

(vii)

j(viii):

21

5

y because

Therefore y = x l .

Suppose the Epstein-Horn subdirect decomposition of L

involves a chain of cardinality

> n. Take 1 > z1 > 2 2 > ... > 2,-1 > 0

in such a chain, for which the left side of (vii) becomes zl)) A 1 = 1, therefore (vii) fails in

L , too.

(

n-2

A i=l

0

The above characterization is suggested by the axiomatization of a special intuitionistic propositional calculus whose characteristic Lindenbaum-Tarski algebra is a finite chain (Thomas [1962], Becchio [1978b]). In t h e case of a three-element chain such a characterization has been given by Lukasiewicz

I19381and the corresponding Heyting algebras are said t o be three-valued (L. Monteiro [1964]): they are those Heyting algebras which satisfy t h e identity

(2.5)

(((2

+ y) + z) A ((y

z)

jz))

2

= 1.

2.17. Theorem (Boicescu [1984]). The equational subclasses of K are

K1CKzC

... C K , C ... C K .

Comment.

See the similar characterization of the equational subclasses of the class of L-algebras, given by Hecht and Katrina’k [1972].

# Kn+l

C,+, E I(n+l K because K contains infinite chains. Now we prove t h a t every equational subclass K’ of K falls Proof. Clearly but

C,+,# K,

K,

C_

Kn+1 and K,

by Proposition 2.16 (ii). Also

because e.g.

K,

C

within (2.6). There are two cases. a)

K, c K‘ for every n. Let R be the set of equations defining K’. It follows that every finite chain satisfies R, therefore the identities R hold in K by a result of Epstein and Horn [1974a]. Thus K’ = K.

311

Applications of the representation theorem

K' for some n. Since Cl is a homornorphic image of every algebra o f K' it follows that K1 C K'. Therefore there exists no such t h a t K, C K' and K,o+l g K'; hence K, g K' for every n > no. This implies directly that no infinite chain can belong t o K'. Therefore every chain of K' has

b) K,

a t most no elements. In particular this happens for the chains occurring

in a subdirect decomposition of an algebra

L E K', which proves t h a t

L E Kn0. Therefore K' = Kn0. 2.18. Theorem.

LMn E KO but LMn

Kn-l, for every n 1 2 .

Proof. Each of the identities in Proposition 2.16 holds in

L,

and its two

sides are LMn-polynomials by (4.3.1) or (4.3.3). It follows from Proposition 2.13 that those identities hold in every L E LMn, therefore LMn Proposition 2.16. On the other hand L, E K, but L,

$Z K,-l

C K, by

by Proposition

2.16 (viii) via Proposition 1.6.

0

In particular for n = 3 we recapture a result of L. Monteiro [1970a]. On

the other hand every Moisil algebra satisfies the identities in Proposition 2.16 for every n 2 2, as was first proved by Becchio [1978b] using elementary methods, then by Becchio and ltturioz [1978] by a proof which utilizes the properties of prime filters in a Moisil algebra. 2.19. Corollary.

The equational class LM3 (M3) is equivalent to the equational class KJ. Proof. Take

L E KB. Then L is a double Stone algebra by Theorem 9.2.10

in Balbes and Dwinger [1974] and its dual. Also,

L is a subdirect

product of

5 3 by Proposition 2.16 and those chains obviously satisfy the identity z A z+ 5 y V y*, therefore so does L. It follows from Theorems 4.4.4 and 4.4.9 L E LM3. The converse holds by Theorem 2.18.0 chains of cardinalities

2.20. Proposition (Boicescu [1984]).

The following conditions are equivalent for L E D01: (i) L f LM3;

312

Representation theorems and duality for LM-algebras

(ii) L is a B-algebra satisfying (2.7)

(2

%

y) V (y

%

2)

V

(2

=% t ) = 1 .

Proof.

(i) + (ii): If L E LM3 then L E K3 hence L is a B-algebra, while (2.7) follows from Proposition 2.16.

(ii) + (i): Taking z = 1 and t = y in (2.7) and noticing that 1 =% y 5 C z y we obtain (y =$ z ) V ( z + y) = 1, i.e. L is a BL-algebra. So L is a P-algebra, hence L E K3 by Proposition 2.16, therefore L E LM3 by Corollary 2.19. For n

> 3 it

is not known whether the inclusion LMn

K, is strict.

However we can prove that LMn is not an equational class of P-algebras.

Take L, x L, E LMn. The product structure is obviously the unique structure of LMn-algebra which can be defined on L, x L,. It is easy to see that M = {(0,0),(c~,cn-2),(l, 1)) is a K,-subalgebra of L, x L, whereas CPZ(CI, en-2)

= (071)

$ M.

We now present certain aspects of th e relationship between the equational class LM3 (M3) and certain classes of Heyting algebras. 2.21. Theorem (Iturrioz [1975], [1976c]).

A Heyting-Brouwer algebra L is an LM3-algebra if and only if it satisfies the following identities: (2.8) (2.9)

2'

A xt+ = 0

,

(x + y) V (y** + x) = 1 .

Proof. Necessity follows easily from Proposition 2.13. Conversely, (2.9) imp-

+ y) V (y + z) = 1 because y 5 y**.

So L is an L-algebra, while (2.8) shows that L is a dual Stone algebra. This is equivalent t o L E K. Thus L is a subdirect product of bounded chains (Epstein and Horn [1974]) and none of them includes L4 = {cg, c1, c 2 , cQ} because L4 fails t o fulfil (2.9): ( c 2 + c1) V (c:* + c2) = c 1 V c 2 = c 2 . Therefore L E KJ by Proposition lies (z

Applications of the representation theorem

313

2.16, consequently L E LM3 by Corollary 2.19.

0

2.22. Definition (lturrioz 119681, (19741). A symmetric three-valued Heyting (Moisil) algebra is a pair ( L ,-), where L is a three-valued Heyting (Moisil) algebra and is an involutive dual endomorphism on L. N

2.23. Lemma. If ( L ,+,-) is a symmetric Heyting algebra, then ( L ,+,-i= where ) (2.10)

x

-+y =-

x

( N

-

j

y)

,

is a Heyting-Brouwer algebra (cf. Definition 4.4.11). Proof. Obvious.

0

2.24. Proposition (Iturrioz [1976b]). The ChSs of symmetric three-valued Moisil algebras is equivalent to the

class of d l pairs ( L , a ) , where L E LM3 and morphism of L.

a!

is a n involutive auto-

-

Proof. If ( L , - ) is a symmetric three-valued Moisil algebra we define a x = N x . Clearly a! is a lattice automorphism. To prove that a commutes with (pi we apply first Theorem 4.4.2 and Lemma 2.23: N

=

(p1X

N

(x++ ) - [(x e 1) += 11 = -N

[-

="

hence (2.11)

-

= (pl N x

-

(N

[NW ( N

(- x

* 0)) * 01

x ) * ] * = (- x)** = ( p 2

= N

2

= ( p 2 x . Therefore

(pix = 9 3 4 3

-

(i E {1,2}) ,

For the last step we remark that ( p ; x V N ( p i x = 1 implies 0 and similarly (pixV N p i z = 1, therefore pix = prove aax = x: N

-

N

N

N p i X= N(p;x. Now we

pix^

Representation theorems and duality for LM-algebras

314

-

L

Conversely, if define

--

E LM3 and a is an involutive automorphism on

x = Naa:. Clearly a is a dual automorphism and

L we

x = x follows

-

o each c her. 0

If the algebra L E LM3 in Definition 2.22 is specialized t o a Boolean algebra (i.e., 'pl = 'pZ = 1 ~then ) we obtain the definition of s y m m e t r i c Boolean algebras (Moisil [1954], A. Monteiro [1966]) and in view of Proposition 2.24 they coincide with the Boolean algebras endowed with an involutive automorphism (A. Monteiro [1980]). In the general case lturrioz [1976b] has defined and studied the s y m m e t r i c n-valued M o i s i l algebras, i.e. the pairs

(L,-) where L E Mn and tisfying a!

cpi

-=-

w

is an involutive dual automorphism on

~ ~ - ori equivalently, ,

is an involutive automorphism on

th e pairs (L,a!) where

L sa-

L E Mn and

L.

2.26. Proposition (L. Monteiro [1970a]).

E v e r y three-valued Heyting algebra is a n L-algebra.

+ x ) = 1we take z such t h a t x j y 5 z and y + x 5 z and show that z = 1. But 1= z + z 5 ( x + y) =+-z , hence Proof. To prove ( x

(2.12) As y

x

(x

y)V(y

* y) * z = 1.

5 x += y

5z

j

and x

and from z

+y 5 z

by (2.10) we get y

+ x 5 z we obtain

5 z , then

z

+x 5 y +

Applications of the representation theorem Now

315

(2.5), (2.12)and (2.13) imply

2.27. Theorem (Iturrioz [1974]). The equational class of symmetric three-valued Moisil algebras is equivalent to the equational class of symmetric three-valued Heyting algebras. Proof. Every symmetric three-valued Heyting algebra

( L ,-)

is an L-algebra

2.26 and a Heyting-Brouwer algebra by Lemma 2.23. Besides it is immediately checked (A. Monteiro [1980])that (x (.y ) A ( y -+z) = 0, therefore L is a P-algebra. As (y + y) =+ x = 1 + x = z, we obtain from (2.5)that

by Proposition

((x =

hence

L

E

* y) * .) * [((z * Y) =+ ).

5

A

=

((Y

* Y) * .)I

=j 2

= 13

K3 by Proposition 2.16. The proof is completed by Corollary

2.19. 2.28. Remark. In view of Theorems 2.27 and 4.4.2the symmetric three-valued Heyting algebras are equipped with certain modal operators: 'plx = x++, ' p z x = x**. In the general case of symmetric n-valued LM-algebras ( L ,-) (cf. Remark

2.25) L need not belong t o Mn. lturrioz [1982],I19831 has studied the latter algebras under the name of symmetric Heyting algebras of order n (SH-algebras of order

TI)

starting from the axiomatization obtained in

1974;cf. Ch. 4,53. It is shown that the subdirectly irreducible (simple) algebras in this class are the subalgebras of S,Z, i.e. of ( L , x L, -), where L, x L, E LMn and (z,y) = ( N y , N t ) . Also Q(Z,Y) = ( y , ~ )is an involutive autornorphisrn on S,z. lturrioz [1983]describes the structure of

-

finitely generated free SH-algebras of order n and the injective objects of this class.

2.29. Corollary (lturrioz [1982]). T h e fol2owing conditions are equivalent f o r a n SH-algebra of order n:

Representation theorems and duality for LM-algebras

316

(i) L i s a Kleene algebra;

(ii) L i s a Moisd algebra of order n.

-

Proof.

-

(i)+ (ii): It suffices t o prove that cpixV cpix = 1and cpixA Fix = 0. But in every Kleene algebra L , f =- x for all x E C ( L ) (A. Monteiro; cf. =- pix. Cignoli [1965]), it follows that (ii) (i): This is Proposition 3.1.13.

0

2.30. Corollary (L. Monteiro (1970aj).

A) T h e following conditions are equivalent f o r a s y m m e t r i c three-valued Heyting algebra: (i) L is a Kleene algebra; (ii)

L i s a three-valued Moisil algebra.

B) W h e n this is the case, cp2x

x

=N

+x.

Proof of B). Taking into account (2.11) we can prove as in Corollary 4.3.5

that

therefore

N

x

=3

x = x V ‘pix V (cpzx A cpzx) = cpzx.

0

2.31. Remark. Becchio and lturrioz [1978] constructed certain generalizations of n-valued Moisil algebras using explicitly Thomas-like axioms and the Kleene axiom. For n 5 4 the new concept reduces t o t h a t of Moisil algebra.

Represent ation of LM -dgebras by continuous functions

317

$3. The representation of Lukasiewicz-Moisil algebras by continuous functions In this section we shall show that every Lukasiewicz-Moisil algebra can

be represented as an algebra of continuous functions defined on a Boolean space. In the finite case Cignoli [1972]gave a representation by means of Post algebras and used it t o obtain a characterization o f the dual of the category of n-valued Post algebras, a characterization of the injective LukasiewiczMoisil algebras and a construction of the free product of Post algebras. The results of this section generalize t h e above mentioned results of Cignoli and are due t o the first and second author. The construction in Theorem 3.6 follows essent ia II y lorgulesc u [1984~).

3.1. Hvpothesis and notation. Throughout this section: a) t he set LLq of increasing functions

f

:

I -+ L2 is viewed either

as the

algebra Lrl E LMN6 constructed in Example 3.1.10.6 or as i t s negationfree part

Li'

E LM29 in Example 3.1.3 (cf. Definition 3.1.8);

b) X denotes a Boolean space; c) t he set (L\q)x = {F I F : X + LLq} is made into an algebra (L\q)x E

LMN6 or ( ~ 5 ; ~ E) ~LM6 obtained from the corresponding structure of

L p (cf.

a)) by obvious pointwise definitions;

d) for each a E J = {0} f,(k) = 1 for k > a; e) t he set J}

L'\

+ I we define

fa

E LL4 by fa(k) = 0 for k 5 a ,

is endowed with the topology having the basis

{[fa)

u ((fa1 I a E J l ;

f) we set (3.1)

C T ( X )= {F E (L\')x

(3.2)

B ( X ) = {F E (Li')x

IF

is continuous} ;

I F ( x ) is constant Vx E X } ;

Ia

E

318

Representation theorems and duality for LM-algebras

g) I is supposed to be regular; cf. Definition 4.6.7. 3.2. Lemma.

In each o f t h e classes L M B and LMNB; (i)

C T ( X ) is a sublugebra o f ( ~ i 4 ) ~ ;

(ii) B ( X ) is a subalgebra o f C T ( X ) ; (iii) B ( X ) = C ( C T ( X ) )= c ( ( L [4),x ). Proof.

(i) T h e constant functions 0 7 1 E C T ( X ) . Suppose F,G E C T ( X ) . Then

and similarly

(F therefore

([fa))

= F-'

([fa))

u G-l

([fa))

FVG E C T ( X ) . One proves similarly t h a t F A G E C T ( X ) .

Further note that

and for every

f

E

@, if a E J - (1)

then

Representation of LM-algebras by continuous functions while (3.5)

fl(k) = 0 (Vk [fa) =

E

319

I),therefore

{f E LL4

and similarly (3.6)

(fa] =

{f E LL4

then from (3.3) and (3.5) we get

and (cpiF)-'([fi))= {Z E

XI YiF E LL4} = X . One proves similarly

that ( Y i F ) - ' ( ( f e ] ) = F-'((fi]) ( a # 0) and (cp;F)-'((f,]) = X . This shows that cpiF E C T ( X ) and one proves similarly that cpiF E

C T ( X ) . Further (3.4)-(3.6) imply

for a

# 1, while ( N F ) - ' ( [ f I ) ) = X .

One computes similarly

( N F ) - ' ( ( f , ] ) and the conclusion is that N F E C T ( X ) . (ii) B ( X ) is clearly a subalgebra of (LF1)xand moreover B ( X ) C T ( X ) because F - ' ( ( f , ] ) , F-'([f,)) E ( 0 , X ) for every F E B ( X ) .

(J~F')~,

(iii) As B ( X ) C C T ( X ) G it suffices to prove that B ( X ) = C ( ( L i 4 ) x ) .But for every F E ( L i 4 ) x ,

F

E

C((LL4)X)u cpiF = F (Vi E I) u

* (cpiF)(.)(k) = F ( z ) ( k )(VZ E X)(Vk,iE I) H * F ( Z ) ( i )= F ( Z ) ( k ) (Vk,iE I) (VZ E X) u * F ( z ) E B ( X ) (VZ E X ) .

320

Representation theorems and duality for LM-algebras

3.3. Corollary.

B ( X ) is a Boolean algebra. 3.4. Lemma.

X is homeomorphic to S p e c B ( X ) . Proof. X is homeomorphic t o SpecKX by Lemma 2.1.16, where K X is the Soolean algebra of all clopen sets of X , by Lemma 2.19.a. Therefore it suffices t o prove t h a t IIX is isomorphic t o B ( X ) .

@ ( F )= F-'({fo}). It follows from [fo) = {fo} is clopen in (L\q)x, hence F-'({fo})is clo-

For every F E B ( X ) set

{fo} =

n

[fit)

$I

that

X , therefore @ : B ( X ) + K X . Clearly 9 is a homomorphism. To prove bijectivity take V E K X and look for F E B ( X ) such that V = @ ( F )= {x E X I F ( z ) = fo}, which holds for a single function: F(x)= fo for x E V , F ( s ) = fo for x E X - V . 0 pen in

3.5. Lemma.

Let

1 be a subalgebra o f C T ( X ) (in LM6 o r in LMN6) such that B ( X ) E

1. T h e n C ( 1 ) = B ( X ) . Proof.

c(L>= i

n C ( C T ( X ) )= 1 n B ( X ) = B ( x ) .

0

3.6. Theorem. Let L E LM6 o r L E LMN6, card L > 1. T h e n there is a Boolean space X , unique u p t o a homeomorphism, such that L is isomorphic t o a subalgebra o f C T ( X ) such that C ( i )= B ( X ) . Proof. Take X = SpecC(L). According t o t h e dual of Lemma 5.2.2, the elements of X are of th e form P' = P n C ( L ) , where P E Spec L. For every a E L define @ ( a ) : X --t Li4 by (3.7)

@(.)(P*)(i) =

P* E X

{

0

7

1)

PiaE

P'

9

via#P')

i E I . Then it is easily checked that @ : L -+(L\q)x is a homomorphism. If a # b then via # p i b for some i E I;if e.g. cp;a $ cpib

where

and

Represent at ion of LM-algebras by continuous functions then there is P' E X such that v i a E

321

P* and cpib # P*, by the

dual of

E

= @ ( L )is a

(Li')x, isomorphic t o L. To prove i C C T ( X )take @(a)E i.Then (@(u))-'([fl))

= X , while

Corollary 1.3.13, therefore @(u) # @ ( b ) . It follows t h a t subalgebra of for a

# 1, using Definition 3.1.d we obtain =

(@(.))-'([fa))

{P* E x I @(u)(P*)(a+) = 1)

= {P* E X

I 'pa+u

=

P'} = s('pa+u) ;

similarly ( @ ( u ) ) - ' ( ( f o ] )= X and ( @ ( u ) ) - ' ( ( f a ] )= s(cpaa). This proves @(a)E

CT(X).

To prove will imply

C(i)= B ( X ) it suffices to show t h a t B ( X )

2 because this

B ( X ) = C ( B ( X ) ) G C ( 2 ) C C ( C T ( X ) )= B ( X ) by Corollary

3.3 and Lemma 3.2 (iii). So take F E B ( X ) . We have seen in t h e proof of Lemma 3.4 that F - ' ( { f o } ) E K X , hence Theorem 2.1.8 implies the existence of an element u E C(L)such that F - * ( { f o } )= s ( u ) . For every P' E X and i E I,

* P* 6

S('p@)

= s(a) H

F ( P * )# fo

therefore @ ( u ) ( P *= ) F ( P * )i.e. F = @ ( u )E Note that

H

F ( P * ) ( i )= 0

i.

2 is unique up t o an isomorphism. Therefore if X '

0

is another

space with the same properties, Lemmas 3.4 and 3.5 imply X ' S B ( X ' ) =

C(i)= B ( X )

= x.

3.7. Definition. The space X constructed in Theorem 3.6 will be called t h e Boolean spec-

trum of L. Our next aim is t o use Theorem 3.6 in order t o obtain a characterization of d-valued Post algebras, under th e same hypothesis that 1 is regular.

Representation theorems and duality for LM-algebras

322

3.8. Lemma. C T ( X ) is a 6-valued P o s t algebra. Proof. We check condition (iii) in Proposition 4.1.6, via Lemma 3.2 (iii). Let Q E ( C ( C T ( X ) ) ) [ ' . Then Q E (C((Li')x))" and it was shown in the

proof of Corollary 4.1.2 that setting F ( z ) ( i )= Q ( i ) ( x ) ( k ) (Vk E I) yields a function F E (Li')x such that 'piF = Q ( i )(Vi E I ) . Now using the same technique as in Lemma 3.2 and taking into account that Q ( i )E C T ( X ) it 0 is easily proved that F E C T ( X ) . 3.9. Definition. Let L be a d-valued Post algebra. A subalgebra Lo of L in LM29 is said to be a P o s t subalgebra of L provided Lo contains the centres of L. 3.10. Lemma. L e t L be a 6-valued P o s t algebra a n d Lo a subalgebra of L in L M 6 .

(i) If Lo

23

a P o s t algebra t h e n it is a P o s t subalgebra of L.

(ii) If, moreover, C(L)E Lo t h e n Lo = L . Proof. (i) The centres of Lo satisfy condition (Cl) in Definition 4.1.5, which characterizes the centres of L as well.

(ii) Clearly C ( L )= LO), therefore Lo = L by Proposition 4.2.9 (ii).

0

3.11. Corollary. L, has no proper P o s t subalgebra. 3.12. Theorem. L e t L, X a n d fulfil t h e hypotheses of T h e o r e m 3.6. T h e n L is a 29valued Post-algebra if a n d o n l y if = C T ( X ) .

e

e

Proof. We have to show that is a Post algebra if and only if 2 = C T ( X ) . The "if" part follows from Lemma 3.8. Conversely, suppose L is a Post algebra, But is an LMd-subalgebra of C T ( X )and C ( C T ( X ) )= B ( X ) =

e

Represent ation of LM -dgebras by continuous functions

323

C ( 2 ) by Lemma 3.2 (iii), therefore 2 = C T ( X ) by Lemma 3.10. Our last aim in this section i s t o prove t h a t the Post algebra minimal extension of the Lukasiewicz-Moisil algebra

L; this

0

C T ( X )is a

will sharpen the

existence of a minimal Boolean extension of a distributive lattice.

3.13. Remark. If f , g : L -+ L’ are two LM6-morphisms such t h a t f I C ( L ) = g I C ( L ) then f = g , because

therefore f(x) = g(x) (Vx E

L ) by the determination principle.

3.14. Lemma.

Let L, L’ be algebras in LM6 (in LMNd) of cardinalities > 1; let X , X’ be their Boolean spectra and @L, Qpv the corresponding isomorphisms constructed in Theorem 9.6. FOT every homomorphism f : L -+ L’ there is a unique homomorphism g : C T ( X ) -+ CT(X’) such that g ( @ L ( a ) ) = @ u ( f ( a ) ) for every a E L.

( C ( L ) .Then h : C ( L )-+ C(L’) and Spec h : X’ cf. Notation 2.1.10. For every F E C T ( X ) the map Proof. Set h = f

(3.8)

g ( F ) = F o Spech : X’

-+

-+

X;

D(L2)

is continuous because Spec h and F are so. It follows t h a t g : C T ( X ) -+

CT(X’) and moreover, g is a homomorphism because Spec h = h-l and the operations on (L\q)x are defined pointwise. For every u E L, PI* E X’ and i E I , it follows from (3.8) and (3.7) t h a t

o

g ( ~ L ( u ) ) ( ~ ’ * ) (=i ) H @L(u)(h-l(P‘*))(i) = H H

o

E h-’(P’*) H v i f ( ~=)f(qia) E P’* H @,y( j ( u ) ) ( P ’ * ) ( i= )

o

proving that g ( @ L ( u ) ) = @ p ( f ( u ) ) .

H

Representation theorems and duality for LM-algebras

324

CT(X)

C T ( X ‘ ) be a homomorphism such that g*(@L(ca)) = @ p ( f ( a ) ) for every a E L. If z E C ( C T ( X ) ) ,then z = @ ~ ( y for ) some y E L. It follows that g * ( x ) = therefore 9* I C ( C T ( X ) )= S*(@L(Y)) = @ U ( f ( Y ) ) = g ( @ d y ) ) = g I C ( C T ( X ) ) ,hence g* = g by Remark 3.13. To prove the uniqueness of g , let g*

:

+

3.15. Theorem. Let L, X , 2 and @ have the meanings in Theorem 3.6. FOT every 19valued Post algebra A and every h o m o m o r p h i s m f : L -+ A there is a unique homomorphism h : C T ( X ) -+ A such that h ( @ ( a ) )= f(a) f o r

every a E L (cf. R e m a r k 4.1.13). Proof. Let

X’ be the

spectrum of

A and

@’

ding isomorphism constructed in Theorem 3.6.

:

A

+

Then

A the corresponA

=

CT(X‘) by

C T ( X ) + CT(X’) be the homomorphism obtained by applying Lemma 3.14 t o the homomorphism f : L -, A . Then h = (a’)-’ o g : C T ( X ) + A is a homomorphism and for every a E L, Theorem 3.12. Let g

:

The uniqueness of h is proved in t h e same manner as the uniqueness of g in Lemma 3.14. 0

3.16. Remark. lorgulescu (1984~1has obtained a generalization of Theorem 3.6 to malgebras. The point is a convenient duality theory for Boolean m-algebras.

PFl(B) is played by a family 3 of prime m-filters of the Boolean m-algebra B such t h a t the intersection of all members of 3 is (1) and every non-zero element of B is contained in a member of 3. The In t h a t theory the role of

corresponding concept of Boolean m-space is defined by the requirements

that every. intersection of a t most m clopen sets be clopen and the lattice of clopen sets include a distinguished prime m-filter. After the construction of this Boolean m-duality, the Boolean m-spectrum of an m-algebra

L is taken

Representation of LM -algebras by continuous functions t o be the above family

F

325

corresponding to the Boolean m-algebra C ( L ) .

Then the proof runs essentially as for Theorem 3.6 and its prerequisites.

326

Representation theorems and duality for LM-algebras

$4. The representation of Lukasiewicz-Moisil algebras by Moisil fields of sets The set-theoretical representation Theorems 4.11 and 4.13 are due t o Filipoiu [1979], [1981]; cf. [1981a]. The starting point is the notion of 19structure, similar to a concept which has been used in the theory of Post algebras; cf. Maksimova and Vakarelov [1972]. 4.1. Definition. Suppose I is a dually well-ordered set, i.e. every non-empty subset of I has a last element. A 19-structure is a couple S =

where T

# 0 and the sets Af

(A,{AF 1 t E T , i E I } ) ,

satisfy the following conditions:

(4.1)

A; #

(4.2)

A: n A: = 0

(4.3)

i<j+AiCA;

(4.4)

A = U {Af I t E T , i E I }

0 Vt E T , Vi E I ;

# t’, V i , j E I

Vt,t’ E T , t

;

W E T , Vi,jEI;

.

We also introduce the following notation and terminology:

I y H vt E T Vi E I ( Z

(4.5)

(z

(4.6)

At = u { A i I i E I} ,

(4.7)

cpi(X) = u { A t I Af

X}

E

A; + y E A;))

vx,y E A ;

V X C A , Vi E I

and we denote by B ( A ) , the family of increasing subsets of A ; cf. Definition 2.3.1. 4.2. Remark. If 2 E A , t is the (necessarily unique) element of T such that z E At and i is the last element of I such that z E Af then { y E A I z 5 y} = A;. 4.3. Lemma. a)

{ A t } t Eis~a partition of A ;

The representation of LM-algebras by Moisil fields of sets

327

b) the relation 5 is reflexive and transitive;

e) X E B ( A ) @ X = U { A f I t E T ,

iEI,

AfEX};

Proof. a)

- d) follow immediately from the above definitions and imply e), +.

e,+)

c

If x E X E B(A) and t, i are as in Remark 4.2 then x E Af X . Therefore X C u { A l I Af c X } E X.

c cpi(X U Y). To prove the converse inclusion we Y. take t E T such that Af C X U Y and prove t h a t Af s X or Af

f) Clearly cpi(X) U vi(Y)

Otherwise there exist a E A f - X and b

E Af-Y. Let i‘and i”be the last indices of I such that a E Af, and b E A;,,, respectively. If i’ 5 i” then b E A:, Af, therefore a 5 b by Remark 4.2, hence a E Y would imply b E Y, a contradiction; this proves t h a t a $! Y, consequently a $! X U Y,

c

again a contradiction. The case i”5 i’yields a similar contradiction.

g) Clearlycpi(

n

n

XX)C_

X€A

cp;(X,). Conversely, t a k e x E

n

cpi(X,)

X€A

XEA

Then

VA E A 3t E T Af & XX& x E At ; but in view of a) it follows that

that

t

we obtain Af C_

n

&A

t

is i n fact independent of A, hence for

X X and x E At, that is x E cp;(

0 XX). AEA

328

Representation theorems and duality for LM-algebras

4.4. Proposition. FOT every d-structure

s, the 8yStem

where cpj and (P; are defined by (4.7)and (4.9)

Cpi(X) = U { A t I A:

X}

VX C A , Vi E I ,

is a complete d-valued LM-algebra.

Proof. It is easy to check (3.1.1)-(3.1.3)

then A; G X, otherwise Af

n A;

and (3.1.5). Then note that if

= 0 whenever A;

CX .

Therefore

which proves (3.1.4) via (4.7) and Lemma 4.3.2. Similarly, if pj(X) = cpi(Y) Vz E I then X = Y follows via Lemma 4.3.e from

4.5. Definition. The subalgebras of the algebras M ( S ) constructed in Proposition 4.4 are called 19-valued Moisil fields of simply Moisil $-fields. The study of the ideals of an LM-algebra (cf. Ch. 5, $2) suggests the construction of an LM-algebra which will be a particular case of the above ones.

4.6. Definition. A d-space is a triple C = (S, 5, { D i } i E ~where ), S # 0, I is a reflexive and transitive relation on S and D; : S -+ S (Vi E I) satisfy

The representation of LM-algebras by Moisil fields of sets (4.10)

D; o Dj = D;

(4.11)

i 5j

(4.12)

,

329

Vi,j E I ;

+ Dix 5 Djx 2 5 y + Dix = Diy

Vx E S , Vi,j E I ; Vi E I , V X ,y

E

S ;

4.7. Lemma.

Suppose I is dually well-ordered and let C = ( S ,5 , { D i } i € ~be) a 9-space. Define x M y ($ (Dix = Diy Vi E I ) , f = { y E S I x M y} and Sf = { y E S I y 2 Dix}. Then M is an equivalence on S and S(C) = ( S ,{Sf I i E 9 = S/ x, i E I } ) is a 9-structure. Proof. Clearly M is an equivalence. Sf are well defined and not empty. Note that (4.12) and (4.10) imply (4.15)

z E Sf

=$

D ~ =zDkDiz = Dkx (Vk E I ) =$ 2 = f

therefore if z E Sf n S,” then f = 2 = 9. Finally (4.3) and (4.4) follow from (4.11) and (4.14), respectively. 0 4.8. Remark. Let 5‘ be the relation defined by (4.5) on the h t r u c t u r e Lemma 4.7. Then x

S constructed in

5 y +- 5 5’ y .

For every L E LM9 l e t us set (4.16)

P 3 ( L ) = U yT1(PF1(L)) . iEI

4.9. Lemma.

Suppose I is dually well-ordered and L E LM9.

The triple ( P 3 ( L ) ,

G, {Ai}iEI), where Ai = c p r l ( p q L ) (Vi E I ) , is a 9-space. Proof. Properties (4.10)-(4.13) are valid even for the functions yY1 applied

Representation theorems and duality for LM-algebras

330

cp;'(PF(L)) C PF(L). Let us check e.g. (4.12). If F E PFl(L) then clearly cp;'(F) = cp~'(F*)and F' is a maximal filter of C(L) by Lemma 5.2.2 and Proposition 1.3.18, therefore if G E PFl(L) and F C G then F* = G* hence cp;'(F) = cp;'(G). To prove (4.14) take F E PF(L). Then F = cp,'(G) for some i E I and G E PFl(L) therefore F = cp;'(cp;'(G)) = Ai(F) E t o elements of PFl(L); in particular (4.10) implies t h a t

u

Ai(PF(L)).

0

i€I

4.10. Corollary. a)

If P E PFl(C(L)) t h e n cp;'(P) E PF(L) (Qi E I ) .

b) FOT every r , y E L such that x $ y there i s F E P F ( L ) such that x E F and y # F ; in particular for z # 1 there i s G E PF(L) such that z $! G. Proof. a) cp;'(~)

=

ai( c p ; l ( ~ ) ) .

$ y then pix $ cpiy for some i E I , hence there is P E PFl(C(L)) such that cpix E P and (piy $! P. Now take F = cp;l(P) and apply a).o

b) If r

Now we prove that if I is dually well-ordered then every d-algebra is

isomorphic to a Moisil $-field: 4.11. Theorem.

Suppose I is dually well-ordered and L E LMd. FOT each x E L let

h ( 2 ) = { F E PF(L) I r E F } . T h e n (4.17)

h : L +M(S(PF(L)))

i s a n injective L M d - h o m o m o r p h i s m . Zom ment .

PF(L) denotes both th e 6-space in Lemma 4.9 and its underlying set, S(PF(L)) is the d-structure in Lemma 4.7 and M(S('PF(L))) is the

The representation of LM-algebras by Moisil fields of sets

331

Moisil 6-field in Proposition 4.4.

B(PF(.L)),therefore (4.17) x $ y hence h ( z ) # h(y) by

Proof. Clearly h ( z ) E and

x

#

y then e.g.

holds. If x,y E L Corollary 4.10.b. It

follows immediately from Remark 1.3.5 and Proposition 1.3.9 that h is a

L and P E P F ( L ) . Note first that for every Q E P F ( L ) , (4.6) and (4.15) imply ( P E PF(L)Q*=+ k = 0) and since P E P F ( L ) l 2 PF(L)' it follows that (PE PF(L)Q+ = 4). Therefore, using also the fact that AiP E PF(L)f and h(x) E B ( P F ( L ) ) ,we get DO1-homomorphism. To prove (3.1.50) take z E

P E v ; ( h ( ~* ) ) 3Q E P F ( L ) P E P F ( L ) Q& & PF(L)f

c h(x) *

4.12. Corollary.

Every LMn-algebra is isomorphic t o a Moisil n-field. The Moisil d-fields M ( S )are complete, but the morphism h in Theorem 4.11 need not preserve the existing infinite meets and joins of certain supplementary hypotheses on the d-algebra sentations of

L

L. Under

L one can obtain repre-

that preserve all existing meets and joins. Thus e.g. the

following result was proved in Filipoiu (19811: 4.13. Theorem.

Suppose card I 5 Xo and let L E L M d . For every set Q of infinite meets and joins that exist in L, cardQ 5 Xo, there exists a d-valued Moisil field M ( S ) and a n L M 9 - m o n o m o r p h i s m h : L --+ M ( S ) such that h preserves all meets and joins in Q.

332

Representation theorems and duality for LM-algebras

$5. The Stone duality of d-valued Lukasiewicz-Moisil algebras In this section we use th e results of Chapter 2, $1,t o obtain a characterization of the dual of the category LMd in terms of Stone spaces endowed with an additional structure. The results were first proved by Cignoli [1969a] for n-valued Moisil algebras then by Georgescu [1971a] for d-valued LMalgebras. 5.1. Definition.

A d-valued Lukasiewicz-Moisil space, is a couple ( X ,{ g i } i E l ) where X is a Stone space and gi : X + X (i E I ) is a family of functions satisfying the following conditions:

(Viyj E I )

g j = gi

(5.1)

gi 0

(5.2)

i 5j

(5.3)

gi is strongly continuous

(5.4)

X

(5.5)

V A ,B E K X : g,'(A) = g,T'(B) (Vi E I ) + A = B .

+ g;l(A) G g;'(A)

- g;'(A)

EKX

(Vi,j E I ) (VA E K X ) , (Vi E I ) ,

(Vi E I) ( V A E ICX) ,

5.2. Notation.

We denote by Std the category of d-valued Lukasiewicz-Moisil spaces, in which Hom((X, { g i } i E l ) , ( X ' , { g i } i E l ) consists o f those morphisms f : X +

X' in St that fulfil (5.6)

f

o gi = 9:

o f

,

(V2 E 1) .

In the sequel we will use a slight abuse of notation. As is well known with each map h : S --t T is associated a map h-'

:

P ( Y ) + P ( S ) . The

same symbol h-' will denote any restriction of the above function.

5.3. Lemma. Let ( L ,A, V, 0, 1, {pi}iEr, {@}iE1) be a d-valued Lukasiewicz-Moisil algebra. Let Spec L denote the space of p T i m e ideals of L endowed with the

The Stone duality of fl-valued Lukasiewicz-Moisil algebras topology { k ( I ) I I E Id L}; (cf. T h e o r e m 2.1.8). is a 19-valued Lukasiewicz-Moisil space.

333

T h e n (Spec L, {cp;'}iE~)

Proof. Spec L is a Stone space by Theorem 2.1.8. Further (p'; : Spec L -+ Spec L by the dual of Proposition 5.2. Then (5.1) follows from 9;' o p;' = ( c p j o q i ) - l = 9;.' To prove the next points recall that the compact open sets of L are given by li(SpecL) = {.(a) I a E L} where s ( a ) = {P E Spec L 1 a # P},cf. (2.1.13) and (2.1.7), respectively. Note that

P

E ( v ; ' ) - l ( s ( u ) ) @ cpf'(P) E s ( a ) H a

P

E (v;')-'(s(u))

# (pF'(P) ,

therefore (5.7)

* via # P * P E

S('piU)

P E Spec L and we will obtain (5.2) (5.5) from (5.7). If i 5 j 'pja hence ( y i u # P + vja $ P).Therefore ( ~ ; l ) - ' ( s ( a ) )C (cp;')-'(s(u)), proving (5.2). Then (cp;l)-'(s(a)) = s ( v i a ) E K(Spec L),

for every

then cpiu 5

which proves (5.3). Further applying Lemma 2.2 (vi) we get

P E Spec L - (p;'>-' ( s ( u ) ) e v i a E P n C(L) H

e cpia # P n C(L) H P E s(cpia) , therefore Spec L - (v ;')-'(s(u )) = s(@a) E K(Spec L) i.e. (5.4) holds. Finally (5.5) is obtained via the injectivity of s (cf. Corollary 2.1.6):

( v ; ' ) - l ( s ( u ) ) = (&)-'(s(b))

(Vi E I)

#

+ via = 'p;b (Vi E 1)j a = b . o

H s(cp;a) = s(cpib) (Vi E I >

5.4. Lemma. Let f : L + L' be a m o r p h i s m in LM19. T h e n (5.8)

{

Specf

:

(SpecL',

{V;-'}i€I)

+

(SpecL, { V ; ' } i a )

7

Spec f(P)= f -'(P>7 is a m o r p h i s m in St19.

Proof. Specf is strongly continuous by Lemma 2.1.11 and it remains t o check (5.6) for Specf, cp{-' and cp;'. But

334

Representation theorems and duality for LM-algebras

5.5. Notation. Let

(5.9)

Spec : LM8

--f

St8

be the functor pointed out in Lemma 5.3 and 5.4.

5.6. Lemma. Let ( X , { g i } i E l ) be a 0-valued Lukasiewicz-Moisil space. Let K X stand for the family of compact subsets of X and (5.10)

(VA E K X ) (Vi E I ) .

gi(A) = X - g,'(A)

Then ( K X ,n, U, 0, X , { g ; ' } i E 1 , algebra.

{ij;}iE1)

is a 29-valued Lukasiewicz-Moisil

(ItX,n,U,0,X) E DO1 by Lemma 2.1.13. Then for each i E I , gt:' : K X + K X by (5.3), while J; : K X -+ K X by (5.4). It is well known that g i ' are endomorphisms of P ( X ) , hence of K X as well. Axiom (3.1.3) is fulfilled by the very definition (5.10). Then (5.1) implies gl:f 0 g7' = (gj o gi)-' = g;' i.e. (3.1.4) and similarly (5.2) and (5.5) 0 imply (3.1.5) and (3.1.6) respectively. Proof.

5.7. Lemma. Let f : ( X , { g j } i E ~+ ) (X',{gi}iE1)be a morphism in St.9 and K X , K X ' the LM-algebras constructed in Lemma 5.5. Then

Kf

:

KX' +K X

(5.11)

Ir'f(A)= f - ' ( A )

,

,

is a morphism an LM8.

K f is a morphism in DO1 by (3.1.50) for Kf,gl-' and g;l; cf. Proof.

Lemma 2.1.13 and it remains t o check Definition 3.1.28 and Remark 3.1.29.

The Stone duality of d-valued Lukasiewicz-Moisil algebras

335

This follows via (5.6) as in the proof of Lemma 5.4. 5.8. Notation. Let (5.12)

K

: Std

4

LMd

be the functor pointed out in Lemmas 5.6 and 5.7. 5.9. Theorem. T h e dual of t h e category LMd is equivalent t o t h e category Std. Proof. We construct the functorial isomorphisms

K

(5.13)

s : idLM8

(5.14)

u : idsts --+ Spec

4

o

Spec, o

K ,

by taking the restrictions of the functorial isomorphisms (2.1.34) and (2.1.35) from Theorem 2.1.17. Thus it still remains to prove that S L and CTX are morphisms in LMd and Std, respectively. But for every P E Spec L , every u E L and every i E I,

* cp;'(P) E % ( a ) a # ~ p r ' ( P* ) via # P * P E

P E (rpr')-'(sL(u)) H

hence (rp;')-' The proof of (gi')-'

@

sL(cpia) >

o S L = S L o cpi

.

o (TX = OX o gi is quite similar.

0

5.10. Definitio n. The definition of a d-valued Lukasiewicz-Moisil pre-space is obtained from Definition 5.1 by dropping condition 5.5. Let Std0 denote the category of 19-valued LM-pre-spaces, where the morphisms are defined as for Std. Thus Std becomes a full sub-category of Stdo. 5.11. Theorem. T h e dual of t h e category LMdOis equivalent t o t h e category Stdo.

336

Representation theorems and duality for LM-algebras

Proof. Follow the proof of Theorem 5.9, including the proofs of all prerequisites, and drop everything which refers to axioms (3.1.6) and (5.5). 0

The Stone duality of &valued LM-algebras with negation

337

$6. The Stone duality of &valued Lukasiewicz-Moisil algebras with negation In this section we use the results of $5 and of Chapter 2, 52, t o obtain a characterization of the dual of the category LMN29 of &valued LukasiewiczMoisil algebras with negation, in terms of appropriate Stone spaces. The results are due t o Cignoli [1969a] for n-valued algebras and lorgulescu [1984] in the general case. 6.1. Definition.

The couple ( X , { g ; } ; € J ) is called a d-valued Lulcasiewicz-Moisil space with negation provided

(6.2)

( X , g o ) E 1st

where d :

I+I

,

is a decreasing involution; cf. Notations 5.2, 2.2.2 and

Definition 3.1.8. 6.2. Notation.

We denote by StN6 the category of &valued Lukasiewicz-Moisil spaces with negation in which Hom((X, { g i } i E J ) ,( X ' , { g : } i € J ) ) consists of those strongly continuous functions

f

:

X

--$

X' that satisfy

6.3. Remark.

It follows from (6.4) that in Definition 6.1 condition (6.3) can be replaced by

338

Representation theorems and duality for LM-algebras

6.4. Remark. For every X,X' E StN6 (6.7)

HomstNd(X,XI) = Hornstd(X,XI) fl Hombt(X,X')

.

6.5. Lemma.

Let ( L ,A , V, N , 0,1, { pi } i El , be a 6-vaZued Lukasiewicz-Moisil aZgebra with negation. Then (Spec L , (gi}iEJ), where SpecL is the 6-valued LM-space constructed in Lemma 5.3, gi = (P;~ (Vi E I ) and go(P) = L - NP, is a Lukasiewicz-Moisil d-valued space with negation. Proof. Conditions (6.1) and (6.2) are fulfilled by Lemmas 5.3 and 2.2.3, respectively. Then for every i E I , P E Spec L and x E L ,

6.6. Lemma.

Let f : L

L' be a morphism in LMN6 and Spec L, Spec L' the spaces constructed in Lemma 6.5. Then

(6.8)

t

Spec f : SpecL'

t

Spec L

defined by Specf(P) = f-l(P), is a morphism in StN.9. Proof. By Lemma 6.5, 5.4, 2.2.4 and Remark 6.4. 6.7. Notation. Let (6.9)

Spec : LMN29 t StN6

The Stone d u a l i t y of 6-valued LM-algebras with negation

339

be the functor pointed out in Lemmas 6.5 and 6.6. 6.8. Lemma.

L e t ( X , {g;}iEJ ) be a 29-valued Lukasiewicz-Moisil space w i t h negation. T h e n ( K X ,n, U, N , 8 , X , {g;'};€I, { S i ' } i c ~ ) where K X is the 0-valued LM-algebra constructed in L e m m a 5.6 and f o r every A E K X , N ( A ) = K X - g o ( A ) , i s a 6-valued Lukasiewicz-Moisil algebra with negation. Proof. K X E LM29 by Lemma 5.6 and ICX E M g by Lemma 2.2.6. It remains to check conditions (3.1.25) and (3.1.30.1) in Definition 3.1.8. For every i E I and every A E K X ,

go(S;'(A))

= 90' (s;'(A)) = (9;

0

go)-'(A) = 9 i 1 ( A )

therefore Ng;'(A) = K X - g;'(A) = gi(A) and similarly g;'(go(A))

=

g z 1 ( A ) via Remark 6.3, therefore g,'N(A) = g ; ' ( K X

-go(A)) = K X -gi' (go(A)) = = gG'(A)

.

0

6.9. Lemma. Let f : (x,{gi};,=~) -+ ( x ' , { g : } ; E J ) be a morphism in S t N 8 and ICX, KX', the 29-valued L M - algebras with negation constructed an L e m m a 6.8. Then the f u n c t i o n (6.10)

Kf

:

KX'

t

KX

defined by I C f ( A ' ) = f - ' ( A ' ) , is a m o r p h i s m in L M N d . Proof. By Lemmas 6.8, 5.7 and 2.2.7. 6.10. Notation. Let (6.11)

(I

:

StN29

--t

LMNd

be the functor pointed out in Lemmas 6.8 and 6.9.

0

Representation theorems and duality for LM-algebras

340

6.11. Theorem. The dual of the category LMN6 is equivalent to the category StN19. Proof. Construct the functorial isomorphisms (6.12)

s : idLMN8 + K o Spec,

(6.13)

0

:

idstNs + Spec o K

,

by taking the restrictions of the functorial isomorphisms (2.1.34) and (2.1.35) from Theorem 2.1.17. In view of Theorems 5.9 and 2.2.9 the functorial isomorphisms are also valid for LMN29 = LM29 n Mg and StN6 = S t 6 n 1St.U Another duality for Moisil algebras was constructed by Georgescu and Vraciu [1969a,b] in terms of sheaf theory.

The Priestley duality of &valued Lukasiewicz-Moisil algebras

341

57. The Priestley duality of &valued Lukasiewicz-Moisil algebras In this section we use the results of Chapter 2, 53, to obtain a charac-

terization of the dual of the category LMO of d-valued Lukasiewicz-Moisil algebras in terms of ordered topological spaces. The results are due to Filipoiu [1980], [1981a]. 7.1. Definition. A Priestley space (X, 7,s) endowed with a family of mappings @i : X + X (i E I ) is said to be a d-valued Priestley space (X,{@;}~Q) provided the following conditions are fulfilled: (7.1)

@pi

E Homp,(X,X)

(Vi E I) ,

(7.2)

@j

o Q j = @i

(Vi7.7- E I) ,

(7.3)

i5 j

(7.4)

X - @ ; l ( Y )E G(X)

(7.5)

(Q;'(Y)

+ @;l(Y)

(W E GX) (Vi,j E I) ,

G Q;l(Y)

(W E G X ) (Vi E I ) , (Vi E

= Qyl(Z)

I) + Y = 2 ) ( W , Z E G X ) ;

cf. Definitions 2.3.1, 2.3.2 and 2.3.4.

7.2. Notation. We denote by Pr6 the category of d-valued Priestley spaces in which Hom((X, {@t)i}iEl),( X ' , {@f}ier)) consists of those isotone continuous func-

tions f : X -+ X' that satisfy (7.6)

f o

@i

= Q{ o f

(Vi E

I).

Recall the construction from Ch. 2, 53; if L E DO1 then

and if Q E HomDol(L,L') then

342

Representation theorems and duality for LM-algebras

7.3. Lemma.

Let ( L ,A, V, 0 , 1, { ' p i } i E r , { @ i ) i € l ) be a 9-valued Lukasiewicz-Moisil algebra. T h e n ( H L ,{ H ' p j } j € ~is) a 19-valued Priesti'ey space. Proof. H L E Pr by Lemma 2.3.8 and H'pi E Homp,(HL,HL) (Vi E I ) by Lemma 2.3.9. Clearly (H'pi o H'pj)(f) = f o 'p, o 'pi = f o 'pi= ( H ' p i ) ( f ) . Thus (7.1) and (7.2) hold.

To prove (7.3) suppose i Ij , Y E G ( H L ) and f E (H'pi)-l(Y). Then 'pi I ' p j and f o 'pi = ( H ' p i ) ( f ) E Y . But f E H L is isotone and Y is increasing, hence f o 'pi 5 f o 'p, E Y ,therefore f E (H'pj)-l(Y). Let t : L --t G ( H L ) be the isomorphism in Lemma 2.3.11, i.e. f E t ( z ) H f ( x ) = 1for every x E L and f E H L . If Y = t ( x ) , (7.4) follows from

f

E H L - (H'p;)-'(Y)#

H f(Yjx) = O H

f

0 'pi

f ( p i ~=) 1

#Y u

f

E

t(pi3) .

To prove (7.5) suppose that for all i E I , (Hyi)-' ( t ( z ) )= (H'pi)-' (t(y)) that is for all v E H L , (H'pi)(v)(x)= 1 u (H'pi)(v)(y) = 1. Then "('pix) = 1 "(cpiy) = 1, i.e. t('piz) = t('piy) (Vi E I), hence pix = 'piy (Vi E I). Therefore x = y. 0

*

7.4. Lemma.

Let f : L --t L' be a m o r p h i s m an LM8 and H L , HL' the spaces constructed in L e m m a 7.3. T h e n (7.9)

Hf

:

HL'+ H L

defined by (7.8) is a m o r p h i s m in Pr8. Proof. Hf is isotone and continuous by Lemma 2.3.9. Moreover, for every i E I and every v' E HL'

=

(H'pj 0

Hf)(v')

.

0

The Priestley d u d i t y of 6-valued Lukasiewicz-Moisil algebras

343

7.5. Notation. Let (7.10)

H : LM6 -+ Pr-9

be the functor pointed out in Lemmas 7.3 and 7.4. 7.6. Lemma.

Let (X, { @ i } ; € ~ )be a d-valued Priestley space. T h e n (GX, n, U, 0,X, { @ ; ' } i € ~ , { 6;'}iE~),where GX is the f a m i l y of clopen increasing subsets ofX and 6;'(A) = X - @;'(A) f o r every A E G X , is a 6 - v a h e d Lukasiewicz-Moisil algebra. Proof. (GX,n,U,!&X) E DO1 by Lemma 2.3.5. The maps @pi' = G@i are endomorphisms of G X by Lemma 2.3.6. Then

6;'

:

GX

--t

G X by (7.7)

and axiom (3.1.3) in Definition 3.1.1 is verified by the very definition of while axioms (3.1.4)-(3.1.6) respectively.

spr',

follow immediately from (7.2), (7.3) and (7.5), 0

7.7. Lemma.

Let f : ( X , { @ i } i E-+ ~ ) ( X f , { @ i } i Ebe~ )a m o r p h i s m in Pr-9 and G X , GX' the 8-valued LM-algebras constructed in L e m m a 7.6. T h e n the f u n c tion (7.11)

Gf : GX'-+ G X

defined by (Gf)(A') = f-'(A'), Proof.

Gf is a morphism in DO1 by Lemma 2.3.6 and

7.8. Notation. Let (7.12)

is a m o r p h i s m an L M 6 .

G

:

Pr6

-+

LM6

Representation theorems and duality for LM-algebras

344

be the functor pointed out in Lemmas 7.6 and 7.7.

7.9. Lemma. FOT every L E LM29 the isomorphism t L : L Lemma 2.3.11is an isomorphism in LMd. Proof. For every z E

(GH% =

{V

= {v

+

GH(L) constructed in

L, since tL(z) = {v E H L I v ( z ) = l}, it follows that

0 tL)(X)

= (HY4-l (tL(.))

=

I

E HL ( H v i ) ( ~ ) (= t ) 1) =

E HL

I v('p;(z))

7.10. Lemma. FOT every X E Pr29 the isomorphism TX : Lemma 2.3.12is an isomorphism in Pn9. Proof. For every z

.

= l} = tL(cp;z)= ( t L o pi)(.)

x + H G ( X ) constructed in

E X , ~ ( z :) GX + {0,1} is defined by .rx(z)(A)=

1H z E A . Therefore we obtain in turn

7.11.Theorem. The dual of the category LM29 is equivalent t o the category Pr29. Proof. Construct the functorial isomorphisms

(7.13)

t

: idLM0 + G o

(7.14)

T

:

H ,

idp,g + H o G ,

by taking t he restrictions of the functorial isomorphisms (2.3.14)and (2.3.15) from Theorem

2.3.13.This

is possible in view of Lemmas 7.9 and

7.10. 0

The Priestley duality of &valued LM-algebras with negation

345

$8. The Priestley duality of &valued Lukasiewicz-Moisil algebras with negation In this section we use the results of $7 and of Chapter 2, $4, to obtain a characterization of the dual of the category LMN29 in terms of ordered topological spaces. 8.1. Definition. A 8-valued PriestZey space with negation is a triple ( X , g , { i P i } i E l ) such that (8.1)

( X , g ) E PrMg

(8.4)

'Pi

?

(Vi E I) ,

o 9 = @pi

where d : I + I is a decreasing involution; cf. Definitions 2.4.1 and 7.1 and Notations 2.4.2 and 7.2. 8.2. Notation. We denote by PrN29 the category of &valued Priestley spaces with negation in which Hom(X,g, {@i}iE~), (X',g', {@:}iE~) consists of those isotone continuous functions f : X + X' that satisfy

(8.5)

9'

(8.6)

@:

0

o

f

=f

f =f

0

9

9

o @i

(Vi E I) .

8.3. Remark. It follows from (8.4) that in Definition 8.1 condition (8.3) can be replaced bY (8.7)

g o

@i

= @di

8.4. Remark. For every X , X ' E PrN6,

(Vi E

I) -

Representation theorems and duality for LM-algebras

346

Using again the notation (7.7), (7.18) introduced first in Ch. 2, 53, we obtain: 8.5. Lemma.

Let ( L ,A , V, N , 0,1, {(pi}iE1, {Cpi}iE1) be a $-valued Lukasiewicz-Moisil algebra with negation. Then ( H L , ~ , { H y i } i ~ where f), 7 : HL H L is --f

defined by (8.9)

(yv)(a) = v ( N a )

(Vv E H L )

,

is a 8-valued Priestley space with negation. Proof. Conditions (8.1) and (8.2) are fulfilled by Lemmas 2.4.5 and 7.3, respectively. In view of Remark 8.3, it remains to prove (8.7) and (8.4). For every

i

E

I , v E H L and a E L ,

8.6. Lemma.

Let f : L + L' be a morphism in LMN29 and H L , HL' the spaces constructed in Lemma 8.5. Then (8.10)

Hf

:

HL' + H L

defined by (7.8) is a marphism in PrN8. Proof. By Lemmas 2.4.6 and 7.4 and Remark 8.4. 8.7. Notation. Let

The Priestley duality of &valued LM-algebras with negation (8.11)

H

: LMNS

347

-tPrNS

be the functor pointed out in Lemmas 8.5 and 8.6. 8.8. Lemma.

Let ( X , g , {@;}iEl) be a 6-valued Priestley space with negation. Then where N A = X - g(A) (VA E ( G X ,n, U, N , 0 , X , {@;l};E~, {6r1}iE1), G X ) and the remainder of G X is defined in Lemma 7.6, is a 19-valued Lukasiewicz-Moisil algebra with negation. Proof. G X E Mg by Lemma 2.4.3 and GX E LM29 by Lemma 7.6. Then for every i E I and A E G X ,

N@F1(A)= g(@T'A) = g-l(@;'A) = (a; o g)-'(A) = @:'A @,'(HA) = X - @;'(gA) = X

- @;'(g-'A)

,

=

= ( 9 o @;)-'(A)= @ i > ( A ) .

0

8.9. Lemma. --t ( X I ,g', {@:}iEl) be a morphism in PrNS a n d Let f : ( X ,g , {@i}iEl) G X , GX' the 19-valued LM-algebras with negation constructed an Lemma 8.8. Then the function (8.12)

Gf

:

G X ' + GX

defined by ( G f ) ( A ' )= f-'(A'), is a morphism in LMN19. Proof. By Lemmas 2.4.4 and 7.7 and Remark 8.4. 8.10. Notation. Let (8.13)

G : P r N S +LMN19

be the functor pointed out in Lemmas 8.8 and 8.9. 8.11. Theorem. The dual of the category LMN19 is equivalent to the category PrN29. Proof. Construct the functorial isomorphisms

0

348 (8.14)

Representation theorems and duality for LM-algebras t :

idLMN0

-+

G

o

H ,

by taking the restrictions of the functorial isomorphisms (2.3.14) and (2.3.15) from Theorem 2.3.13. In view of Theorems 2.4.10 and 7.11 the functorial isomorphisms are also valid for LMNQ = LM9 fl Mg and PrN9 = Pr9 n

PrMg.

0

Representation of n-valued M o i s i l algebras by 3 - v d u e d algebras

349

$9. The representation of n-valued Moisil algebras by three-valued algebras As the three-valued Moisil algebras are t h e richest i n “good” properties,

it seems a natural idea to relate n-valued Moisii algebras to three-valued ones. A. Monteiro [1964],L. Monteiro and Coppola [1964]and L. Monteiro

[1978]have constructed three-valued Moisil algebras starting f r o m monadic Boolean algebras; a simpler and more general construction was given by Boi-

119701. In this section we present the above constructions and severat related results obtained in Boicescu [1984].

cescu

9.1.Definition. Let L E Mn and i E {l,

...,

[:I}.

For every x , y E L set

9.2. Lemma. L e t L E Mn and i E { 1 , ..., [S]}. T h e n pi is a congruence of t h e algebra ( L ,A, V, N , vi,vn-i). Proof. Routine; e.g. if x pi y and

k E {i,n - i}

Nvn-ky = (PkNy, hence N x pi N y .

9.3.Theorem. L e t L E M n and i E {1, ...,

[%I}.

then v k N z = N q n - k x = 0

Then

9.2. W e use Proposition 1.2.9to prove that L;E Mg. For every x , y E L and k E {i,n - i}, Proof. T h e algebra

t i

exists by Lemma

therefore 2 A N ( N 2 A NC) = 2 , i.e.

(1.2.13),while x V 0 = 5 and x

(1.2.12)holds. One proves similarly

V 1= 1.

Representation theorems and duality for LM-algebras

350

We have thus checked axiom (3.3.0) in the axiom system S2 for threevalued Moisil algebras given in Proposition 3.3.2. Further from 'pk(Pn-;(x A y ) = (Pk((Pn-;x A cpn-iy) we obtain (3.3.1). For (3.3.8) we compute

the results are identical for k = n-i, while for k = i we get 0 = y i x A N ' p n - i x because i 5 7~ - i hence 'pix 5 cpn-;x and apply (1.2.19'). Finally (3.3.9) follows via (1.2.19") from

9.4. Remark. The identity cp; = Ncpn-iN holds both in L and L;, so that the endomorphisms of the latter algebra are cp; and ' p n - ; . As a matter of fact the algebra

Ll coincides with

be constructed starting with the weak implication

+ M

Monteiro (cf. Cignoli (1969a1).

9.5. Definition. In every algebra L E Mn we set

(9.3)

z -+ y = ( X + y ) A ( N y + N x )

(9.4) (9.5)

, x ny = N ( N z U N y ) ,

(9.6)

zpy%z+y=l

M

x

uy

Note that

M

=(x +y) +y

and y + z = l .

an algebra that can

,

introduced by A.

Representation of n-valued Moisil algebras by 3-valued algebras

A

351

v VlNY v 'ply) .

(Vn-lX

9.6. Prooosition. p = P1. Proof. If x p y then

1 hence

'pix

I 'ply

+ y)

and

= 1 i.e. ( N p 1 ~ v ' p l y ) A ( \1-1 Y v N\ 1 - 1 2 ) = 5 'pn-ly via (1.2.19"); we obtain by sym-

( P ~ - ~ X

-

metry the converse inequalities, therefore 'pix = 'ply and 'pn-l = y ~ ~ - ~ y . Conversely, z p l y implies z

y = N v l y Vy

2 N'plyVcply

= 1 and si-

M

milarly N y + N x = 'p,-lxVNx M and y

--+

= N'plNxVNx

2 1, hence x

z = 1 by symmetry.

9.7. Lemma. F o r every x , y E

+y =1

0

L E Mn,

(9e10)

'pk(x

u 9) = v k x v 'pky

(k E (1,n - 1))

(9.11)

'pk(x

f l y ) = v k x A 'pky

(k E (1,n - 1)) .

7

Representation theorems and duality for LM-algebras

352

Proof. We use Proposition 9.6, (9.8) and (1.2.18):

9.8. Lemma. The relation p is a congruence of the algebra ( L ,U, n, N , (PI, Vn-1). Proof. In view of Lemma 9.2 and Proposition 9.6, it remains to verify the substitution property for U and n. This follows readily from Lemma 9.7; e.g. if z p d and ypy'then for k E ( 1 , . - 1) we get

9.9. Corollary. The algebra ( L / p ,n, U, N , 0 , 1,cpn-l) coincides with L1. Proof. From Proposition 9.6 and the fact that Lemma 9.7 implies 'pk(xUy) =

pk(z V y) and

cpk(a:

r l y) = ( P ~ (Az y) for k f (1, n - 1 ) .

0

9.10. Corollary. (i)

If L is an n-valued Post algebra, n Post algebra.

# 2,

then L l p is a three-valued

Representation of n-valued Moisil algebru by 3-valued algebras

353

(ii) If n i s even, L/pttl i s a Boolean algebra.

.,>I:[

(iii) C ( L )E pi) ( ~E i{ 1, ...,

If x E C ( L ) t h e n ( 2 ) p ,= {x}.

(v) Proof.

From Proposition 4.1.8 because (plZ1 =

(i)

h

(Pn-lC1

(

~

=3 0 ~and (P,,-~E~ =

= 1.

(ii) If n = 2k then

(Pkx

= ( P k ( P k x implies z p k C p k z (Vz E L ) .

(iii) Let p ; E Mg(L,L / p ; ) be the canonical surjection and q; = pilC(L). Then from (x = ( P k x -%- f = yk2) we deduce easily that q; E B ( C ( L ) ,C(L/p;))and qi is a bijection. (iv) From (iii) and Propositions 5.1.23 and 5.1.14.

The above constructions have been extended by lorgulescu [1984a,c] t o monadic Lukasiewicz-Moisil algebras; cf. Ch. 8, §3.

9.11. Proposition. Let L E M n and i E { 1 , ..., T h e n the pair ( & , p i ) , where ei = L/pi and pi : L + L; is the canonical surjection, is unique u p to a n isomorphism, with the following properties:

.}I;[

(ii) for every A E M3 and f E Mg(L,A) such t h a t f ( ( P k S ) = ( P k f ( 2 ) (z E L, k E { i , n - i}) there is a unique g E M3(Li,A) such that

f

=9

0

Pi.

Representation theorems and duality for LM-algebras

354

Proof. Property (i) holds by th e construction of ei and Theorem 9.3. Further

A and g as in (ii). Define g by g ( p i ( x ) ) = f(z). If pi(.) = p i ( y ) then z pi y i.e. ( P ~ Z= (pky ( k E {i,n - i } ) ,hence (Pkf(z) = f ( c P k 2 ) = f(cPky) = cpkf(y) (k E { i ,n-i}), therefore f(x) = f ( y ) by the determination principle. take

Thus g is well defined. Moreover, g is obviously unique, g E M g ( L i , A ) and g(PkPi(x)) = g(Pi((Pk2)) = f((Pkx) = Y ' k f ( s ) = (Pkg(Pi(x)) (k E

{ i ,n - i}). The uniqueness of the pair (&,pi) is shown in a canonical way (see e.g. the proof of Proposition 7.3.1).

9.12. Corollary. Let L E M n , i E (1,..., ii E M3, f i E M g ( L , i i ) such that fi((~k~= ) ( ~ k f i ( x (k ) E {i,n - i}). T h e n the following conditions are

,}I:[

equivalent:

(ii)

(E;, f i ) satisfies

the eztension property in Proposition 9.11;

(iii) fi is surjective and ji

I C(L)E B(c(L),

~ ( i i ) )is

injective.

Proof.

(i) H (ii): By Proposition 9.11. (ii) + (iii): Shown in the proof of Corollary 9.10 (iii). (iii) + (i): According to Proposition 9.11 (ii) applied t o & and f;there is g E M 3 ( L / p i , i i ) such that f i = g o p i . Then g is surjective because so is fi. For injectivity suppose p i ( z ) # pi(y). This implies the existence of

k E { i , n - i } such that

(pkx

= fi(PkZ)# f i ( y ) = g(pi(Y)).

# P ~ Yhence , qkfi(z)

q k f i ( y ) , therefore g ( p i ( x ) ) = fi(x)

+

9.13.Corollary. For every

f

i E {l,..., o pi = pi o f .

E M n ( L , L ' ) and every

fi E M 3 ( L / p ; ,L'/pi) such that f;. Proof. Apply Proposition

9.11 (ii) t o pi

o

f.

[:I}

fi(PkY)

= 0

there is a unique

CI

355

Representation of n-valued Moisil algebras by 3-valued algebras 9.14. Definition.

i E {1, ..., [ S ] } let F; : Mn + M3 be the functor defined by L H L/p; and f H A; cf. Corollary 9.13. For each

We recall that Mg and Mn are equational categories (cf. Theorem 3.2.4) therefore their rnonomorphisms coincide with t h e injective rnorphisrns by Proposition 1.6.6.

9.15. Corollary.

The functors F;preserve monomorphisms and surjective morphisms. 0

Proof. Similar t o th e proof of Remark 7.1.9.

9.16. Corollary. The functors F;reflect monomorphisms. Proof. From Corollary 9.10 (iii) and the fact that the functor reflects monomorphisms.

C

: Mn

4

B 0

9.17. Theorem. For every L E Mn there is a pair (A,f) such that A E M3, f E Mg(L,A) and f is a monomorphism. Proof. Take A =

n {L/p; 1 i = 1,..., [ S ] } , Then A E M3 and on the other

}I:[

L/pi I i = 1, ..., is a direct product in Mg by Remark 1.6.10. Therefore since pi E Mg(L, L / p ; ) ( i = 1, ..., [;]), there exists f E Mg(L,A) such that pi = A; o f (i = 1, ..., If f ( x ) = f ( y ) then hand

{T;

:

A

--t

.)I;[

p i ( x > = p i ( y > (i = 1,..., [:]I, x =y.

Let

n

I:[ stand for

i

n

i.e. Cpkx = (pky

(i = 1,...,n - 11,therefore 0

.

i=l

9.18. Corollary.

For every f E Mn(L, L’) and i E { 1, ...,

}I;[

there esists

356

Representation theorems and duality for LM-algebras

. f E M3(

(i)

f

0

n

(Lip,),

i

g =h o

n

( L ' / p i ) ) such that:

i

f, where g E Mg(L,

n i

h E Mg(L',

n

(Lip;))

and

(L'/pj)) are the embeddings in Theorem 9.17;

i

(ii) f is a monomorphism no m o rp his m;

(iii)

J

surjective

ail

fj

($

all f;. are monomorphisms H

f is

a mo-

are surjective.

Proof (cf. Fig. 6.1). Let pi

:

L

+ L/pi and p: :

L'

L'/pi be the

--t

f

Fig. 6.1.

canonical surjections, iq :

n

( L / p i ) + L/pi and a: :

n

i

L'/pi the canonical projections. Let further g

(L'lpi) +

i

:

L +

n

( L / p i ) and

1

h : L' +

n i

( L ' l p i ) be the embeddings constructed in Theorem 9.17,

Representation of n-valued Moisd algebras by 3-valued algebras

357

f;.

f=

: L/pi -+

L'/p; the morphisms constructed in Corollary 9.13 and -

ni 5, i.e. xi' o f =

-

fi 0

xi (Vi).

(i) r: o f o g = f; o ,ri o g = f;: o therefore f o g = h o f.

pi

= p : o f = ,r: o h o f (Vi),

(ii) and (iii) From Corollaries 9.15, 9.16 and r: o

f = f;: o

9.19. Proposition.

ri.

[;I.

Let B E B, I = 2 = (0, l}, n E I?, n 2 2 and p = There ezists a De Morgan subalgebra L of the three-valued Post algebra (BI21)Psuch that (i) L is an n-valued Post algebra, and

B.

(ii) C ( L )

Proof. Suppose n = 2 p + 1, p 2 2. Let c be the centre of B[2](cf. Corollary 4.1.12) and consider the following elements of A = c1 = (c, 0, ...,0 ) , c2 = ( C , C , O , ...,O ) , ...,cp = (c,c, ..., c ) , CP+l = (c,c, ...,c, l),...,q p - l = (c, 1, ..., 1). Let L be the set of all elements of A of the form (9.12)

2

= (a1 A

~ 1 V)

...V ( ~ 2 ~ -A1 czP-1)

V ~2~

,

where ai = ( b i , ..., b,) E A , b; E C ( B [ 2 ]and ) 15 i (6 bi

5 j 5 2p

jai

2 aj

2 bj).

Clearly z,y E L =$ x V y E L and also 1 E L; to prove that L is a De Morgan subalgebra of A it remains to show that x E L + Na: E L. To do this we write (9.12) in several equivalent forms, taking into account Corollary 4.2.19:

x =(blAc,O

,...,O ) V ( b z A c , b 2 A c , O ,..., O )

V

... V

Representation theorems and duality for LM-algebras

358

N z = ( ( N b , V c) A Nbzp,...,( N b pV c) A Nbp+l) , (9.14)

Ns = ((NbzpA c ) V Nbl, ...,(Nbp+lA c) V Nb,> ,

where Nbi E C(B['])and Nbzp 2 ... 2 Nbl, so that (9.14) is of the form (9.13), proving that N z E L. To prove (i) we use Definition 4.1.1, so that it remains to verify the uniqueness of the representation (9.12) or equivalently, of (9.13). But (9.13) is expressed by the system of equations xi = (bi A c ) V bzp-i+1 (2 = 1, . . . , p ) in BIZ].Taking into account that c ( 1 ) 5 4 2 ) and c(1) = c(2)(because c = N c ) , we obtain c(1) = 0 and 4 2 ) = 1, therefore q ( l ) = bzp-i+l(l) and 342) = bi(2) V &+j+1(2) = bi(2) because i < 2 p - i 1. Then bZp-i+1(2) = bzp-i+l(l) and bi(1) = b i ( 2 ) because all bk E C(B[']). For (ii) we use (i), Corollary 4.2.8 via Theorem 4.2.17 stating that the axes coincide with the centres, and (9.12):

+

5

E C ( L ) # z = cp;z = a; (2 = 1,...,2p)

#

3b E C(B['])a; = ( b , ...,b) (i = 1,..., 2p) , therefore

C ( L )= { ( b , ...,b ) E A I b E C ( B [ ' ] ) 2 } C(B['I) B

.

In the case n = 2 p the proof is similar.

0

9.20. Corollarv. Let n E N , n 2 2 and B a Boolean algebra. Let L be the corresponding Post algebra constructed in Proposition 9.19. The Mn-subalgebras S of L having the center C(S)E B exhaust, up t o a n isomorphism, all n-valued

Moisil algebras L' having the center C(L') E B . Proof. Let L' E Mn such that C(L') E B . We will use the injective Mn-homomorphism (4.1.3) and Proposition 4.1.6 via Proposition 9.19 (i). Thus L' EZ FLf((L')hence C(Ftf(L')) S C(L') EZ Mn-subalgebra of

(C(L'))["-*]g

B["-11 g

B and FEf(L') is

)y]

(C (I ,

EL

,

an

0

359

CHAPTER 7 CATEGORICAL PROPERTIES OF LU KAS IEWICZ-MOIS IL ALGEBRAS

This chapter presents several important properties of the category LMd and provides intrinsic characterizations of certain distinguished objects of

this category.

81. Some adjoint functors In this section we construct some adjoint functors related to the category of Lukasiewicz- MoisiI a Igebras.

1.1. Remark. Although d-algebras cannot in general be defined by equations, some categorical constructions are standard. Since LMd is obviously a category with direct products and subalgebras, the existence o f free d-algebras is ensured (see e.g. Pierce [1968], Theorem 4.1.6). From this we can prove in a standard way that monomorphisms and injective morphisms coincide in LM19 (see e.g. the proof of Theorem 1.20.1 in Balbes and Dwinger [1974]).

1.2. Notation. Let

(1.1)

C

:

LMd + B

be the functor defined by

C(L)= {z

E

L I cpix = 2, i E I} and C(f)=

f l q ~for) f E LMd(L, L'). Let (1.2)

T : B +LM19

be the functor defined as follows: T ( B ) = B[q and for g E B(B,B'),

T ( g ) : T ( B )+ T(B')is the mapping defined by T ( g ) ( u )= g u E T(B).

o u for any

Categorical properties of LM-algebras

360

1.3. Definition. Let P6 be the full subcategory of LM6 having as objects the &valued Post algebras (6. Remark 4.1.13). Let T : B -+ P6 be the functor for which T ( B )and T ( g ) are constructed in Definition 1.2. 1.4. Remark.

We recall that for every L E LM6 and B E B we have constructed the monomorphism (4.1.3), i.e. (1.3)

PL

:

L

, PL(x)(~) =Y ~

+T(C(L))

X

and the isomorphisms in Proposition 4.1.2 (i),i.e., (1.4)

RB

:

C ( T ( B ) )+ B ,

RB(u)= ~ ( i )(Vi E I).

Proposition 1.5 and Corollary 1.6 below are due t o Georgescu and Vraciu

[1970]. 1.5. Proposition. The functor T i s a Tight adjoint of C . Proof. We apply Definition 1.6.1 to the functors (1.1) and (1.2). For any

L E LM6 (1.5) (1.6)

and

B E B let

= Y ( L ,B ) : L M ( ~L ]T ( B ) )-+ B(c(L),

6 = S ( L , B ) : B(c(L),B)

B)

-+ LM~(L,T(B))

be defined by (1.7)

y ( t ) = RB o C(t) ,

(1.8)

6(s) = T ( s ) o

PL ,

for any

t E LM6 (L,T ( B ) )

for any s E B ( C ( L ) , B ).

One can prove that the maps y(L, B ) and 6(L,9)are natural in

L

and

B. We shall prove that 6 is the inverse of y. Let s E B(C(L),B) and 6. = S(s) : L + T(B). By definition, 6,(z) = T(s)(PL(z)) = s o PL(s), hence 6 , ( ~ ) ( i = ) s(P~(z)(i))= s ( ' p g ) , for any z E L and i E I. For every z E C(L)we have the equivalences:

Some adjoint functors

361

1.6. Corollary.

C is faithjul and T is fully faithful. Proof. Apply Proposition 1.6.25.b, where

F,IL =

PL and E B = Rg by Remark

1.6.22.a.

0

1.7. Remark. Proposition 1.5 and Corollary 1.6 allow the transfer of some properties from the category of Boolean algebras t o the category of &valued LukasiewiczMoisil algebras, as will be illustrated in what follows.

1.8. Corollary. C preserves colimits and epimorphisms and T preserves limits and monomorp his ms. 1.9. Remark. It is easy t o see directly that both functors

C

and

monornorphisms and epimorphisms. Thus e.g. if

C(f)

o u

= C(f) o w for some A E

every a E A and every

iEI

B

f

T

preserve and reflect

is a monomorphism and

and u, w E

B ( A , C ( L ) ) then

for

Categorical properties of LM-algebras

362

1.10. Corollary. T h e category Pd of 6-valued Post algebras is equivalent to the category B of Boolean algebras. Proof. For any L E P6 and B E B, PL and R B are isomorphisms.

0

1.11. Corollary. T h e category P6 is a reflective subcategory of L M 6 . Proof. We shall prove that the functor

P =T

C

: LM6 +

P6 is LM6. For any L E LM6, o

Pd -+ L’ E Pd and f : L + L’, morphism in LM6, the diagram i n Fig. 1.1is a left adjoint t o t h e inclusion functor

commutative:

fL

f , ,, ’

I

/

” ’’ 0

k

0

0

0‘

’

L’ k L‘

+

PL’ Fig. 1.1.

It is easy such that

3 see tha f = Pfilo f o PL = f.

T ( C ( f ) ) is the uniqu morphism of Pd 0

1.12. Remark. The morphism in the proof of Corollary 1.11will be used many times in what

363

Some adjoint functors follows. We remark that, for any u E

T ( L ( L ) ) ,the following equivalences

hold:

1.13. Remark. If { c i } i E I L’ are the centers of by the relation: (1.9)

f(u> =

v

L’then the

( f ( u ( i ) )A c;)

&I

,

morphism

f can be expressed

for any u E T ( c ( L ) )

This follows by Remark 1.12 and using Cpj(

v (f(+))

tEI

A

Ci))

=

v

(Ipj(f(4i)) A

Wi))

=

iEI

= f(u(j,)

1

j E 1.

Corollary 1.11can be also proved using directly this expression of

f.

1.14. Corollary (The Moisil representation theorem; cf. Corollary 6.1.8).

FOT any 1’1-algebra L there exists a non-empty set morphism L + (LL4)X.

x and

an injective

Proof In accordance t o the Stone representation theorem for the Bool-

C ( L ) there exists X # 0 and an injective Boolean morphism s : C ( L ) + P ( X ) S Lf (cf. Corollary 2.1.6 and Proposition 1.2.23). Since T preserves direct products, we have an isomorphism g : ( L f ) [ q+ (L, [.rl)x .

ean algebra

Now it suffices t o consider the following injective morphisms in LM19:

1.15. Definition. A category C has the amalgamation pmperty (AP) if for any pair of mo9 nomorphisms A 5 A’, A + A” of C there exist the monomorphisms

Categorical properties of LM-algebras

364

A’ Y B , A” A B such that

u o

f =v

o g.

The following result is well known (see e.g. Balbes and Dwinger [1974]). 1.16. Lemma. The cutegory B has AP. 1.17. Corollary. The category LM29 has AP.

Proof. Let us consider in LM29 the monomorphisms L L’, L 5 L”. By Lemma 1.16, there exist in B two monomorphisms hl : C(L’) --t B , h2 : C(L”)+ B such that hl o C(f)= h2 o C ( g ) . In LM29 we have the commutative diagram in Fig. 1.2.

Fig. 1.2.

Some adjoint functors But T ( h l )o proof.

365

PLI,T(h2) o P L ~ are monomorphisms, which completes the

1.18. Remark. The previous results remain true for th e category LMNd.

1.19. Remark. Let L be a 9-pre-algebra and

(1.10) L/

x N

N

y

w pix = piy

,

the following equivalence relation on L : for any i E

I .

is a d-algebra and the canonical surjection XL : L

--t

L/

-

is a

morphism of d-pre-algebras.

1.20. Proposition (Betnea [198l]). If L i s a d-pre-algebra, L' is a d-algebra and f : L

.--)

L' a morphism of

d-pre-algebras, then there exists a unique morphism of d-atgebras g : L' such that g o XL = f.

L/ -+

x E L, let P be the equivalence class of x. If we put g ( P ) = f ( x ) , for any x E L , then we get a morphism g : L / -+ L' in 0 LMd. The rest of the proof is straightforward. Proof.

For any

1.21. Remark.

-

1.20 any rnorphism of d-pre-algebras f : L -+ L' induces a unique morphism of &algebras f : L / -4 L'/ such t h a t f o X ~= A,yof. The assignment L H L / -, f H f yields a covariant functor from the category of d-pre-algbras t o LMd. Thus LM29 is a reflective subcategory of the category of 8-pre-algebras; cf. e.g. Balbes and Dwinger [1974],Theorem 1.18.2. By Proposition

1.22. Remark. Suppose L E LMN9, L' E LMN9 and f E LMd(L, L'). Then for every x E L and i E I,

Categorical properties of LM-algebras

366

1.23. Proposition. LMN6 i s a fuZ1 wbcategory of LMB. Proof. Let

L,L’E LMN6

and f E LMd(L,L’). Remark 1.22 implies

(Vi E I) (Vx E L ) therefore f ( N z ) = N f ( z ) (Vz E

L).

0

1.24, Definition (Boicescu [1984]; cf. Pierce [1968]). A free LMN6-eztension of an algebra L E LM6 is a pair (A,f), where A E LMN6 and f E LM6(L,A), such that for every g E LM6(L,B), B E LMNQ, there is a unique h E LMNd(A,B) such that h o f = g. 1.25. Proposition (Boicescu [1984]). Suppose the set I is endowed with an antitone involution d. Every algebra L E LM6 has a free LMN6-eztension (U(L),PL),unique up to an isomorphism. Proof.

Let PL be t h e LM6-morphism given by (1.3) and U ( L ) the De

Morgan subalgebra of T ( C ( L ) )generated by PL(L).

1) Using distributivity and th e identities f ( u l ) A ... A f(u,) = f ( u l A ...A u,) and N f ( b l ) A ...A N f ( b , ) = N f ( b l V ...Vb,), it follows that the elements of

U ( L ) are of th e form

2) Let B E LMN6 and g E LMd(L,B). Using Remark 1.22 we obtain

Some adjoint functors

367

3) It follows from 1) and 2) with B = U ( L ) and g = PL that z E A implies

therefore U ( L )E LMN29. 4) Let (1.12) and

be two representations of z E U ( L ) . Then

by (1.14), hence P

m

by the injectivity of

for all

PL and from (1.13) and (1.16) we get

i E I,therefore

Categorical properties of LM-algebras

368

5) According t o 4) we can define

and it is straightforward t o check that h E LMNd(A,B), h o

PL = g

and h is unique with these properties.

6) To prove the uniqueness of (U(L),PL)suppose (B,g)is another free LMNO-extension of L. Then we obtain the morphisms h, h’ such that h o PL = g and h’ o g = PL. It follows that h‘ o h o PL = PL and since l u ( ~ o) Pt = Pt,the uniqueness of the homomorphic extension of PL : L + U ( L ) implies h‘ o h = lqL). Similarly h o h’ = lg,therefore h is an isomorphism. 0 1.26.Corollary. FOT every L,L’ E LMI9 and f E LM.S(L,L’) t h e w is a unique U ( f ) E LMN29(U(L), U(L‘)) such that (1.19)

PL‘ 0 f = U ( f ) 0 PL

1.27. Notation. Let U : LMI9 + LMN6 be the functor for which the assignments L H U ( L ) and f H U ( f ) are those constructed in Proposition 1.25 and Corollary 1.26,respectively. 1.28. ProDosition. T h e reflector U is faithful and preserves and reflects monomorphisms and epimorphisms. Proof. If U ( f ) = U ( g ) for f , g : L + L’ then

PL’ 0 f = U ( f ) 0 PL = U(g)0 f = g by the injectivity of PL,. Further apply the functor C t o (1.19):

therefore

PL = PL’ 0 g

Some adjoint functors

369

C ( U ( L ) ) = PL(C(L)) = C ( P L ) ~ ( L ) ) , hence C ( P L ) : C ( L ) + C ( U ( L ) ) and C ( P L ~ ): C(L')+ C U(L')) are isomorphisms. Therfeore (1.20) implies that C ( U ( f ) ) is a monomorphisrn (epimorphism) if and only if C ( f )is a monomorphism (epimorphism). and note that (1.14) implies

The proof is completed by Remark 1.9. 1.29. Notation Let LMn' denote th e full subcategory of LMn having as objects the prealgebras L satisfying 'pox 5 x

5 'p1z (Vx E L ) .

1.30. Proposition (Boicescu [1984]).

LMn is a reflective subcategory of LMn' and the reflector S, : LMn* + LMn preserves monomorphisms and surjective morphisms. Proof. Let

L

E

L/

LMn* and

-

-

the equivalence (1.10). Then from

is an object of LMn. Set S,(L) = L /

-

L -+ S,(L) be the canonical surjection. For any L' E LMn and f : L + L' in LMn' we define the map g : S,(L) + L' by putting g(f) = f(x) (Vx E L ) . g is th e unique morphism of LMn such that g o 7 r ~= f. We define Sn(f) : S,(L) + S,,(L') by Sn(f)= 7rLt o g. It follows that Sn(f)0 7rL = 7rL8 0 f . For any L E LMn', T L I C ( L ) : C ( L ) --t C(S,(L)) is an isomorphism. y, hence 'p,-lz= Indeed, for any z , y E C ( L )such that f = jj we have 2 ( ~ , , - ~ yBut . piz = 'pjz ( V i , j E I) by the same proof as for (i) + (v) in Proposition 3.1.5 and since 'plx5 x 5 'pn-lz it follows that 2 = 'pn-lxand we see that

and let

7 r ~:

-

similarly y = 'pn-ly, therefore z = y. Using the commutative diagram in Fig. 1.3, it is straightforward t o prove t h a t S, preserves monomorphisms and surjections.

370

Categorical properties of LM-algebras

Fig. 1.3.

Injective LM-algebras and injective hulls

371

$2. Injective Lukasiewicz-Moisil algebras and injective hulls In this section we shall characterize the injective objects and the injective hulls in the categories of Lukasiewicz-Moisil algebras and of axled Moisil algebra; cf. Definitions 1.6.27, 1.6.33, 1.6.8 and 1.6.5. The characterization of the injective objects in a category is a general problem. In order to find the injective Lukasiewicz-Moisil algebras we need two results in Boolean algebras: 2.1. Proposition (Sikorski [1948], [1964] and Halmos [1963]). A Boolean algebra is injective iff it is complete.

2.2. Proposition (Banaschewski and Bruns [1968]). T h e MacNeille completion B of a Boolean algebra B i s a n essential extension of B . 2.3. Proposition.

FOTany L E LM6 the following assertions are equivalent: (i) L is a n injective object ofLM6; (ii) L is a retract of any of its extensions; (iii) L has n o proper essential extensions; (iv)

L is a 8-valued Post algebra and C(L)as complete;

(v)

L is a 9-valued complete Post algebra.

Comment. For 19 = 3, the equivalence (i) H (v) was given in L. Monteiro [1965]. For finite 9, the previous theorem was obtained independently by Cignoli [1969a], [1972] and by Georgescu and Vraciu [1970]; cf. Cignoli [1969a]. Another proof can be found in Boicescu and Georgescu [1970]. For some extensions see L. Monteiro [1970b], Badele and Boicescu [1969] and Boicescu [1979].

Categoricd properties of LM-algebras

372 Proof.

(i) + (ii): By Remark 1.6.29.b. (iii): Let f : L + L' be an essential extension of L. By (ii), (ii) there exists g : L' 4 L such that g o f = 1 ~But . g o f o g = g o l L f r and g is a monomorphism, hence f o g = l,y, therefore f is an isomorphism. (iii) + (iv): Let Q be the MacNeille completion of the Boolean algebra C ( L )and g : C ( L ) + Q the canonical embedding. T preserves monomorphisms, hence L % T ( C ( L ) ) T(s! T(Q)is a monomorphism in LM6. We shall prove t h a t T ( g ) o PL is essential in LM6. Consider a morphism

f In

: T ( Q ) + X in LM6 such that f o T ( g ) o PL is a monomorphism. B we have the commutative diagram in Fig. 2.1(cf. Remark 1.4 and

- [ - 1% -

Proposition 1.5) therefore

C(PL1

C(L)

C(T(g))

c(T(W)))

C(f 1 c(T(Q)) -T(X)

RW)

Q

C(L)

9

Fig. 2.1.

and since C preserves monomorphisms, C(f) oRg' o g is a monomorphism of

B. But g is essential, therefore C(f) o R i l

is a monornorphism, hence C(f)

is a monomorphism, hence T ( C ( f ) )is a monomorphism too, therefore the commutative diagram in Fig. 2.2 implies that hence f is a monomorphism.

PX

o

f

is a monomorphism,

Injective LM-algebras and injective hulls

373

Fig. 2.2. We have proved that T ( g ) o PL is essential, therefore by (iii), T ( g ) o PL

T ( g ) and Pr, are isomorphisms, hence L is a d-valued Post algebra by Proposition 4.1.6. It is easy to prove, using Corollary 1.6, t h a t g is an isomorphism, so C ( L ) is a complete Boolean is an isomorphism. It follows t h a t

algebra. (iv)

e (v):

(v)

j

morphism

By Corollary 4.5.8.

(i): Consider in LMd the morphism f

i

:

X’ + X .

Since

C

:

X’ 4 L

and the

preserves monomorphisms and

is injective there exists a morphism g

:

C(L)

C ( X ) + C ( L ) such t h a t g

o

C ( i ) = C ( j ) . In LMd we have the commutative diagram in Fig. 2.3 with m = Pzl o T ( g ) o Px. Thus L is injective.

Categorical properties of LM-algebras

374

i Fig. 2.3.

2.4. Proposition.

The following conditions are equivalent for L E P6:

L is injective in P6;

(i)

(ii) L is injective in LM6; (iii) C ( L ) is complete; (iv)

C ( L ) is injective in B ;

(v)

L is complete.

Proof.

(ii): Take in LM6 a monomorphism f : A ---f B and a morphism g : A + L. Then we have the commutative diagram in Fig. 2.4, where g' = Pzl o P ( g ) . But L , P ( A ) , P(B)belong to the full subcategory P6 of LM6 and P ( f ) is a monomorphisrn by Proposition 1.6.24, therefore there

(i)

j

Injective LM-algebras and injective hulls

375

Fig. 2.4.

exists h’ such that h’ o P ( f ) = 9’. Then h = h’ o PB is a morphism of LM6 such that h o f = h’ o PB o f = h’ o P(f) o PA = g‘ o PA = Pi10 P(g) 0 PA = P i 1 0 PL 0 g = g . (ii)

+ (i): Because PQ is a full subcategory of LM6.

(ii) w (iii): Follows from Proposition 2.3. (iii)H (iv): By Proposition 2.1. (iii) e (v): By Corollary 4.5.8. 2.5. Remark. For L E LMQ consider the MacNeille completion Q of the Boolean algebra C ( L )and the canonical embedding f : C(L)+ Q . Then T ( Q )is injective in LM29 (because TRQ o PT(Q = ~ T ( Q by ) Proposition 1.6.23) and the morphism

L

3 T(C(L)) 2 T(Q)

is a monornorphism of LM6. This ensures that any

19- valued Lukasiewicz-

376

Categorical properties of LM-algebras

Moisil algebra may be embedded in an injective object of LM6. 2.6. Notation.

Let AMn be the category having as objects the axled Moisil algebras and as morphisms those LMn-homomorphisms that preserve the axes. 2.7. Remark. Any LMn-homomorphism between two axled algebras is necessarily a De Morgan homomorphism in view of Theorem 4.2.10 and formula (4.2.9).

2.8. Lemma. Lz is injective in AMn. Proof. Take a rnonornorphism f : A + B and a rnorphisrn g : A + Lz in

AMn. Let e : LZ --t L, be the inclusion mapping. Inasmuch as L, is injective in LMn by Corollary 4.1.10 and Proposition 2.3, there is h : B --t L, in LMn such t h a t h o f = e o g ; cf. Fig. 2.5. Moreover, in view of

le Fig. 2.5.

Remark 2.7 all the arrows in Fig. 2.5 are morphisms in Mn. Further let and

bi

be the axes of A and

B, respectively (i

E {1,..., n

ai

= 1,..., n - 2). Then, for

- 2 ) we have h(b,) = h ( f ( a i ) ) = e ( g ( a i ) ) = g ( a i ) = 0 because 0 is the only axis of L2. This shows that h preserves the axes and

each

i

implies also, via Corollary 4.2.8, that

377

Injective LM-algebras and injective hulls

h ( z ) = h(cp1z) = cplh(3) E C(L,)= L2

(Vz E B ) .

B + L2 by h’(z) = h ( z ) (Vz E B ) and it 0 follows t h a t h’ is a morphism in AMn such that h‘ o f = g . Therefore we can define h’ :

2.9. Lemma. L, is injective in AMn. Proof. Let f : A + B be a monomorphism and g : A + L, a morphism

L, is injective in LMn by Proposition 2.3, there is a morphism h : B --f L, in LMn such that h o f = g. But f and g preserve the constants of the Post algebras, therefore so does h, i.e. h E Pn(B, L,). 0 in AMn. Since

2.10. Proposition.

An algebra L E AMn is injective if and only if it is complete.

A x B for some A E Pn and B E B. In view of Corollary 6.1.11 there is an embedding el : A + LE in Pn and an embedding e2 : B + Lf in B = P2. But el and e2 preserve the constants of the Post algebras (0 for B ) by Remark 4.1.13. Since the axes of A x B are (cj,O), where c j are the centers of A (6. the proof of Theorem 4.2.26), it follows that el x e2 : A x B + L’ = L: x Lg is a monomorphism in AMn. Thus we obtain an extension e : L + L’ of L in AMn. Besides L’ is a direct product of complete lattices, therefore it is Proof. First we apply Theorem 4.2.26:

L

complete.

If

L

is injective then it is a retract of

L‘

by Remark 1.6.29.b and since

L’ is complete this implies easily that L is complete (see e.g. Balbes and Dwinger [1974], Lemma 5.9.1). Conversely, suppose L is complete. Then A and B are complete, therefore they are injective by Proposition 2.4, hence they are retracts of el and e2, respectively, again by Remark 1.6.29.b. Let

fl be the Pn-morphism such that fi o el = 1~and f2 the B-morphism such that f2 o e2 = lg. Then f = fl x f2 : L’ + A x B is an AMn-morphism . shows that L is a retract of L‘ in AMn. But such t h a t e o f = l A x ~ This L’ is injective in AMn by Lemmas 2.8, 2.9 and Remark 1.6.29.c, therefore 0 L is injective by Remark 1.6.29.a.

378

Categorical properties of LM-algebras

2.11. Proposition. FOT a n y L E L M d , the canonical m o r p h i s m PL : L + T ( C ( L ) ) i s a n essential extension. Proof. Let us consider a morphism f : T ( C ( L ) )--+ L' of LM29 such that

f OPLis a monomorphism. Suppose t h a t f ( u ) = f ( v ) with u , v E T ( C ( L ) ) , hence f(cp;u) = cpjf(u) = cpjf(v) = f(cpiv) for any i f I. We remark that p L ( u ( i ) ) ( j ) = ( ~ j ( ~ ( i )=) ~ ( i=) v i ( u > ( j ) l j E 1,SO ~ " ( " ( i )=) ~ i ( u ) for any i E I; analogously, P~(v(i)) = cpi(v), i E I . Then we get

f ( P L ( u ( i ) ) ) = f(rpju) = f(cpiv) = f ( P L ( v ( i ) ) ) , hence u ( i ) = v(i) for I7 any i E I , so u = v and f is injective. 2.12. Proposition (Georgescu and Vraciu (19701).

Suppose L E LMd and let assertions are equivalent: (i) E as isomorphic t o

E be one

of its extensions.

T h e following

T(Q),where Q is the MacNeille completion

of

C(L); (ii) E is a n injective hull of L ;

(iii) E is a n injective extension of L not including properly a n y injective extension of L ; (iv) E is a n essential extension of L n o t included properly in a n y es-

sential extension of L. Proof.

(i) =+ (ii): T(Q)is injective by Proposition 2.4 and it was established in the proof of the implication (iii) =+ (iv) in Proposition 2.3 that T ( Q ) is an essential extension of

L.

(ii) =+ (iii): Let i : L --t E be an injective hull, cf. Definition 1.6.30. Let j : L --t E' be an injective extension of L , included in i , i.e. there is a monomorphism f : E' + E such t h a t i = f o j; cf. Definition 1.6.8. Since E' is injective there is a morphism g : E --t E' such that g o i = j . Then g o i is a monomorphism and since i is an essential

Injective LM-algebras and injective hulls extension it follows that g is a monomorphism. Therefore

379

f og

:

E

an extensionjet us prove it is essential. Take a monomorphisrn h o

+ E is

(f o 9 ) ;

j = h o f o g o i is a monomorphism and since i is an essential extension it follows that h is a monomorphism. Thus f o g is an essential extension of the injective object E E LM8, therefore f o g is an isomorphism by Proposition 2.3. This implies that f is surjective, hence

then h o

i =h

o

f

o

an isomorphism.

(iii) j (iv): We prove first that E is an essential extension of L. Let

L + T(Q)be th e canonical embedding, where Q is the MacNeille completion of C(L).Also, let i : L + E be the monomorphism which defines E as an extension of L. Since E is injective there is a morphism g : T(Q)+ E such that g o f = i. But T(Q)is an injective essential extension because (i) =+(ii), therefore the inclusion g is not proper by hypothesis, i.e. g is an isomorphism. Thus E E T(Q)is an essential extension. If E' is another essential extension of L and if there is a rnonomorphism E + El, then it may be easily checked that E' is an essential extension of E. Since E is injective it follows by Proposition 2.3 t h a t E and E' are

f

:

isomorphic.

+ (i): Let g

E + T(Q')be the canonical embedding, with Q' the MacNeille completion of C ( E ) . Since T(Q)is an essential extension of E and since E is an essential extension of L , we can infer that T(Q')is an essential extension of L. By an immediate inference we get that g is an (iv)

isomorphism.

:

0

2.13. Proposition.

T h e injective hull of a n axled Moisil algebra i s its MacNeille completion. 2.14. Remark.

It was proved in Boicescu [1984] that any complete n-valued Post algebra is injective in LMn'.

380

Categorical properties of LM-algebras

$3. Free Lukasiewicz-Moisil algebras One of the most important applications of the Moisil representation theorem of the n-valued LM-algebras and Moisil algebras is the construction of

the free algebras with c generators, where c is a finite cardinal. In the sequel we follow closely Cignoli [ 1969a1, [ 19751. Note first the following property of the general concept of free algebra.

3.1. Proposition. Let A and A' be free algebras in the class

K: (cf. Definition

1.5.22). If

there i s a bijection between their sets X and X' of free generators then A and A' are isomorphic.

i

and

tion and g =

f-l.

Proof. Let:

i' be the

X -+ X' a bijecInasmuch as A is free in K: and A' E K: there is a canonical injections,

i

f

:

i X-A

A'

X' -b

i'

Fig. 3.1.

A' in K: such t h a t Q o i = i' o f and similarly there is $ : A' + A such that 11, o i' = i o g (cf. Fig. 3.1.a). Then $ocpoi = $oil0 f = i o g o f = i. But 1Aoi = i therefore $09= 1A because the map i : X + A E K: has a unique homomorphic extension t o t h e free K:-algebra A (cf. Fig. 3.1.b). Similarly cp o $ = l ~ tcompleting , the proof.0 morphism

Q

:

A

+

381

Bee Lukasiewicz-MoisiJ algebras 3.2. Definition.

For each equational class K and each cardinal number c, we denote by FK(c) the free K-algebra with c generators, i.e. the free K-algebra having a set of free generators of cardinality c.

Definition 3.2 is justified by the following result.

3.3. Corollary. For each equational class K and each cardinal n u m b e r c, FK(c) exists and is unique u p t o a n isomorphism. Proof. By Theorem 1.5.30 and Proposition 3.1. 3.4. Corollary.

For each n E

PT, n 2 2, and each cardinal number c, the algebras

FLM,,(c), F M n ( C ) , Fpn(c) and FAMn(c>exist and are unique. Proof. By Theorem 3.2.4, the easy remark that

AMn is an equational class,

Remark 4.1.14 and Corollary 3.3.

0

We are going t o construct the above free algebras. 3.5. Lemma.

Suppose K is a subcategory of LMn such that L, E K and Fx(c) exists. Let G be the set of free generators of FK(c), cardG = c. T h e n there is a bijection K : L f + PFln(FK(c)) . Proof. For every u :

G -P L,

L, be t h e morphism such

let h ( u ) : FK(c) -+

that h(u)IG = ZL. Then obviously h : L: + Hom(F'(c),L,) is bijective and the required bijection is K = ker o h, where ker : Hom(FK(c), L,) -t PFln(FK(c)) is the bijection constructed via Theorem 5.2.14 and Corollary 4.6.13.

3.6. Lemma. S a m e hypotheses as in L e m m a 3.5. Let u : G

0

-+

L, and A a subalgebra

382

Categorical properties of LM-algebras

of L,. Then

A

FK(c)/K(u) if and only if u(G) E A and u(G) $ A'

E

for any (mazimaf) proper subalgebra A' of A . Proof. Note that

FK(c)/~'(u) = &(c)/(ker

o h ) ( u ) Ei h(u)(F~(c)) by

Proposition 1.5.16 via Remark 5.1.16. But

by Corollary 6.1.30. Finally the lemma follows from t h e fact that u(G) =

h ( u ) ( ~is)a set of generators for h ( u )(~K(c1).

0

3.7. Lemma. Same hypotheses as in Lemma 3.5. If, moreover, c E DJ - (0) then card &(c)

5 nnc.

Proof. In view of Corollary 6.1.9 (where the number o f factors is given in Proposition 6.1.5) and Lemma 3.5 we have

cardFK(c)

5

n

card(FK(c)/ri(u))

.

0

uEL,O

3.8. Corollary. Under the hypotheses of Lemma 3.7,

Proof. From Corollary 6.1.19 (which can be applied in view of Lemma 3.7) and again Proposition 6.1.5 and Lemma 3.5.

0

3.9. Theorem (Cignoli [1969a]). Let n = 2 m and for each k = 1,...,m let p ( k ) =

( ( n- 2)/2 )

and Akj k-1 ( j = 1, ...,p( k)) be the subalgebras of L, of cardinality 2k. Let c E J V - { O.} . k-1 . k-1 and a ( c , k ) = 2' )(k - ')i (k = 1,..., m). Then (-l)'( i=O

i

383

n e e Lukasiewicz-Moisil algebras

m

(3.2)

cardFMn(c)=

n

(2k)R(c9k)P(k) .

k=l

Proof. The number of subalgebras Akj is p(k) by Theorem 6.1.31. FM~(= c ) L and, using Corollary 3.4 and Lemma 3.6, let

Set

Fkj = {u E Lf I L / K ( u ) = Akj} ( j = 1,...,p(k)), (k = 1,...,m ) .

Then Lf =

u k=l

P(k)

U Fkj and the

sets

Fkj are disjoint, hence Corollary

j=1

3.8 reads

and if we succeed to prove that (3.3)

CmdFkj = a(c,k)

( j = 1, . . . , p ( k)), (k = 1,...,m )

this will imply (3.1) and hence (3.2). As in the proof of Theorem 6.1.31 we can see that the maximal proper subalgebras of Akj are the (2k - 2)-element subalgebras of Akj and their k-1 (2k - 2)/2 number is ( - ) = k- 1; l e t ALh ( h = 1, ..., k-1) be ) = ( k-2 these algebras. Let further G i = (u E Lf I u(G) E Akj} and Gjkh = {u E k-1

Lf

I u(G)

ALh}. Then we get in turn Fkj = G i

3.6, cardGi = (2k)",

-

u

h=l

Gkh by Lemma

Categorical properties of LM-algebras

384

k-1

u

card

Gkh

=

h=l

= a(c,k).

0

3.10. Theorem (Cignoli [1969a]).

Let n = 2 m + 1 and for each k = 1, ...,m let q ( k ) = ( m - 1

) and A k j , k-1 B k j be the subalgebras of L, of cardinalities 2k and 2k 1, respectively k-1 k-1 ( j = l,...,q(k)). L e t c E n V - { O } a n d a ( c , k ) = 2 " c ( - 1 ) ' ( )"(2k-

+

i=O k-1

2i)", b(c, k) =

k-1

i

i=O (k = 1, ...,m). Then

) ( 2 k - 2i

m

(3.5)

cardFin(c)=

n k=l

+ l ) " , d(c, k ) = b(c, k ) - U(C, k )

+

(2k)"(clk)9(k)(2k l ) b ( c * k ) q ( k ) .

Proof. Similar to that of Theorem 3.9.

i

n e e Lukasiewicz-Moisil algebras

If we accept that

( 0 ) = 1then for n = 2 we obtain F'(c) 0

385

E LX",which

is well known, while for n = 3, the result is F M ~ ( cE) Li" x L:"'". The latter formula was obtained by A. Monteiro (unpublished; cf. A. Monteiro [1980]) and for c = 1 by Moisil [1940]. 3.11. Theorem (Cignoli [1970], Dwinger [1972]).

If c E IV - (0) then

L f and card Fpn(c) = nnC.

Fpn(c)

Proof. By Corollaries 3.4, 3.7 and the remark that Fp,(c)/K(u)

L,

(Vu E L f ) in view of Lemma 3.6 and Corollary 6.3.11.

0

3.12. Proposition.

TC (FMn(c))

=~Pn(c).

G 4 FM,(c) = L be the canonical injection. Then i ( G ) generates L and PL(L)generates T C ( L ) in P n by Remark 4.1.15, therefore (PLo i ) ( G )generates T C ( L ) in Pn. Now take A E P n and f : G --t A. Then there is g E Mn(L,A) such that g o i = f . Further TC(g)o PL = PA o 9, therefore h = Pi1o T C ( g )E Proof. Let i :

I

Fig. 3.2.

Pn(TC(L),A) and h o Pr, o i = Pi1 o TC(g) o PA o i = g o i = f. Finally if h' is another morphism such that h' o PL o i = f then h' o PL = g by the uniqueness of g, hence PA o h' o PL = PA o g = TC(g) o PL,i.e.

Categorical properties of LM-algebras

386

PA o h’ 1 PL(L) = TC(g)I PL(L)and since PL(L)generates T C ( L ) the latter equality implies PA o h’ = T C ( g ) ,therefore h’ = Pi1o T C ( g )= 15.0 3.13. Theorem (Cignoli [1975]).

For each k = 2, ...,n let r ( k ) =

( nk -- 22 ) and let S k j ( j = 1,...,r ( k ) ) be

the subalgebras of L, of cardinality k. Let c E lN - (0) and e(c, k) =

n

(3.7)

cardFLMn(c) =

n

ke(clk)’(k)

k=2

Proof. As for Theorems 3.9 and 3.10.

0

3.14. Proposition. The free M n - e x t e n s i o n (cf. Definition 1.24) of FLM~(c) is isomorphic t o FMn(C).

Proof. Similar to that of Proposition 3.12.

0

The next theorem was given by A. Monteiro [1980] for n = 3, but the proof is extended easily t o arbitary n. 3.15. Theorem. FAMn(C)

FPn(C)

x FB(c).

Proof. In view of the representation F A ~ , , ( c2 ) (‘pn-lal]x [‘pn-lal)given i n Theorem 4.2.26 it suffices to show that (cpn-lal] = Fpn(c) and [‘pn-1al) = F’(c). The proofs being similar, we check below only the latter equality. Let i : G [cpn-lal) =

-, FAM~(c) = A be the canonical injection

B be the

and g : A -+

surjective morphism defined by g(x) =

x V ‘pn-lal.

n e e Lukasiewicz-Moisil algebras

387

G1 = g ( i ( G ) ) generates B. Let j

G1

B be t h e inclusion mapping and g1 : G + G1 th e function defined by j o g1 = g o i. Now take f : G1 --+ B‘ E B. Then f o g l : G + B‘ E AMn hence there is a unique ho E AMn(A,B’) such th a t hooi = f o g l . Further we prove there

Then

:

4

i

Fig. 3.3.

is a unique map h : B -+ B’such th a t hoj = f . Every element of B is of the form g(z) wit h z E A and we shall define h ( g ( z ) ) = ho(z). If g(z) = g(y) then z V v n - l a l = Y V v n - l a l hence h o ( z ) V v n - l h o ( a l ) = h o ( y ) V v n - l h o ( a l ) ; but ho(ul)= 0 because ho preserves th e axes, therefore ho(z)= ho(y). Th e map h satisfies h o j o g1 = h o g o i = ho o

i = f o g1 and since

g1 is a surjection this implies h o j = f . Th e next step is to prove t h a t h is a Boolean morphism. Let p = z V

( P ~ - ~ U Q ~=,

yV

( ~ ~ - 1 ~B1;

E

then

PA

4) = h ( ( z A Y> v vn-lal) = A Y> = ho(z)A ~O(Y) = h ( ~A )h(q)Taking int o account (4.2.29) and (1.2.18”) we also have h ( p ) = h ( N p V cpn-lal) = h ( N z V

- vn-lal) = h o ( N z ) = ho(z) = h(p).

Further let h’ be

another morphismsuch that h‘oj = f. Then h ’ o g o i = h ’ o j o g l = f o g l , therefore h’og = ho by the uniqueness of ho, hence h’ = h by t h e uniqueness of h.

To conclude th e proof it remains to show t h a t cardG1 = c , i.e. t h a t

388

Categorical properties of LM-algebras

g1 is a bijection. But gl is obviously surjective. Let z,y E G, such that gl(x) = gI(y) = x V cpn-lal = y V

( ~ ~ - 1 ~ 1Let .

B” be a Boolean al-

gebra having a free set o f generators Go and cardGo = c a r d G . There exists a bijection fo

:

G +

Go

and

k

o

fo

:

G + B” where

B” is th e canonical injection. Because A is free, there is a morphism go : A + B”, such that golG = k o fo. It follows that

k

: Go

--t

go(xVcpn-la1) = SO(Y Vcpn-lal) = go(~)vcpn-lgo(al) = go(Y)Vcpn-lgo(al). But go(a1) = 0, because go preserves the axes. Hence go(x) = go(y). Thus

f o ( s ) = fo(y), and since fo is injective, it follows t h a t

2

= y.

0

3.16. Corollary . If c E IV - (0) then FAM~(c) 2 L;‘ x LiC and cardF’Mn(c) = nnC- 22c. Proof. From Theorems 3.15 and 3.11.

0

Epimorphisms and projective LM-algebras

389

54. Epimorphisms and projective Lukasiewicz-Moisil algebras In this section we describe the epimorphisms and the projective algebras in the categories Pn and

LM19 and obtain several related results.

4.1. Lemma.

Let f : L + L' be a m o r p h i s m of axled ( P o s t ) n-algebras. If C(f) is surjective t h e n f is surjective. Proof. Every x' E L' is represented in the form (4.2.4), i.e.

x' = V(b{A a:),

where b: E C(L') and u: are the axes of L'. But 6: = f ( b i ) for some

bi E C ( L ) and a: = f ( a i ) where a; are the axes of L, therefore x' = f(x) 0 where x = V ( b i A Ui). 4.2. Proposition [Georgescu and Vraciu [1969e], Georgescu [1970a]).

T h e following conditions are equivalent for a m o r p h i s m f E Pn(L,L'):

(i)

f is a n epimorphism;

(ii) C(f) is a n epimorphism;

(iii) C(f) is a surjection; (iv) f is a surjection. Proof.

*

(i) (ii): By Corollary 1.10. (ii) + (iii): By Corollary 5.8. I Balbes ant Dwinger [1974]. (iii) + (iv): By Lemma 4.1. (iv) + (i): By Theorem 1.14.1 in Balbes and Dwinger [1974].

0

4.3. Proposition.

In the category AMn of axled n-algebras epamorphisms coincide w i t h S U T jections. Proof. The non-trivial part is t o prove t h a t if h E AMn(L, L') is not surjective then h is not an epimorphism.

Categorical properties of LM-algebras

390

It followsfrom Lemma 4.1 that C ( h )is not surjective, whence Proposition 4.2 implies that C ( h ) is not an epimorphism in B. Take B E B and u , v E B(C(L‘),B) such that u # v and u o C ( h ) = v o C ( h ) . Now t a k e 2 E C(L’) such t h a t u(x)# ~ ( z ) Then . there is a prime filter F of B such that u ( z ) E F and v(x) @ F or conversely. Further let k E B(B,L2) be the characteristic function of F. Then f = k o u # k o v = g U

B : C(L)

2 C(L’)

B

k

4L2

V

Fig. 4.1.

L, E P n and C(L,) = L2, Proposition 4.1.6 implies the existence of f ’ , g ‘ E LMn(L’,L,) such that C ( f ’ ) = f and C(g‘) = g. Then f’ # g’ and C(f’ o h ) = C(g’ o h ) , which implies f’ o h = g’ o h by Corollary 1.6. See Fig. 4.1. The last point is t o prove that f’ and g‘ are morphisms in AMn. The axes of L and L’ fulfil h(ai) = a:, therefore f’(a:) = g’(a:) = a;” (i = 1,...,n - 2). Now we apply Theorem 4.2.16: t h e axes a; satisfy (4.2.15) for a certain a’ E C(L‘) in the role of a , therefore a;” satisfy (4.2.15) for a” = f‘(a’) E C(L,) = L2. Thus either but f o C ( h ) = g o C ( h ) . Since

a) d’= 0, in which case all ai” = 0, hence using the representation (4.2.7) it follows t h a t f ‘ , g ’

p)

E AMn(L‘, Lz), or

a’, = 1, in which case

ai”

AMn( L’, L,).

are the centres of L,, therefore f ’ , g ’ E 0

4.4. Corollary.

The following conditions are equivalent for a morphism f E AMn(L, L’):

(i)

f

is

an epimorphism;

Epimorphisms and projective LM-algebras

391

(ii) C(f) i s a n epirnorphism; (iii) C(f) is a surjection; (iv)

f is

a surjection.

Proof.

(i)

(iv): By Proposition 4.3.

(ii)($ (iii): By Corollary 5.8.2 and Theorem 1.14.1 in Balb s and Dwinger [1974]. (iv)

+ (iii): If y

E

C(L') then from y

= f(z),where

2

E

L , we get

f ( W )= (PiY = Y, (iii) + (iv): By Lemma 4.1.

0

In t h e categories LMd and LMNt9 there exist epimorphisms that are not surjections.

4.5. Examples.

C ( L ) --t L is an epimorphism, by Remark 1.9 applied t o C(i) : C ( L )4 C ( L ) ,but need not be a surjection (cf. Balbes and Dwinger [1974], Exercise 11.8.4) if L # C(L).

a) The canonical embedding i :

b) In particular if L is a subalgebra of L;', L

i : L2

# L2, then

the embedding

L is a non-surjective epimorphism.

c) Similarly, the canonical monomorphism

Pr, : L

epirnorphism and it is surjective if and only if

L

+

T ( C ( L ) ) is an

is a Post algebra.

4.6. Remark. More generally, every injective epimorphism f in Pt9 is a surjection (because

C(f) is an epimorphism by Corollary 1.10 hence a surjection therefore an isomorphism, which implies that f is an isomorphism again by Corollary 1.10). 4.7. Remark.

The concept of weakly projective algebra is obtained from the classical concept of projective algebra (Definition 1.6.28) by replacing epimorphisms by

392

Categorical properties of LM-algebras

surjective morphisms (6. Balbes and Dwinger [1974], Definition 1.20.13). This definition makes sense because non-surjective epimorphisms do exist in certain classes of algebras (e.g. in L M 6 , as we shall see below). We shall need the following result: in an equational category an algebra is weakly projective if and only if it is a retract of a free algebra (ibid, Theorem 1.20.14). 4.8. Corollary. A n algebra of Pn (of AMn) is projective if and only if it is a retract of a free algebra of Pn (of AMn). Proof. By Theorem 1.20.14 in Balbes and Dwinger I19741, taking into account that in our case weak projectivity is equivalent to projectivity by Prop0 osition 4.2 (4.3). 4.9. Corollary. If L E P n then:

(i) L is projective if and only i f C ( L ) is projective in B; (ii) If L is finite then L is projective. Proof.

(i) By Corollary 1.10. (ii) By (i) and Corollary 5.7.6 in Balbes and Dwinger [1974].

0

4.10. Corollary. Every finite axled algebra L is projective. Proof. L 2 LP,x LQ2 by Corollary 6.1.20, hence it follows from Corollary ) p 5 ncand q 5 2". Now apply 3.16 that L is a retract of F A M ~ ( cprovided 0 Corollary 4.6. Propositions 4.2 and 4.3 cannot be extended t o LM6 or L M N 6 : 4.11. Proposition. The following conditions are equivalent f o . a morphisrn f E LM6(L, L')

(f E LMN6( L,L')):

393

Epimorphisms and projective LM-algebras

(i)

f is

(ii)

C(f) is

an epimo~phiam;

(iii)

C(f)is

a surjection;

(iv)

i : f ( L ) + L‘ is an epimorphism;

(v)

T ( C ( i ) ) : T ( c ( ~ ( L ) )+ ) T(c(L’)) is

(vi)

C (i> : c ( ~ ( L )-, ) c(L’)is

(vii)

C(L’) C f(L);

(ix)

Fln f(L)

an epimorphism;

= F2 n f(L)

an isomorphism;

an isomorphism;

=+ Fl= F2 (W,F2E PFlO(L‘)).

Proof.

(i) j(ii) H (iii): As for Proposition 4.2. (iii) + (iv): If u , v E Hom(L‘,L”) satisfy u o i = v o i then u I f(L) = v(f(L) and since C ( L )= f(C(L)) E f(L) it follows that ulC(L)= vIC(L) hence u = v by Remark 6.3.13. (iv) =+ (i): Use the decomposition f = i o fo where fo is a surjection. (iv) =+ (vi): Because C ( i ) is an injective epirnorphism by Remark 1.9. (vi)

=+ (v): Obvious.

=+ (iv): By Remark 1.9 applied t o the epirnorphism T ( C ( i ) ) . (iii) + (vii): Because C ( V ) = f(C(L)).

(v)

(vii)

=+ (viii):

(viii)

+ (ix):

Because C(L’)

C(f(L)).

If F t n f ( L ) = F 2 n f ( L )then F l n C ( f ( L ) )= F z n C ( f ( L ) )

therefore FlnC(L‘) = F2nC(L‘),which is equivalent t o Fl = F2 by Remark 5.2.4.c. (ix)

+ (iv):

Suppose i : f ( L ) + L’is not an epimorphisrn. Then there

v but u I f ( L ) = v I f(L). Take x E L‘ such that u(x)# v(x). There is i E I such that u(y) # v(y) where y = pis. Suppose e.g. that u(y) $ v ( y ) ;then according t o Remark 5.2.6.b

exist u , v E Hom(L‘,L”) such that u

#

Categorical properties of LM-algebras

394

F on L" such t h a t u(y) E F and w(y) # F . It follows that u - l ( F ) and v - l ( F ) are distinct prime &filters on L' although u-'(F) n f(L) = v-'(F) n f(L) (because u I f(L) = w I f(L)). This conthere is a prime &filter

tradicts (ix).

0

To state t h e next result, recall that t h e Post algebra Lhq is injective be-

E LM.9 and an injective morphism f : L + L'. For each morphism g : L + Li4 there is a morphism h : L' + LIZ']such that h o f = 9 . Then:

cause it is complete. Now t a k e L,L'

4.12. Corollary. Under the above hypotheses, h is unique f o r each g if and only iff a's a n ep imo rp his m.

4.11 f is not an epimorphism if and only if there are F1,F2 E PF16(L') such t h a t Fl n f(L) = F2f l f(L) and Fl # F2. In view of Theorem 5.2.13the l a t t e r condition is equivalent t o the existence 0 of hl,h2 E Hom(L',L\') such t h a t hl o f = h2 o f and hl # h2. Proof. In view of Proposition

Now we turn to th e construction of projective algebras in

LMd.

4.13. Lemma. L e t L E LMd. FOT every x E L - C(L) there is a p r i m e 9-filter F such that x # F but (pix E F . Proof. Since x

# [(plx) (otherwise z = (plx E C(L)) and [cplx) is a &filter,

we obtain the desired result by Proposition

5.2.5.

O

4.14. Lemma.

If L is a projective algebra in LMd t h e n L i s a Boolean algebra. E LMB - B. Take x E L - C(L). According t o Lemma 4.13 there is F E PFld(L) such that ' p 1 x E F and x $! F , hence (pox # F . Let f : L + Liq be the morphism associated with F in Theorem 5.2.13.Then f(x)(l) = 1 and f(z)(O) = 0, therefore f(x) # C(LLq). On the other hand Proof. Let L

Epimorphisms and projective LM-algebras

395

the inclusion i : L2 + Li4 is an epimorphism by Proposition 4.11 (iii). For

L + L2 and every z E L we have i ( g ( z ) ) E C(Lh4) 0 hence i ( g ( z ) )# f(z) for the above z, therefore i o g # f .

every morphism g :

4.15. Proposition.

The following conditions are equivalent f o r a n algebra L E LM8 (i) L is projective in LM8; (ii) L E

B and is projective in B;

(iii) L E B and is a retract of a free Boolean algebra. Proof.

(i) + (ii): L E B by Lemma 4.14. Then clearly L is projective in B, via the remark t h a t every epimorphism of B, being surjective, is an epimorphism of LM8, too.

(ii) H (iii): Well known (e.g. Halmos [1963], Corollary 31.2). (ii) + (i): Let f : L’ --t L” be an epimorphism and g : L --f L” a morphism in LM6. Then C(f) : C(L’) --t C(L”) is an epimorphism in B, g(L) E B and g : L -+ g(L) is a morphism in B. Therefore there is a Boolean morphism h : L + C(L’) such that f ( h ( z ) ) = g ( x ) for every 0 z E L , where h : L --+ L’ is a morphism in LM8. 4.16. Lemma.

Let n E {2,3} and p , q E LV satisfy 0 < p 5 q. Then LP, is a retract of L; . Proof. The map i : L2 + Li-p+’ defined by i(co) = 0 and i ( c l ) = 1, is a monomorphism. Define a monomorphism j

:

Lg

--f

L; setting j = i or

j = i x 1,-1 where 1,-’ is the identity of L;-’, according as p = 1or p

> 1.

Since Li is injective by Proposition 2.3, it follows from Remark 1.6.29.b that

L; is a retract of Li. For n = 3 the proof is similar, with i(cg) = 0, i(c2) = 1 and i ( c l ) = c, 0 the center of Li-p+’ E P3.

Categorical properties of LM-algebras

396

4.17. Corollary. The finite projective algebras in LM9 are the finite Boolean algebras, i. e. the algebras of the form Lg, p E lV - (0). Proof. Finite Boolean algebras coincide with the algebras Lg e.g. by Corollary 6.1.18. Now Proposition 4.15 implies that every finite projective algebra in LM9 is a finite Boolean algebra and every algebra L;, being a retract of

L? by Lemma 4.16, is projective in LMO.

0

4.18. Proposition (Boicescu [1986a]). The following conditions are equivalent for a finite algebra L E M3 or L E M4:

(i) L is weakly projective; (ii) L2 is a direct factor of L ; (iii) S ( L ) # (1). Proof for M3 (for M4 the proof is similar). (i) (ii): If (ii) fails then it follows from Corollary 6.1.18 that L is of

+

the form L = L;, p E N .Take the surjective morphism x1 : LP, x L2 + LP, and the morphism 1 : L3p -+ L;. Let h : L3p + L3p x L2 be a morphism such that ?rl o h = 1. If c is the center of Lz E P3 then c = N c by Corollary 4.2.19, therefore h(c) = N h ( c ) , which is impossible in L; x Lz. (ii) (iii): By Proposition 6.2.12. (ii) (i):According to Corollary 6.1.19, L is of the form L = Lp x L!, where rn > 0 and p 1 0. If p = 0 then L is projective by Corollary 4.17, hence weakly projective. If p > 0 take c E llV such that rn 5 2c and q 5 3' - 2". Then it follows from Lemma 4.16 and Theorem 3.10 that L is a retract of the free algebra Lie x LgC-2e= FM,(c),therefore L is weakly 0 projective by Theorem 1.20.14 in Balbes and Dwinger [1974].

+

Direct sums

397

$5. Direct sums In this section the Priestley duality is applied t o t h e construction of direct sums in D01,

Mg,LM29

and LMN29. The results are due to Cornish and

Fowler [1977] for DO1 and

Mg

aRd t o Beznea [1981] for LM29.

A construction of direct sums for n = 3 was given by Cignoli [1979]. 5.1. Proposition. If X , X ' E Pr then their direct product X X' in Pr exists and as the Cartesian product X x X' endowed with the product topology and the product order.

npr

Proof. Routine.

0

5.2. Theorem.

If L, L' E DO1 then their direct s u m in DO1 ezists and is given b y L

(5.1)

LIDO1

L' F G(HL

np,.

HL')

.

Proof. From Proposition 5.1 and Theorem 2.3.13.

5.3. Lemma. Let Ll,Lz E D01. Then the direct s u m

L1

image jl(L1) U j z ( L z ) of its injections

jz.

Proof. Let

j1,

U

Lz is generated b y the

be the subalgebra generated by jl(L1) U jZ(Lz), j

z

:

L

+

L1 LI Lzthe inclusion mapping and k, : L, + the morphisrns such Lz is a direct sum there is a that j o k, = j , ( T = 1,2). Since L1 morphism g : L1 IJ Lz + L such that g o j , = k, (r = 1,2). Then go j o

k,

= g o j , = k,

(T

= 1,2), hence the rnorphisms g o j , 1~ :

z z +

coincide on the set of generators kl(L1) U kZ(Lz), therefore g o j = 1~ by algebraic induction on

2.

Now l e t us prove that (z,kl,kZ) is a direct sum

of L1 and L2,which will imply that

zS L1

Lz.

Let f , E DOl(L,, L') ( T = 1,2). Then there is a morphism h : L1 LI L 2 + L' such t h a t h o j , = f, ( r = 1,2) (cf. Fig. 5.1). It follows that 6 = h o j : L + L' satisfies h o k, = h o j , = f, ( T = 1,2)

Categorical properties of LM-algebras

398

Fig. 5.1. -

and h is unique because h‘ o

1,2) imply h‘ o g o j, = f, ( r = 1,2), therefore h’ o g = h then h o j = h‘ o g o j = h‘. 0

k,

=

(r

ft.

I-

5.4. Corollary.

If

X,X ’ E P r

then m

(5.2)

G(X X X‘) = { U k=l

(uk X

vk)I Uk E G ( X ) , Vk E G ( X ‘ ) , k = G

EN}.

G ( X x X‘) = GX GX’ is generated by G(p)(GX)UG(p’)(GX’), where G(p)(U)= p-’(V) = U x X‘ and G(p’)(V)= p’-’(V) = X x V . Therefore if A E G ( X x X’)there are Uk E G X , v k E GX’ (k = 1 , ) such t h a t Proof. It follows from the previous three results that

uDol

m

m

5.5. Lemma.

Let L E DO1 and L1, L2 two sublattices of L such that L1 U L2 generates L and for every x l , y l E L1 and x 2 , y 2 E Lz,

Direct sums

Then L

E L1

399

Lz.

Proof. Applying the definition of the direct sum t o t h e embeddings i, :

L ( T = 1,2)we obtain

a morphism f :

L,

L1 1l. Lz --t L such that f o j ,

= i,

m

( r = 1,2). Every z E L = L1 U LZis o f the form z = V

(i~(xk)AG(yk)),

k=l

m

which proves that z = f ( u ) where a =

V (j1(xk)Aj2(yk)). Thus f is surk=l

jective and if we prove that f ( u ) 5

f(b)

+ u 5 b it will follow that f is injecV

tive, hence an isomorphism. Write b in the form b =

A

h=l

( j l ( t h ) V&(vh)).

5.6. Notation. Let X E Pr and L E DO1 endowed with the discrete topology. Set

(5.4)

C ( X ,L ) = {f : X

+L

I f continuous and isotone} .

5.7. Remarks. a) Every f E C ( X ,L ) has finitely many values (from

X = U f-'(x) and XEL

compactness).

b) C ( X ,L ) is a sublattice o f L x . 5.8. Notation. Let X E P r and L E D01. For every D E G X , l e t X(D) : X 4 ( 0 , l ) stand for the characteristic function of D. This defines a function

Categorical properties of LM-algebras

400

C ( X , L ) . Also, define (Va E L ) (VS E X ) .

X

:

GX

--t

c

:

L

-+

C ( X , L ) by c(u )(s ) = a

5.9. Remark.

X

The functions

and c are well defined. Besides

because the right side of (5.5) at a point

2

E X is

5.10. Theorem.

I f X E Pr and L E DO1 then

Proof. W e check the hypotheses of Lemma 5.5. Clearly

GX

2

{ X ( U )I U

E

G X } = L1 and L S {c(d) I d E I;} = L2; since f-'([d)) E G X , Remark 5.9 shows that L1 U L2 generates L ( X , L ) . Now suppose cjd1) A X(Ul) 5 c ( d 2 ) V X ( U 2 and ) X(Ul) $ K ( U 2 ) .Then there is z such that X(Ul)(x) = 1 and X ( U 2 ) ( 5 )= 0, hence dl = (c(d1) A X(Ul>)(.) 5

(c(d2)

v X(U2))(4 = d2 -

0

5.11. Corollary.

IfL1,I;z E DO1 then (5.7)

where

L1 tL1 :

U D o l L2 =DO1

(C(HLl,L2),tLr

x,c)

7

L1 DO^ GH(L1) is the isomorphism in Lemma 2.3.11.

5.12. Lemma.

If ( X ,g), ( X ' , 9') E PrMg then their dzrect product ( X ,g)

nPrMg ( X ' , 9')

Direct sums

401

in PrMg exists and is ( X

np,

X ' , g x 9').

Proof. Routine via Proposition 5.1.

0

5.13. Theorem. If L , L' E Mg then their direct sum an Mg exists and is given b y (5.8)

L

uMg

L'

EM^

G(HL

nprMg

HL')

.

Proof. From Lemma 5.12 and Theorem 2.4.10.

0

5.14. Lemma. Let ( X ,g ) E PrMg and ( L ,N ) E Mg. FOTevery f E C ( X ,L ) (cf. Definition 5.6) set ~ ( f = ) N o f o g. Then ( C ( X , L ) , T )E Mg and the maps X and c (cf. Notation 5.8) are De Morgan homomorphisms. Proof. C ( X ,L ) E DO1 by Remark 5.7.b. For every f E C ( X ,L),the functions N , f, g are continuous, f is increasing, while N and g are decreasing; therefore ~ ( f E) C ( X ,L ) . It is easy to check that T is an involutive dual endomorphism of C ( X , L). To prove the second part, let N' be the negation of G X ,i.e. N * A = X - g ( A ) , where g(A) = { g ( u )I a E A } , for every A E G X ; cf. Lemma 2.4.3. Then for every 2 E X , 7-

(XW) (4= N ( X ( A ) g ( 4 ) = N

(.

(gw)

(4)=

= ( X ( X - g ) ( A ) ) ( z= ) X(N*A)(4 7 hence T o

X =X T(C(U))(.,

o

N * . To prove T

o c =c o

N , take a E L:

= N ( c ( u ) g ( x ) )= N u = C ( N U ) ( $ )

.

0

5.15. Theorem. If ( X , g ) E PrMg and L E Mg then (5.9)

GX

U M L~

?M~

(c(x,L ) ; x , C ) .

Proof. The direct sum exists by Theorem 5.13. Taking also into account Lemma 5.14, we obtain in Mg the commutative diagram depicted in Fig. 5.2.

Categorical properties of LM-algebras

402

Fig. 5.2. But it follows from Theorems

5.2,5.13 and Lemma 5.12 that G X

uMgL

DO1 as well. Therefore the whole diagram that f i s the morphism obtained i n Theorem

is t h e direct sum of G X and L in in Fig. 5.2 holds in

D01, so

5.10 by applying the construction in Lemma 5.5,hence f is a bijection.

0

5.16. Corollary.

If L1,L2E Mg t h e n

where t ~ :, L1

EMg

GH(L1) is the i s o m o r p h i s m constructed in L e m m a

2.4.7. 5.17. Definition. Let LMO' denote the category o f &valued LM-pre-algebras (cf. Definition 3.1.1)and LMN19' the category of &valued LM-pre-algebras with negation. Let further Prd' be the category of d-valued Priestley pre-spaces, i.e. of those objects ( X ,{ Q i } i E 1 ) which satisfy conditions (6.7.1)-(6.7.4)in Definition 6.7.1.Let also PrN1S' denote the category o f 4-valued Priestley prespaces with negation, i.e. o f those objects ( X , g , {@;}iE1) which satisfy conditions (7.8.1),(7.8.3),(7.8.4)in Definition 6.8.1and ( X ,{@;}iE1) E Pr29'. The morphisms of Pr8' and PrNd' are those constructed in Definitions 6.7.2 and 6.8.2,respectively.

Direct sums

403

5.18. Remark. Condition (6.7.5) in Definition 6.7.1 is the dual of condition (3.1.6) in Definition 3.1.1. Therefore the results of $56.7-6.8 imply that PrB' (PrNB') is equivalent to the dual of LM9' (LMNB'). 5.19. Lemma.

If (x, {@j}jEI),

(x', { @ : } ~ E I )E PrB'

exists and is ( X

npr

X',{@jx

then their direct product in Pro',

@:}jE1).

Proof. Routine via Proposition 5.1. For example let us check (6.7.3) and (6.7.4) using Corollary 5.4. If i 5 j and A E G ( X x X ' ) then

X x X ' - ( @ j x @:)-'(A) =

E G(X x X ' )

.

0

5.20. Theorem.

If L , L' E LM8' then their direct sum in LMd' exists and is given b y (5.11)

L

ULMSO

L'

E L M ~GO( H L n p d o

HL')

.

Proof. From Lemma 5.19, Theorem 6.7.11 and Remark 5.18. 5.21. Lemma.

Let (X, {@pi}i,I)

E Prd' and

( L ,{ ( p j } i C ~ ,

E LM9'.

FOT every

404

Categorical properties of LM-algebras

f E L ( X , L ) (cf. Definition 5.6) and i E I s e t $if = cpi o f o ' P i , $ i f = Cpi o f o ai. Then ( C ( X ,L ) , { $ ; } ; € I , { $ i } i E ~ )E LM6' and the maps X and c (cf. Definition 5.8) are LM6O-morphisms. Proof. Similar to the proof of Lemma 5.14. We only check here the second part, which reduces to X o a~~= $i o X (cf. Lemma 6.7.6) and

c

0 Cpi

= $; o c. For every A E G X , x E X and a E L,

5.22. Theorem.

If ( X , { @ ; } ; G I ) E Pr6' and ( L , {Yi}icf,{Cpi}ici) E LM6' then

Proof. From Lemma 5.21 by arguments similar to those in the proof of Theorem 5.15. U 5.23. Corollary. If L1, L2 E LM8' then

( L l ) is the isomorph&srnconstructed in Thewhere tLl : L1 S L MG~HO orem 6.7.11 via Remark 5.18. 5.24. Lemma.

If (x, g, PrN6',

(x', g', { @ { } i E ~ )E

PrN6' then lheir direct product dn

Direct sums ezists and is ( X

405

npr

X', g x g',

{Qi

x

Q:}iEI).

Proof. From Lemmas 5.12 and 5.19 plus the easy verification of (6.8.3) and (6.8.4). 0 5.25. Theorem.

If L, L' E LMNS' t h e n their direct s u m in LMN.9' ezists and is given b y

Proof. From Lemma 5.24, Theorem 6.8.12 and Remark 5.18.

0

5.26. Lemma. Let ( X , g , { Q i } j E l ) E PrN6' and ( L , N ,{Pi}iCl, {Cpi}jcl) E LMNS'. T h e n (C(X,L),7,{ $ ; } i E i , { $ i } i E 1 ) E LMN6', where 7 and &, & are defined in L e m m a s 5.14 and 5.21, re8pectively. Besides, the maps X and c (cf. Definition 5.8) are LMN6'- morphisms.

Proof. From Lemmas 5.14 and 5.21 plus the easy verification of the identi0 ties 6if = .($if) and ($i ~ ) ( f = ) +di)(f). 0

Proof. By Theorems 5.15 and 5.22.

(T 0

0

5.28. Corollaw. If L1, Lz E LMN6' t h e n

where t L , : L1 2 ~ ~ ~G H8 ( L0I ) is the isomorphism constructed in Theorem 6.8.12 via Remark 5.18. The technique used so far cannot be applied to the construction of direct sums in LM6 or LMN6 because Pr6 and PrN6 are not closed with respect

Categorical properties of LM-algebras

406

to Cartesian products. However the following result holds:

5.29. Theorem. A) If L1, L2 E LMd t h e n their direct s u m in LM6 exists and is given by

where ( L , U L M ~ OL 2 , j l , j 2 )i s their direct s u m in LMdO, while N and X are the equivalence o n L1 U L M , p L2 and the corresponding canonical surjection, respectively, constructed in R e m a r k 1.19.

B ) Similar result f o r LMN6. Proof of A). By Theorem

L1

5.20 and Proposition 1.20 (cf. Fig. 5.3).

/

L2

’1

\

ULMBO 0’

Fig.

5.3.

5.30. Corollary. A)

If L1,L2 E LMd t h e n

where t L 1 : L1 Theorem 6.7.11.

~ L M B GH(L1) is

the isomorphism constructed in

407

Direct s u m s B) Similar result f o r LMNI9. Proof of A). By Corollary 5.23 and Theorem 5.29.

17

As a matter of fact there are several possibilities of constructing direct

~ AL M 6 sums of arbitrary (not necessarily finite) families of algebras ( L A ) A in or in LMN6. Thus e.g. set

A* = { F E A 1 F finite} and LF =

U

LA for each

XEF

F E A*. Then { L FIF E A*) is a filtred inductive system of objects in LM6 (in L M N S ) and

where the inductive limit in the right side is constructed on the corresponding inductive limit in Set as underlying set, the LMd-operations (LMN6operations) being defined in a natural way. Another technique uses the equivalence

T : B -+ PI9 (cf. Corollary

1.10). If (Ax)XEA C_ PI9 then

this construction was given for I9 = n by Cignoli [1972] and Dwinger [1972]. Now we can use direct sum in P6 t o construct direct sums in LM6.

5.31. Theorem.

LMd-sztbalgebra of A generated by

U jh(PLA(Lh)) and j , = j X€A

(Vx E A), where j : S 3 A is the inclusion mapping. T h e n

o

kx

408

Categorical properties of LM-algebras

H o r n L M d ( L x , L ) (VX E A). Let f : A + T C ( L ) be the unique P6-morphism such that f o j , = T C ( f x ) (VX E A). From f o j , o P L = ~ T C ( f x )o PL,, = PL o fx we infer f ( ( j x ( P ~ , , ( L x )C) ) PL(L), hence f(S) C PL(L),therefore there is g E HornLMff(S,pL(L)) such that f o j = i o g, where i : PL(L) + T C ( L ) is the inclusion mapping. Also, PL = i o QL, where QL : L + PL(L)is an isomorphism. Proof. Let

L E LM6 and fx

E

Thus we obtain the commutative diagram in Fig. 5.4, which shows that

f

PL Fig. 5.4.

h = QZ1 o g : S + L satisfies PL o h o kx o PLx= PL o fx therefore h o (kx o PL,,) = fx (VX E A). Now suppose h' : S + L satisfies h' o kx o P L = ~ fx (VX E A). Then

= f o j , o PL,,= f o j o kx o P L ~ o=j ~ o hence QL o h' o QL 0

kx o PL,,

kx o PL,,= g o kx o PL,therefore

h' 1 j x ( f ' ~ ~ ( L x ) ) = g I j x ( P ~ , ( L x ) )

which implies QL o h' = g or h' = Ql;

o g.

(VX E A)

, 0

Direct sums

409

5.32. Corollary.

Let U : LM6 + LMN29 be the functor in Definition 1.27. Then for every (LA)A€A C LMN6, (5.21)

LI LMNS LA = u( LI LM9 LA) -

A€A

A€h

Categorical properties of LM-algebras

410

$6. Free Post and m-Post extensions In this section we construct free extensions of n-valued algebras and of compl etely ch rysi p pia n 6-va Iued a Igebras.

6.1. Definition. a) Let: -r be a type o f algebras, A a .r-algebra and C a category of .r-algebras

(6. Definition 1.6.2). A free C-eztension of A is a pair (B,f) where

(i)

B E C,

(ii) f : A + B is an injective .r-morphism,

(iii) f(A)

E A,

(iv) f(A) generates (v)

for every

B and

C E C

and every .r-morphism g

h E C ( B , C )such th a t h

o

:

A + C there is

f = 9.

b) Let: D be a subcategory of DO1 or more generally a category of distributive bounded lattices, m an infinite cardinal number and C = mD a subcategory of m-complete objects of D and m-complete D-morphisms between them. T h e concept o f free mD regular estension is obtained f r o m Definition 6.1.a applied to mD by requiring t h a t f,g (and necessarily h ) be m-complete. W e are interested here in th e cases D = B and D = LMn and we shall use the words Boolean and Post instead of B and Pn, respectively. T he free m-Boolean extensions (regular extensions) of Boolean algebras were constructed independently by Yaqub [1962] and Sikorski [1963], cf. [1964]. T h e above general definition 6.1.a is a slight modification of a concept due to Pierce [1968].

6.2. Remarks. a) T h e morphism h constructed i n Definition 6.1.a (v) is unique in view o f (iv) and h I f(A) = g.

411

n e e Post and m-Post extensions

/3) Let (vi) be the variant of (v) requiring uniqueness of h. If C is closed with respect t o subalgebras then in Definition 6.1.a one can replace (iv) and (v) by (vi).

Suppose (vi).

This implies easily that

f o satisfies

Definition 6.1.a. therefore satisfies (vi) as noted in the above remark a. Now a standard argument shows that

fo

B (see e.g. Pierce (19681.

Proposition 4.1.2). The converse holds by a).

y) Since LMN6 (Mn) is a full subcategory of LMO (LMn) by (3.1.52), whenever it is true that every algebra L E LMd (LMn) has a certain type of free extension, it follows that every algebra L E LMNd (Mn) has that type of free extension.

6 ) Since P29 is a full subcategory of LMd by Remark 4.1.13, the problem of looking for free Post extensions of LM29 algebras makes sense.

6.3. Theorem. Every algebra L E LMn (or L E Mn) has a free Post extension. Proof. Take

(TC(L),PL).Property (iv) in Definition 6.1.a follows from the

representation

of t he elements of

T C ( L )given in Remark 4.1.15, while (v) holds by Corol-

lary 1.11.

0

Other constructions of the free Post extension were given by Cignoli

[1972](6. Theorem 6.3.15) and Balbes and Dwinger [1974]. 6.4. Lemma.

Suppose that L E LMIJ LIE P6. Then: a) Every h E

and

(OT

L E LMN6) is completely chrysippian and

B(C(L),C(L')) has a unique estension g E LMS(L, L')

Categorical properties of LM-algebras

412

b) g satisfies

(ii) the dual property. Proof.

V

The morphism g was constructed in Proposition 4.1.6: g ( z ) = (h(cp;z)A q). It follows from the representation 4.1 (T4) of g ( z )

iEI

that g is unique. To show (i) recall that y i g ( x ) = h(cpiz), as was noted in

the proof of Proposition 4.1.6, (iii)

+ (iv).

It follows that for every i E I ,

6.5. Corollary.

If h is m-complete (complete) then g is m-complete (complete). 6.6. Corollary. If L E LMn (or L E Mn) and L’ E P n then every m-complete (complete) morphism h E B(C(L),C(L’)) has a unique m-complete (complete) extension g E LMn(L, 1;’). Proof. By Corollary 6.5 and Theorem 4.5.12.

0

A particular case of Corollary 6.6 was obtained by Cignoli [1984]. The next result is a generalization of the Rasiowa-Sikorski Lemma 1.3.32, 6.7. Corollary (Cignoli [1984]). Suppose L E LMn, card L > 1, a E L and let X,, Y, G L such that in€Xn and supY, exist (Vn E nV). The necessary and suficient condition for

h e Post and m-Post extensions

413

the existence of a morphism g : L 4 Ln preserving all inf X n and sup Y, and such that g ( a ) = 1 is cpla # 0 . Proof. If g ( a ) = 1 then g(cp1a) = cplg(a) = 1.

To prove sufficiency set a, = inf X , , bn = supY, (Vn E N ) and apply Lemma 1.3.32, via Proposition 1.3.9, t o the element cpla and the countable families of subsets Xni = { c p i ~I x E Xn} and Yni = {ViY I y E Y n } of C ( L ) : there is h E B(C(L), L2) such that h(cpla) = 1 and

h(cpian) = 1 H h(cpi~) =1

(VV~X E X,i)

h(cpia,) = A { h ( c p i X ) 1 z E X n } and also h(cpibn) = V {h(cp;y) I y E Y n } . Further apply Lemma 6.4 via Theorem 4.5.12 t o L, L, and h: there is g E LMn(L,Ln) such that g(an) = A{g(x) 13 E X,} and g(b,) = i.e.

V{g(Y)

I Y E Yn}. Finally g(a)

0

>_ g ( ~ 1 a=) h ( ~ l a=) 1.

6.8. Theorem. Every algebra L E LMn (or L E Mn) has a free m-Post extension. Proof. Let (Bm,fm) be the free m-Boolean extensions of

C ( L ) ;we will

( T ( B m ) , T ( f mo) PL)is the required extension. Taking into account that (TC(L),P') and ( B m , f mare ) free extensions and T(B,) is prove that

m-complete by Proposition 4.6.5, we see that conditions (i)-(iii) in Definition

6.1.a are verified. Thus it remains t o prove (vi); cf. Remark 6.2.p.

Let L' E Pn be m-complete and g E LMn(L, L'). According t o Theorem

6.3 there is a unique h E Pn(TC(L),L') such that h o PL = g. But

Categorical properties of LM-algebras

414

C ( h ) E B(C(L),C(L‘)) hence there is a unique m-complete morphism k E B(B,, C(L‘)) such that k o fm = C(h). Further Corollary 6.5 yields an m-complete extension g E LMn(T(B,), L’) of k,such that cpig(s)= k(cp;z) (cf. Fig. 6.1). Taking also into account Definition 1.2 we obtain, for a E TC(L) and i E I,

vig(T(fm)(a)) = =

which proves that tj o

k ( ~ i ( f m0

a ) ) = cpi(k(.m(a>))

cpi(C(h)(4)= V i ( W )

T(f,) = h.

=

9

From this and h o PL = g we get

g o T(fm) o PL = g . If g’ E LMn(T(B,), L’) satisfies g’ o T(fm) o PL = g then g’ o T(fm) = h , hence C ( g ’ ) o fm = C ( h ) ,therefore C ( g ‘ ) = k whence

g’ = g.

6.9. Remark.

L E LM6 (or L E LMN6) then for every m-complete algebra L‘ E P6 and every morphism g E LM6(L,Lt) there is a unique m-complete morphism g E LM6(T(Bm),L’) such that One can prove in a similar way that if

g

0

T(fm)0 PL = 9 .

6.10. Theorem. E v e r y algebra L E LMn (or L E Mn) has a free m - P o s t regular extension. Proof. Similar to the proof of Theorem 6.8. The required extension will be

(T(B&), T(fk)o PL),where (I?&,f:) is t h e free m-Boolean regular extension of C(L). The morphism PL is complete by Proposition 4.5.11 and Theorem 4.5.12, therefore h is m-complete, while the m-completeness of fk implies obviously t h a t T(fG)is m-complete. Therefore T(fA)o Pr, is m-complete.

LMn(L,L’) t o be m-complete. Then C ( g ) is mcomplete and since C(PL) : C(L) --f CTC(L) is an isomorphism, it follows that C ( h ) = C ( g ) o (C(PL))-’ is m-complete. Therefore we can apply Now suppose g E

the definition of

(Bk,f;)

and obtain an m-complete morphism k satisfying

k o :f = C(h). The remainder of the proof proceeds as for Theorem 6.8.0

R e e Post and m-Post extensions

415

6.11. Remark.

The above construction of the m-complete extension 3 of the m-complete morphism g can be carried out for completely chrysippian algebras L E L M 6 . 6.12. Corollary.

For every cardinal m, the category mPn is a reflective subcategory of each of the categories L M n , M n , the category of L M n - algebras (Mn-algebras) and m-complete morphisms. Similar constructions can be done when m-completeness is replaced by completeness. Thus Day [1965] has shown that for every Boolean algebra A, there is a free complete-Boolean extension of A if and only if A is superatomic. Using this fact and Corollary 4.5.24 one obtains the following result.

6.13. Proposition. Let L E L M n (or L E M n ) . T h e n L has a free complete-Post eztension if and only if L is superatomic. For every Boolean algebra A there is a free complete-Boolean regular extension of A,namely i t s MacNeille completion. 6.14. Corollary (Georgescu [1971d]).

If L E LM19

L E LMN19) is completely chrysippian t h e n its injective hull is also the free complete-Post regular extension of L. (OT

Proof. As for Theorem 6.10, using Proposition 2.12.

0

6.15. Corollary (Cignoli [1984]).

If L E L M n ( o r L

E M n ) then

L has a free complete-Post regular exten-

szon. 6.16. Corollary. (i) T h e category of complete P o s t n-algebras and complete m o r p h i s m s is a reflective subcategory of the category of Lukasiewicz-Moisil (Moi-

416

Categorical properties of LM-algebras

sil) n-algebras and complete morphisms.

(ii) The reflector preserves and reflects monomorphisms. Proof. Follows from a result of Banaschewski and Bruns (19671; cf. Balbes 0 and Dwinger [1974], Theorem 12.3.4.

417

CHAPTER 8 MONADIC AND POLYADIC LUKASIEWICZ-MOlSlL ALGEBRAS

The monadic d-algebras and the polyadic d-algebras reflect algebraically the properties of the d-valued predicate logic in the same way as the monadic Boolean algebras and the polyadic Boolean algebras are algebraic structures imposed by the study of the classical predicate calculus. This chapter is concerned with the basic properties of the monadic dalgebras and polyadic d-algebras. Most results presented here are natural generalizations of those given in the Boolean case. We do not include some important results in the theory of monadic and polyadic d-algebras (see e.g.

the L. Monteiro theorem on the structure of finite free 3-valued Moisil algebras, the duality theory of monadic d-algebras etc.; cf. Georgescu [1971e],

L. Monteiro [1974]). $1. Monadic Lukasiewicz-Moisil algebras In this section we present some basic concepts and results on monadic Lukasiewicz-Moisil algebras. The main result is the representation theorem 1.20, which extends the Halmos representation theorem for monadic Boolean algebras. The content of this section is taken from Georgescu [1971e] and

L. Monteiro [1974]. 1.1. Definition. Let L be a &valued Lukasiewiu-MoisiI algebra. An existential quantifier on L is a function 3 : L + L such that, for every z,y E L , t h e following axioms hold:

(1.1)

30=0 ;

(1.2)

5

5 33 ;

Monadic and polyadic Lukasiewicz-Moisil algebras

418

(1.3)

3(x A 3y) = 3~ A 3 y ;

(1.4)

3vi = 'pi3 ,

for every

iEi

An 3-monadic 8-valued Lukasiewicz-Moisil algebra or an 38-algebra is a pair

( L ,3), where L is a &algebra and 3 is an existential quantifier on L .

1.2. Definition. Let ( L ,3), (L', 3) be two 38-algebras. A morphism of 38-algebras f : ( L ,3) -+ (L',3) is a morphism of 8-algebras f : L --f L' such t h a t f3 = 3f. The category of 38-algebras will be denoted by 3LM8. 1.3. Definition. A universat quantifier on a 8-algebra L is a function V : L + L having the following properties:

(1.5)

V l = 1;

(1.6)

Vx

(1.7)

V(x V Vy) = Vx V Vy

(1.8)

Vv;

5x ,

= 'piV

for every

,

xEL;

,

for every

for every x, y E

i

E

I

L;

.

A V-monadic 8-valued Lukasiewicz-Moisil algebra or a V8-algebra is a pair (L,V), where L is a 8-algebra and V is a universal quantifier on L. We shall denote by VLMO the category o f V8-algebras.

1.4.Definition. An 3V-monadic 8-valued Lukasiewicz-Moisil algebra or an 3v8-algebra is

a triple ( L ,3,V), where ( L ,3) is an Walgebra, (L,V) is a V8-algebra and

(1.9)

Vx = 3a:, for every x E C(L).

The category of 3V8-algebras will be denoted by

3VLM23.

419

Monadic Lukasiewicz-Moisil algebras 1.5. Remarks.

(a) Let L be a d-algebra with negation and let 3 be an existential quantifier

on

L.

Then V = N 3 N is a universal quantifier on

of 3 and (L,3,V) is an 3Vd-algebra.

L , termed the dual The structure ( L ,3,V) of this type

will be called a monadic d-algebra with negation and their category will

be denoted by Mond.

(b) Let L be a d-algebra and 3 : L --t L, V : L --t L be two functions which satisfy (1.4), (1.8) and (1.9). Using the determination principle we can prove that ( L , 3 ) is an 39-algebra iff (L,V) is a VTJ- algebra. (c) Let (L,3,V), (L’,3,V)

f

L’ be a morphism of d-algebras. Then f is a morphism of 32’1-algebras iff f is a be two 3Vd-algebras and

: L -+

morphism of Vd-algebras.

1.6. Example. l e t L be a complete d-algebra and

and x E

X be a non-empty set. For each p E Lx

X we denote:

In this way we obtain the functions 3 : Lx + Lx and V : Lx +

Lx.

If L is also completely chrysippian, then 3 is an existential quantifier on Lx and V is a universal quantifier on Lx. We shall verify only the axioms (1.7) and (1.8). Using Proposition 4.5.9, we have for every p , q E

Lx and y

EX

:

Monadic and polyadic Lukssiewicz-Moisil algebras

420

Since L is completely chrysippian, we have for every p

E Lx,y E X and

i E I that

Consequently, ( L x , 3 ) is an 319-algebra and ( L x , V ) is a Vd-algebra. Since

C ( L x ) = ( C L ) x ,it is easy t o prove t h a t ( L x ,3,V) is an 3Vd-algebra. If L is a &algebra with negation, then (Lx,3,V) is an object of Mond. An 36-sublagebra of ( L x , 3 ) will be called a functional L-valued 329-

algebra. We define similarly the functional L-valued Vd-algebras and the functional L-valued 3Vd -algebras.

1.7. Proposition. In every 329-aZgebra ( L ,3 ) the following properties hold: (1.11)

31 = 1 ;

(1.12)

33 = 3 ;

(1.13)

z E 3(L) H 2 = 32 ;

(1.14)

2

5 y + 32 5 3y ;

(1.15)

2

E C ( L )+ 32 E C(L);

(1.16)

3 ( V~ y) = 3x V 3y ;

(1.17)

3pi3z = pi32 ,

for each

i E I.

(1.11)-(1.14) follow by definitions; 5 E C ( L ) implies (pi32= 3y;z = 32,for every i E I , thus (1.15) holds. By (1.15) C ( L ) is a monadic Boolean algebra. Hence, for every i E I,we can write: Proof.

Properties

Monadic Lukasiewicz-Moisil algebras

421

Then, by the principle of determination 3(z V y) = 32 V

3y and thus (1.16)

holds. From pi32 A cpi3z = 0 it follows, for every j E

I , that

and therefore cpj3pi3x I qjcpi3x = cpjpi3x. By the determination principle we obtain that

3pi32 5 @32. The converse inequality follows by (1.2),

thus (1.17) holds.

1.8. Remarks. (a) In every Vd-algebra

(L,b') the properties 3V - dual t o (1.11)-(1.17) hold,

where duality preserves

5 and:

(b) It follows from (1.4) t h a t in every 329-algebra 3(C(L)) = C ( 3 ( L ) ) . (c) Properties (1.2), (1.12) and (1.14) show that every existential quantifier is a closure operator. Proposition 1.9 below is an analogue of Remark

1.1.15 for existential quantifiers. 1.9. Proposition (lorgulescu [1984c]).

(A) A subset M of an algebra L E LMd is of the form M = 3(L), where 3 is an existential quantifier, if and only if the following conditions hold:

(i) M is a Moore family on L ; (ii) M is a 6-subalgebra of L ; (iii)

E M 12 and every i E I .

(Pi

A {z

I Z}

= A{z E MI

5

Q ~ X

Z} f o r

every

2

EL

(6) When this is the case, 3 is uniquely determined b y 3x = A { z E

M Ix 5 z } . Proof. Let 3 be an existential quantifier on

L and M = 3(L). Then Remark

1.1.15 implies (i) and (B), hence (iii) is a mere translation of (1.4). To prove

Monadic and polyadic Lukasiewicz-Moisil algebras

422

(ii) we use (1.13),which shows that 0 E 3(L) by (1.1) and 1 E 3 ( L ) by (l.ll), while if x , y E 3 ( L ) then x A y E 3 ( L ) and x V y E 3 ( L )follow from

(1.3)and (1.16),respectively. Conversely, suppose (i)-(iii) hold and define 3 by (B). Then

(1.1) follows

from (ii) and

(1.4) from (iii), while 3 is a closure operator by (i), therefore (1.2)also holds. Property (1.3) is satisfied for x , y E 3 ( L ) because x A y E 3 ( L ) implies 3 ( x A 3 y ) = 3 ( x A y) = x A y = 3x A 3y. This implies further that (1.3)holds for arbitrary 2,y E L because v i 3 ( A ~3y) = 0 3(CpiX A 3p;y) = 3viX A Ycpiy = ( p i ( 3A~ 39) for all i E I. 1.10. Proposition. In every 3Vt9-abgebra ( L ,3, V ) , the following equalities hold: (1.18)

3v=v; v 3 = 3 ;

(1.19)

V ( L )= 3 ( L ) ;

hence by the determination principle 3V = V . Similarly El = 3.

The equality (1.19)is obtained from (1.13),its dual and (1.18). For each x E L and i E I , V@Vx = CpiVx (by t h e dual of (1.17)), therefore CpiVx E V ( L )= 3 ( L ) . Thus 3CpiV~= CpiVx and, similarly, Vqi3 = Cpi3. 0

1.11. Proposition. If ( L ,3,V) is a monadic 9-algebru with negation, then 3(L) = V ( L ) is a 9-subalgebra (with negation) of L. Proof. We shall prove that x E 3 ( L ) implies N x E 3(L). Indeed, if 3x = x , then, for every

i E I,we have:

423

Monadic Lukasiewicz-Moisil algebras

It follows t h a t 3Nx = N x , hence N z E 3(L).

0

1.12. Remarks. Let (L,3) b e an 3d-algebra and M be a 6-ideal of L. M is called a monadic 9-ideal if x E M + 3s E M. Let us consider t h e d-congruence associated w i t h the 9-ideal M, i.e. the 6-congruence generated by =:

x

y (mod M) @ there exists a E M such t h a t a V x = a V y

Then

3

is a congruence of the M-algebra

a V x = a V y, a E M + 3 a V

( L , 3 ) ,since

3x = 3 a V 3y, 3a E M .

Then the quotient d-algebra L/M has a canonical structure o f 319-algebra. B y duality if (L,V) is an Vd-algebra and F is a d-filter of L, then F will be called a monadic 19-filter if x E F

+ Vx E F ; t h e d-congruence associa-

ted with F is a congruence of t h e V19-algebra (L,V) and L/F is a V19-algebra. 1.13. Proposition. Let (L,3,V) be a n gvd-algebra and M a monadic d-ideal of (L, 3). Then the d-congruence associated with M is a congruence of the 3Vd-algebra (L,3,\s?. A dual result holds for the monadic 19-filters of (L,V). Proof. It suffices to show t h a t xsy(modM)+Vx-Vy(modM).

If x V a = y V a , a E M , then x V 3a = y V 3a, 3a E M; by (1.18),(1.7) we have

= V(y V 3a) = Vy

therefore Vx 3Vd-algebra.

= Vy(modM).

v 3a ,

Hence, L/M has a canonical structure of 0

Monadic and polyadic Lukasiewicz-Moisil algebras

424 1.14. Remark.

If (L,3,V) is in Monz9 and M is a monadic &ideal of (L,3), then L/M is also in Mon9.

1.15. Definitio n. A monadic &ideal M of an Walgebra (L,3) will be called m a s i m a l if it is a maximal element in the ordered set of the proper monadic &ideals of

( L ,3). 1.16. Lemma. If (L,3) is a n 38-algebra and X is a n o n - e m p t y subset of L, t h e n the monadic d-ideal of ( L ,3 ) generated by X is the following set: {z E L I there exist n 2 1 and yl,...,yn E X such that n

1.17. Proposition. For every 39-algebra (L, 3), there exists a n isotone bijection between the set of monadic 9-ideals of (L,3) and the set of (Boolean) ideals of

C(3W).

1

M n C(3(L) is an ideal of the Boolean algebra C ( 3 ( L ) ) .For each ideal N of C(3(L) , denote by the monadic &ideal of (L,3) generated by N . Using the previous lemma Proof. If M is a monadic &ideal of (L,3), then

n

it is easy t o prove that

M n C(3(L)) = M and C(3(L)) n

=N

.

0

1.18. Corollary. FOTevery proper monadic 6-ideal M of a n 36-algebra (L,3 ) the following assertions are equivalent:

(i)

M is m a x i m a l in (L,3);

(ii) z,y E c(~(L)), z ~ E yM

+zE M

ory E M ;

Monadic Lukasiewicz-Moisil algebras

(iii)

2

425

c ( ~ ( L )+ ) 2 E M or 5 E M .

E

M

Proof. In accordance with Proposition 1.17,

is maximal in

( L , 3 ) iff

M n C ( 3 ( L ) )is a maximal ideal of C ( 3 ( L ) ) . 1.19. Corollary. A n y 319-algebra i s semisimple, i.e. ideals is { o}.

0

the intersection of its m o n a d i c d -

1.20. Remark.

Let ( L ,3) be an %-algebra, M a monadic &ideal of ( L ,3) and MI a &ideal of the &algebra 3 ( L ) . Then

L

12

5 39,y E M I } is the

M n 3 ( L ) is a &ideal of 3 ( L ) ,

monadic &ideal of

rnn3(L)= M ,

(L,3)

=

(3

E

generated by M I and

3(L)n@ = M~ .

This proves that there exists an isotone bijection between the monadic &ideals of

(L,3)

and the &ideals of 3(L).

The previous results can be dualized to monadic &filters in an Vd-algebra. 1.21. Remark.

If ( L ,3) is an jd-algebra, then ( C ( L ) ,3) is a monadic Boolean algebra; if

f : ( L , 3 ) + (L',3) is a morphism of %-algebras, then C(f) : C(L)+ C(L')is a morphism of monadic Boolean algebras. Thus we obtain a functor from 3LMd to M o n B denoted by Cg,where M o n B stands for the category of monadic Boolean algebras. Similarly, we define the functors

CV : VLM6 + M o n B

,

C~V : 3mM19+ M o n B

1.22. Remark. Let ( B , 3 ) be a monadic Boolean algebra. Let us consider the d-algebra B[q. For every f E B[q, let 3f : I + B be the map defined by ( 3 f ) ( i )= 3 f ( i ) , for each i E I . We can prove that 3f E B[q and the

Monadic and polyadic Lukasiewicz-Moisil algebras

426

B[q-+ B[q thus defined is an existential quantifier on B[q. If u : ( B , 3 )+ (B',3) is a morphism of monadic Boolean algebras, then the 29-morphism u[q : B[q -+ B'[q is a morphism of Walgebras. Hence we obtain a functor MonB -+ 3LM9, which is a right adjoint of C3.

function 3 :

1.23. Representation theorem. Any 329-algebra is isomorpha'c to a functional 3d-algebra.

( L ,3) be an arbitrary 3-algebra. By applying t h e Halmos repre3), there sentation theorem 1.4.15 t o the monadic Boolean algebra (C(L), exists a non-empty set X , a complete Boolean algebra B and an injective Proof. Let

morphism of monadic Boolean algebras CP : For each a E

L,x E X , l e t

C(L)--t BX.

us consider the function q ( a ) ( x ) :

ItB

defined by

(1.21)

9 ( a ) ( x ) ( i )= iP(v;a)(x),

for every i E I .

The function Q ( a ) ( x )is isotone, hence (1.21) defines a map

9 : L + (B[qX . We observe that B[q is completely chrysippian and 9 is a rnorphism of bounded lattices. For each a E L , x E X , and i , j E I we have:

Thus

cpj9

= Qvj for every j E

I,t h a t

is 9 is a morphism of 29-algebras.

Now, we shall prove that 9 commutes with 3. Indeed, for every a E L ,

x E X and i E I,we have

Monadic Lukasiewicz-Moisil algebras

427

Consequently !P is a morphism o f 38-algebras. By the determination principle the injectivity of !P is immediate.

0

1.24. Remark. T h e similar representation theorems for VI9-algebras, 3VI9-algebras, etc. can be proved in the same way.

Monadic and polyadic Lukasiewicz-Moisil algebras

428

32. Modal operators on Lukasiewicz-Moisil algebras In this section we explore the structure o f Lukasiewicz-Moisil algebras with a modal operator.

2.1. Definition. Let L be a bounded lattice. A modal operator on L is a function P : L

--f

L

which verifies th e following axioms:

(2.1)

PO = 0 ;

(2.2)

P ( x V y) = P x V P y

,

for every x , y E L

A dual modal operator is a function Q : L

(2.3)

Q1 = 1 ;

(2.4)

Q(x A y) = Qx A Qy

,

4

.

L such t h a t

for every x , y E

L

.

2.2. Definition. A closure operutor on a bounded lattice L is a modal operator P satisfying the following axioms

(2.5)

x 5 Pa: , for every x E L

(2.6)

PP = P .

,

An interior operator on L is a dual modal operator Q such that

,

(2.7)

Qx I x

(2.8)

QQ = Q .

for every x E L ;

2.3. Remarks. If P is a modal operator (a closure operator) on a Boolean algebra B and Q x = i??, for every 3 E B , then Q is a dual modal operator (an interior operator) on B .

Modal operators on Lukasiewicz-Moisil algebras 2.4. Lemma. Let P be a modal operator o n a bounded lattice L.

429

T h e n f o r every

x,y E L, we have: (i)

x5y

(ii)

2

E

* Px 5 Py;

P(L)H x = Pz;

(iii) P(L)is a sublattice of L.

A dual result is true f o r dual modal operators. 2.5. Definition. A modal Boolean algebra ( a topological Boolean algebra) is a pair

(B, P),

where B is a Boolean algebra and P is a modal operator (a closure operator) on

B.

A topological Boolean algebra can be also defined as a pair ( B ,Q), where B is a Boolean algebra and Q is an interior operator on B . 2.6. Examples.

(i) A modal structure is a pair ( K , R ) ,where K is a non- empty set and R C I<'. For any modal structure X = ( K , R ) we define a map

P'

:

P ( K ) --f P ( K ) by

P'C = {x E K I there exists y It is easy t o see t h a t denote by

Bx

P'

E C such that

is a modal operator on

the modal Boolean algebra

(2,y)

E R) .

P(K).We

shall

( P ( K ) , P * )associated with

X . If R is reflexive and transitive then Bx is a topological Boolean algebra. If we identify the Boolean algebras P ( K ) and 2 K , then we have

for every f E 2" and

k

E K.

Monadic and polyadic Lukasiewicz-Moisil algebras

430

(ii) If X is a topological space, then we consider the map Qx : P ( X ) + P ( X ) defined by: Qx(C)= int C,for every C X . QX is an interior

P ( X ) . Conversely, an interior operator on P ( X ) (or closure operator) defines a topolgy on X .

operator on

a

2.7. Lemma. Let B be a Boolean algebra, A 2 B and x E B. I f A U {Z} is inconsistent then x E [A). Proof. Recall that a subset of a Boolean algebra is said t o be consistent if

it generates a proper filter. If A U {Z} is inconsistent then 0 E [ A U {Z}), hence there exist al, ...,a, E A such t h a t a1 A ... A a , A 5 = 0. It follows 0 that al A ... A a, 5 x, therefore x E [ A ) . 2.8. Lemma.

If B is a Boolean algebra, A C B and x

E

B , t h e n the following assertions

are equivalent:

(i) FOTa n y ultrafilter M of B we have A 2 M

+ x E A,

(ii) x E [ A ) . Proof.

(i) =$ (ii): If x

#

[ A ) then, by Lemma 2.7, A U {5} is consistent, then there is an ultrafilter M of B such t h a t AU{Z} E M. But A G M + x E A , hence we have a contradiction.

(ii) =$ (i): By a well-known argument.

0

2.9. Proposition (Lemmon [1967]).

Let ( B ,P ) be a modal Boolean algebra. T h e n there exists a modal s t m c ture = (I<,R) such that ( B ,P) is isomorphic t o a modal subalgebra of

x

( B x ,P*). Proof. It is easily seen, imitating the proof of Corollary 2.1.6, or by a direct reasoning, that t h e map (2.9)

d : B +P(PFl(B))

, d(x) = { M E P F l ( B ) I x E M }

Modal operators on Lukasiewicz-Moisil algebras is a morphism in

D01,hence a

Boolean morphism by Proposition 1.2.23.

Define further a binary relation

(2.10)

( M , N )E R

431

R on K putting

* (x E N * Px E M )

K . We shall prove that P*d(x) = d ( P t ) for every x E B. If M E P*d(x), then there is an ultrafilter N such t h a t x E N and ( M , N ) E R. By (2.10), Px E M , hence M E ~ ( P z )Conversely, . suppose M # P*d(x),then, for every N E h',we have

for every M , N E

( M , N )E R

* N # d(t) . B IQy

e.

M } , where Q y = Let N be an ultrafilter of B. By (2.10), ( M , N ) E R iff for every y E B , Q y E M implies N implies 2 E N , y E N . Since N is an ultrafilter, it follows t h a t A therefore by Lemma 2.8, J: E [ A ) . Thus there exist al,..., a, E A such that Let us put A = {y E

al A

... A a, 5 3, hence

But

&a1

M

,...,&a,

E M , then

E

Pa:

E M.

It follows that Px $ M , i.e.

# d(Px).

2.10. Corollary.

If ( B , P ) is a topological Boolean algebra t h e n there exists a modal structure X = (I(, R), with R reflexive and transitive, such that ( B ,P ) is isomorphic to a subalgebra of the topological Boolean algebra ( B x ,P*). 2.11. Definition. Let L be a &algebra. A modal d-operator on L is a modal operator P on

the lattice L , such that ( P ~= PP v ; , for every i E

I.

Similarly, we define the dual modal 8-operators, the closure d-operators and the interior d-operators on a &algebra. 2.12. Remark.

If L is a d-algebra with negation and P is a modal d-operator on L, then

Monadic and polyadic Lukasiewicz-Moisilalgebras

432

the function Q : L + L given by Qz = N P N z , for every z E L , is a dual modal d-operator on

L.

2.13. Definition. A modal 29-algebra (a dual modal 29-algebra) is a pair ( L ,P ) (a pair ( L ,Q)), where L is a &algebra and P is a modal &operator (Q is a dual modal 6operator) on L. Similarly, we define the closure $-algebras and the i n t e r i o r 8-algebras.

2.14. Remark.

If ( L , P ) is a modal &algebra (a closure &algebra) then ( c ( L ) , P l q ~is) ) a modal Boolean algebra (a topological Boolean algebra).

2.15. Example.

X = ( K , R ) be a modal structure every f E Lx and k E K we put Let

and L X the &algebra (L\')K. For

i E I and the rest of the proof is obvious. If R is reflexive and transitive, then ( L x ,P*) (then ( L x ,&*)) is a closure 6algebra (an interior &algebra). We shall prove, for example, that P*P*(f)= P * ( f ) ,for every f E L x . For each k E K we have: Hence y i p * = P*v; for every

=

V { f ( k " ) I (k,k')

E

R and (k', k") E R} 5

Modal operators on Lukasiewicz-Moisil algebras

433

2.16. Proposition.

Let ( L , P ) be a modal 6-algebra. T h e n there exists a modal structure X = ( K , R ) such that ( L ,P ) 23 isomorphic t o a modal 6-subalgebra of (Lx, P*). Proof. By applying Proposition 2.9 to the modal Boolean algebra

( ( C ( L ) ) , P ~ ~there L ~ )exists , a modal structure X = (I<,R)and an injective morphism d : C(L) + Bx of modal Boolean algebras. Let us define a function @ : L -+

L X by

@ ( z ) ( k ) ( i )= d ( c p i 4 ( k ) for every z E

L , k E I< and i E I. It is easy t o see that @ is an injective

morphism of bounded lattices. For every z E A ,

k

E

K and i , j E I we have:

@ ( ~ j z ) ( k ) (= i > d(cpiqjz)(k)= d ( ~ j z ) ( k = ) = @(4(k>(j)= ( c p j Q ( z ) ) ( k ) ( i ) and

@ ( P z ) ( k ) ( i= ) d(cp;Pz)(k)= d(P)cPiZ)(k)= P*(d(cpiZ))(b) =

v {d(cp;z)(k')I (k,k t ) E R} = = v { @ ( z ) ( k ) ( i )I (k,k') E R} = (P*@(4)(k)G) =

It follows that @ is a morphism of modal &algebras.

0

2.17. Corollary.

If ( L , P ) is a closure d-algebra, t h e n there exists a modal structure X = ( K ,R), with R reflexive and transitive, such that ( L ,P ) is isomorphic t o a closure 6-subalgebra of ( L x ,P*). 2.18. Remark.

The previous results can be dualized in an obvious way.

Monadic and polyadic Lukasiewicz-Moisil algebras

434

2.19. Proposition (McKinsey, Tarski [1944]).

For a n y topological Boolean algebra ( B ,Q ) there exist a topological space X and a n injective m o r p h i s m of topological Boolean algebras from ( B ,Q )

to (?(XI, Q x ) . Proof. Let d be the mapping (2.9) in the proof of Proposition 2.9 and for every

CCX

define

Q x ( C )= u { d ( a ) I d(a) E C and a = Q a }

.

It is easy t o see t h a t Qx is an interior operator on P ( X ) . We take the topology on

X

given by this interior operator. We shall prove t h a t d ( Q x ) =

~ x ( d ( x ) )for , every x E B . Since

Qx (d(s)) and Q x

= U{d(a) I a

I x and a

= Qa}

5 x, QQx = Q x , it follows that d ( Q 2 ) C Q x ( d ( x ) ) . If a 5 x

a = Qa, then Q a

I :Q x ,

therefore d(a) = d ( Q a )

and

d ( Q x ) . Consequently

Qx (4x1) E d ( Q x ) .

0

2.20. Lemma.

Let X be a topological space and L the d-algebra (Li')x. T h e n there exists a n interior 19-operator Q> o n L such that the topological Boolean algebras ( C ( L ) Q , > ~ C ( Land ) ) (2x,Q x ) are isomorphic.

f E L and i E I we define the map f; : X t L2 by f4x) = f(z ) ( i ) ,for each x E X . If i 5 j then f j 5 fj, hence, for every x E x: Proof. For every

Thus we can define a function Qk : L

for every f E L , x E X and

i E I . QK

--+

L

by putting:

is an interior d-operator on

L . We

shall only verify Q > ( ( p j f ) = ( p j Q > ( f ) for every f E L and j E I. For every f E L , x E X and i , j E I we have

435

Modal operators on Lukasiewicz-Moisil algebras

2.21. Proposition.

Let (L,Q) be a n interior 6-algebra. T h e n there exist a topological space X and a n injective morphism of anterior d-algebras from ( L ,&) t o Proof. By Proposition 2.19, there exist a topological space

X and an injec-

tive morphism of topological Boolean algebras:

d : (C(L),Qlc(r,))

--t

( L f , Q x ).

Let us define a function @ : L + (Lh')x by putting

@(a)(.)(;>= d(cpia)(z)

X and i E I. @ is an injective morphism of interior &algebras. We shall prove, for example, that @ Q = Q>@. For every a E L , x E X and i E I we have for every a E L , x E

therefore @ ( a ) ;= d(cp;a). Then we get

2.22. Remark. Any universal quantifier V on a &algebra L is an interior 6-operator. For any interior &algebra ( L ,V) the following assertions are equivalent:

Monadic and polyadic Lukasiewicz-Moisil algebras

436

(i) V is a universal quantifier on L; (ii) V is a universal quantifier on the Boolean algebra C(L). 2.23. Proposition. If ( L , v ) is a n interior d-algebra, t h e n the following assertions are equivalent: (i)

(L,v)is avo-algebra;

(ii) V ( L ) is a d-subalgebra of L;

(iii) V~piVx= Cp;Vx, f o r every i E I and x E L. Proof.

(i) =$- (ii): This implication is the dual of Proposition 1.9. (ii) + (iii): Suppose i E I and x E L. Since V ( L ) is a 9-subalgebra of L, Cp;x E V ( L ) and therefore V@Vx = CpiVx. (iii) =$- (i): For every i E I and x , y E L we have CpiX

= Pix V (CpiVY A PiVY) 2 (Pix V CpiVY) A CpiVy

7

hence

CpiVx = VCpiX

1 V(l(i0iXU ViVy) A QpiVy = CpiV(x V Vy) A Cpi'tly

Then, in the Boolean algebra C ( L ) we get

c ~ ~ VV ViVy X 2 c ~ ~ VV( Vy) X Using the determination principle it follows t h a t Vx VVy

2 V(x V V y ) . Since

the converse inequality is true in any interior &algebra, we have:

V(x v Vy) = v x

vvy .

0

2.24. Remark.

This proposition, in a dual form, was proved by L. Monteiro [1974] for 3valued Moisil algebras and by lorgulescu [1984c] for &algebras with negation.

2.25. Corollary (Halmos [1962]). If V is a n interior operator o n a Boolean algebra B, t h e n t h e following properties are equivalent:

M o d a l operators on Lukasiewicz-MoisiJ algebras

437

(i) V is a universal quantifier o n B ;

(ii) V ( B )i s

a

Boolean subalgebra of B ;

(hi) 4' % = t/a, ~

O every T

a E B.

2.26. Corollary.

Let (L,V) be a n interior 9-algebra. T h e n the following assertions are equivalent:

(i) ( L , v ) is a vd-algebra; (ii) V ( C ( L ) ) i s a Boolean subalgebra o f C ( B ) . Proof. By Proposition 2.23 and Remark 2.22.

0

The remainder of this section contains some natural generalizations of the results obtained by lturrioz [1977] for the case of 3-valued Moisil algebras. 2.27. Definition. Let

Q

L. A filter F of L will E L , we have a E F + &a E F .

be an interior operator on a bounded lattice

be called a &-filter if, for every a 2.28. Definition.

A Q-filter is Q - p r i m e if, for every a, b E L:

QaVQbEF+QaEF

or Q b E F .

The following three lemmas are valid in a bounded lattice L with an interior operator

&.

2.29. Lemma. If A is a n o n - e m p t y subset of L, t h e n the &-filter of L generated by A as given by

[A)Q= {x E L I there exist al, ...,a, E A, Qal A ... A &a, 5 x}

.

438

Monadic and polyadic Lukasiewicz-Moisil algebras

2.30. Lemma.

Any maximal &-filter is Q-prime. 2.31. Lemma. a)

If F is a Q-prime Q-filter of L , then F n Q ( L ) is

a prime filter of

Q(L). b) If F' is a prime filter of Q ( L ) , then [F')Qis a Q-prime Q-filter of L . c)

The functions F H F n Q ( L ) , F' H [F')Qestablish an isotone bijection between the &-filters of L and the filters of the lattice Q ( L ) .

2.32. Remark. Let ( L , V ) be an interior 0-algebra. Then Q = Vvl is an interior operator on the lattice L and Q ( L ) C ( L ) . A 0-filter F of L is a Q-filter iff for any a E A , a E F implies Va E F . 2.33. Proposition. Let ( L ) V ) be an interior 0-algebra. Then the following assertions are

equivalent: (i) Any Q-prime Q-filter of L is a maximal &-filter; (ii) ( L , V ) is a V0-algebra. Proof. By applying Lemma 2.31, Nachbin's theorem 1.3.19 and Corollary 2.26 the following assertions are equivalent: (a) Any Q-prime Q-filter of

L is maximal;

(b) Any prime filter of the lattice Q ( L ) is maximal in Q ( L ) ; (c) Q ( L ) = V ( C ( L ) )is a Boolean subalgebra of C ( L ) ;

(d) (L,V) is a Vd-algebra.

Modal operators on Lukasiewicz-Moisil algebras

439

2.34. Corollary.

If V is a n interior operator o n a BooKean algebra B, t h e n the following assertions are equivalent: (i) A n y V - p r i m e V-filter of ( B , V ) is mazimal; (ii) V is a universal quantifier o n B. 2.35. Lemma.

Let Q be a n interior operator o n a lattice L such that Q ( L ) C C(L).If F is a &-filter, t h e n the following assertions are equivalent: (i) F is a maximal &-filter;

(ii) For each a E L , exactly one of the elements &a, Proof. Straightforward.

&. belongs

to F. 0

2.36. Definition. An interior d-algebra is said to be Q-semisimple if the intersection of its maximal Q-filters is(1). 2.37. Proposition.

FOT any interior d-algebra ( L , V ) the following assertions are equivalent: (i) ( L , V ) is Q-semisimple; (ii) ( L , V ) as a Vd-algebra. Proof.

(i) j (ii): Let ( L , V ) be Q-semisimple. Suppose there exists a Q-prime

F which is not maximal and take a maximal Q-filter Ml including F and an element x E Ml - F . Then Q x E Adl - F and 2 # 0. Since ( L , V ) is Q-semisimple, there exists a maximal Q-filter M2 such that x # M2, hence Q X $! Mz. By Lemma 2.35, &z $ MI and &. E Mz, therefore Q&. E M2 - M I . If Qx V Q&. = 1, since F is Q-prime, we have Q x E F or Q&. E F . Q-filter

440

Monadic and polyadic Lukasiewicz-Moisil algebras

The contradiction Qx E M I or Q&. E M I , obtained in this case, implies Qx V Q&. # 1. Then there exists a maximal Q-filter M3 such that Qx V Q&Z # M3. It follows that Qx # M3 and Q&. # M3. By applying Lemma 2.35 we get &. E M3, hence &&. E M3, in contradiction with Q&. $$ M3. Consequently any Q-prime Q-filter is maximal, and therefore by Proposition 2.33, (L,V) is a VO-algebra. 0 (ii) j (i): This implication is the dual of Corollary 1.16. 2.28. Corollary (Halmos [1962], A. Monteiro [1971]). For a n y topological Boolean algebra (B,V) the following assertions are equivalent: (i) (B,V) is V-semisimple; (ii) V is a universal quantifier on B.

Construction of a 3-valued M-algebra from a monadic LM-algebra 441 53. A construction of a 3-valued Moisil algebra from a monadic Lukasiewicz-Moisil algebra A.Monteiro [1964], L. Monteiro and Coppola [1964] constructed a 3valued Moisil algebra

L ( A ) from

a monadic Boolean algebra A .

If L is a 3-valued M o i l algebra, then there exists a monadic Boolean algebra A such that L ( A ) and L are isomorphic (see A. Monteiro [1964] and L. Monteiro [1978]). In this section we shall present some extensions of these results, obtained by lorgulescu. We closely follow lorgulescu [1984a,c]. See also Nadiu [1967]. Throughout this section by a monadic &algebra we shall understand a monadic &algebra with negation.

3.1. Preliminaries. Let (A, A, V,

-, 0,1,

{ y i } j G 3) ~,

be a monadic 8-algebra. Recall that the universal quantifier V is defined in this case by Vx = -3 - x , for every x E A. For every x , y E A we shall denote

z

y

( 3 ~ 1=s 3'cply and Vyox = Vyoy)

.

We get immediately

3.2. Lemma. FOT every x,y E A, we have: (3.3)

-(Zny)=-z

u

-y,

-(zuY)=-z

n

-y.

442

Monadic and polyadic Lukasiewicz-Moisil algebras

The other equality is obtained similarly. 3.3. Lemma.

In any monadic 0-algebra A the following properties hold:

0

Construction of a 3-valued M-algebra from a m o n a d i c LM-algebra

The other two relations follow via Lemma 3.2. 3.4. Lemma. T h e relation -j

443

0

= is a congruence o n A with respect t o the operations n, U,

3p1, VPO.

Proof. We only prove that for every x,x', y, y' E A: 2 3

If x

E

x', y

G

x',y e y'

* x u y = x'u y' .

y' then

VVOY = VVOY' therefore, by Lemma 3.3,

3V1("

v 3y1y' = 3yY,(x' u y') .

y) = 3VlX v 3VlY = 3VIX'

Similarly, Vqo(x u y) = Vqo(x' U y'), hence x U Y

2'

IJ Y'.

3.5. Notation. Consider the following operations on the quotient set A /

i n 6 = ( Z n yr, i u 6 = ( X u y r ,

0

=:

i = (-xT;

(3.8)

012 = (VVoXr;

02?

= (3qlZr.

3.6. Lemma.

(,C'(A) = A /

Z,n, U,

-,b, i) is

a D e Morgan algebra.

Proof. By Lemma 3.3.

0

3.7. Proposition (lorgulescu [1984a,c]).

(,?(A), n, U, N , 0, i,ol,u 2 ) is a 3-valued Moisil algebra. Proof. It suffices to check that for every hold:

Z,

y E A , the following equalities

Monadic and polyadic Lukasiewicz-Moisil Agebras

444

The proof uses Lemma 3.3. Let us verify e.g. the second equality:

It follows that z n -x

= -x n 3 ~ 1 2 hence , the second equality is proved.

0

3.8. Remark. In the particular case of a monadic Boolean algebra ( A ,A, V, lations (3.1) reduce t o

-, 0,1,3)

re-

The 3-valued Moisil algebra L ( A ) obtained in this case was constructed by A. Monteiro [1964], using the theory of Nelson algebras, and, directly, by L.

[1964]. The generalization L'(A) of lorgulescu follows the line of the latter paper.

Monteiro and Coppola

3.9. Notation.

( L ,n, U, -,O, 1,rl,r 2 ) be a 3-valued Moisil algebra. Define the operations A, V, -, 3, sl,s2 on L x L by Let

Construction of a 3-valued M-algebra from a monadic LM-algebra 445

3.10. Lemma. (Lx L, A, V, -, ( O , O ) , (1,l), s1,52,3) i s a monadic 3-walued Moisil algebra 3UCh that V(z,y) = (z n y,a: n y) for every z,y E L and the function h has the following properties:

(3.11)

h(z fl y) = h ( z ) A h(y), h(z U y) = h ( z )V h(y)

(3.12)

h(r1~) = Vsih(2)

(3.13)

h(r22) = 3 ~ 2 h (=~szh(2) ) = 3h(s) ,

(3.14)

h(-s)

(3.15)

h(z U y) = h ( z ) U h(y) on L x L

,

(3.16)

h(z n y) = h ( z )n h(y) on L x L

,

(3.17)

h is injective ,

where

G

,

-h(z) on L x L ,

n, U are the operations on L x L

Proof. We check only

h(r1z) = =

using (3.13) we get

,

defined by

(3.1).

(3.12)-(3.15):

(r1z,r2r15)

= ( r p , 7-12) = V(r12, 1 . 2 5 ) =

V(r12, r1r25)

= VSl(5,

1.25) = Vs,h(z)

,

Monadic and polyadic Luhsiewicz-Moisil algebras

446

=

352(75,

r p x ) = 3 4 7 5 , 7 r 2 x ) = 3s2 - h ( r ) ,

and similarly Vs,h(-s) = Vsl - h(x). Then

3s2(h(x) u h(y)) = 3s2h(x) v 3s,h(y) = h(r25) v h(r2y) = = h(r2(x

u y))

= 3s2h(r u y)

,

and similarly Vsl ( h ( z )U h(y)) = Vslh(r U y).

0

3.11. Remark. Let us consider the 3-valued Moisil algebra given by the diagram r2x -

i' e

I

e

c

0

O

1 1

1

0

We have h ( L ) = {(O,O),(c,l),(l,l)} and (c,l) E h ( L ) , but -(c,l) (c,O) @ h ( L ) . Consequently, h ( L ) is not closed under the operation -.

=

3.12. Proposition.

Define si = Vs1, si = 392 and N h ( x ) = h(7x) for every r E L. T h e n ( h ( L ) ,A, V, N , (O,O), (0, l), si,sh) is a 3 - d u e d Moisil algebra and h : L + h ( L ) as a n isomorphism. Proof. By Lemma 3.10.

0

3.13. Lemma.

h ( L ) is closed u n d e r (Is2-) and (Vsl-)

on L x L.

Proof By applying Lemma 3.10 we get, for every 3s2

- h ( s ) = 3s2h(1x) = h(rz--.x),

VSl

- h ( z ) = Vslh(1z) = h(?-112).

x' = h ( r ) E h ( L ) :

Construction of a 3-valued M-algebra from a monadic LM-algebra 3.14. Lemma. h ( L ) is closed under

447

n, u on L x L.

Proof. By Lemmas 3.10 and 3.11.

0

3.15. Remark. ( h ( L ) ,A, V, N , (O,O), (1,l),si,s& 3) is not a monadic 3-valued Moisil algebra, because, for example, there exists x E

L such t h a t V h ( z ) = ( x , x ) $!

h(L). 3.16. Remark. Let ( A ,A, V, -, 0,1, sl,s2) be a 3-valued Moisil algebra. It is easy t o prove t hat if A. is a sublattice of A w i th 0 , l E Ao, then t h e 3-valued Moisil subalgebra generated by A. is th e set of all elements of t h e form:

where rn E

RV

and c;, bi,zi, yi, ui,vi E Ao, i = 1,...,m. In particular the

elements of the 3-valued Moisil subalgebra S o f

( L x L , A, V, -, ( O , O ) , ( l , 1),s1,s2)generated by h ( L ) ,have the form

where rn E

RV

and ci,bi,z;,v; E h ( L ) ,

i = 1,...,rn. Note also t h a t

(S,A,V,-,(O,O),(l, l),sl,sz) is not a monadic 3-valued Moisil subalgebra of L x L because, for example, V h ( L )= { ( z , ~ )I z E L } S.

Monadic and polyadic Lukasiewicz-Moisil algebras

448

3.18. Lemma. If d , b', x', v' E h(L), then (3.19)

3S2(C' V -b' V six' V ~2

(3.21)

C'

V -b' V s ~ x ' VSZ - V'

E

(c' V Nb' V

X'

- v')

= 3 ~ 2 (V~Nb' ' V X' V NU') ;

V 3 ~ 2 N v 'A) ( 3 ~ 2V~Nb' ' V 392X' V NU') .

Proof. Let c' = h(c), b' = h(b), x' = h(x), v' = h(v), c, b,x,v E L and t = c' V -b' V slx' V s2 - v'. By Lemmas 3.10, 3.13 and Proposition 3.12, we obtain:

3 ~ 2= t 3 ~ 2 h ( cV) 3.52 - h(b) V 3 s l h ( x ) V 3 ~ 2 h(v) = = 3 ~ 2 h ( cV) 3 ~ 2 N h ( bV) 3 ~ 2 h (V~3)~ 2 N h ( v )= = 3S2(C' V Nb' V X' V NU') ,

since 3slh(x) = 3(rlx,r2x)= ( ~ ~ 2 , 7 3= 5 )h(r2x) = 3s2h(x).Then

Construction of a 3-valued M-algebra from a monadic LM-algebra 449

Using Proposition 3.12 we obtain

3 ~ 2 ( (V~Nb' ' V X' V 3 ~ 2 N u 'A) ( 3 ~ 2V~Nb' ' V 3 ~ 2V~N'U ' ) ) = = ( 3 . ~ 2V~ '3S2Nb' V A

3522'

( 3 ~ 2V~3~2Nb' ' V

' V = 3 ~ 2V~3~2Nb'

3522'

3925'

= 3 ~ 2 ( V~ Nb' ' V X' V

V ~ s ~ N uA' ) V 352NU') =

V ~ s ~ N u='

NU') = 3 ~ 2 t

and

Vs1((c' V Nb' V X' V 3~2Nb')A ( 3 ~ 2V~Nb' ' V 392,' V N U ' ) ) = = (Vslc' V VslNb' V VSld V ~ s ~ N uA' ) A

(3~2c'VVslNb'V~ s ~ x ' V V S ~ N=UV 's)l t .

0

3.19. Notation. Let ( L , f l ,U, i , O , 1,r l , 1'2) be a 3-valued Moisil algebra and (3.22)

(C'(L x L ) = ( L x L ) / ~,fl,U,~,(O,Or,(l, lr,~1,02)

be the 3-valued Moisil algebra obtained from Definition 3.5 with A = L x L ; cf. Proposition 3.7. Let further H : L -+ ( L x L)/ be the map defined

=

by (3.23)

H ( x ) = (h(x))^= ((x,r 2 x ) ) ^ , for each x E L

Monadic and polyadic Lukasiewicz-Moisd algebras

450 3.20. Proposition.

T h e m a p H defined by (3.23) is a n injective m o r p h i s m of 3-valued Moisil algebras and H ( L ) = S/ =. Proof. Using Lemma 3.10, we obtain

H(. n Y) = (h(x

n y))= ( w ) n

(h(y$=

H ( 4 n H(Y)

and similarly

H ( z ) = H(y)

H h(x)

3

h(y) H 3s2h(x) = 3s2h(y) and

Vslh(z) = Vslh(y) H h(r2x) = h(r2y) and

rlx = r l y H x = y by the determination principle. Thus H is an injective morphism of 3-valued Moisil algebras. m

Any element a of S has the form a =

A

ai, where

i=l

Using Lemma 3.18 (3.21), there exist d: E

a. = d!, i = 1,...,rn, therefore 1 a

h ( L ) , i = 1,...,rn such that

Construction of a 3-valued M-algebra from a monadic LM-algebra

If we put

20

=

(

m

77%

n

u;)

n 3s2u, z = ( lJ

d:)

n 352U

then by Lemmas

i=l

i=l

3.14 and 3.17 we get w

451

= z , z E h(L) and by Lemma 3.3:

m

and

= Vslu A 3s2u = Vsla . It follows that w E a, therefore z

H ( t ) = (h(t))^=21.

G

a . Since z = h ( t ) , t E L , we obtain

n

Monadic and polyadic Lukasiewicz-Moisil algebras

452

$4. Polyadic Lukasiewicz-Moisil algebras

The polyadic Lukasiewicz-Moisil algebras are natural extensions of the polyadic Boolean algebras. The representation theorem of polyadic 29-algebras (Proposition 4.11) is the algebraic version of the completeness theorem for the $valued predicate calcutus. The results of this section are taken from Georgescu [1971]. 4.1. Definition.

A polyadic 29-valued Lukasiewicz-Moisil algebra or polyadic 29-algebra is a structure (L,U,S,3,V) where L is a d-algebra, U is a non-empty set, S i s a map from Uu to the set of endomorphisms of L and 3, V are two maps from P(U) t o LL, such that the following axioms hold: (4.1)

S(1u) = 1~ ;

(4.2)

S(QT)= S(Q)S(T),

(4.3)

3(0) = V(0) = 1L ;

(4.4)

3(K U K ' ) = 3(K)3(K') and V ( K U K')= V(K)V(K') ,

for every (4.5) for every (4.6)

for every Q , T E U u ;

K,II' E U ; S ( a ) 3 ( K )= S(7)3(Ii) and S ( u ) V ( K )= S( T) V( K),

K E U and for every 0,r E U u such that

QlU-K

=T~U-K;

3(K)S(a)= S ( o ) 3 ( ~ - l ( K )and ) V( K ) S (Q ) = S(a)V(o-'( K ) )

for every (4.7)

II E U and for every Q E U u such that For every

K

Q ~ ~ - I ( is K injective; ~

U ,( L ,3(K),V(K))is an 3Vd-algebra .

If ( L ,U,S, 3,"), (L',U,S, 3,") are two polyadic 29-algebras then a polyadic 29-morphism from (L, U,S, 3,V) t o (L',U,S, 3,V) is a morphism of 29algebras f : L + L' such that fS(a) = S(a)f,f 3 ( K ) = 3(K)f and

Polyadic Lukasiewicz-Moisil algebras

453

p ( K )= V ( K ) f , for every a E Uu and K C U. 4.2. Remark. If ( L ,U,S, 3,V) is a polyadic &algebra, then C ( L ) can be endowed with a canonical structure o f polyadic Boolean algebra. Every polyadic 29-morphism

(Lf,U,S,3,V) induces a morphism of polyadic Boolean algebras C(f) : (C(L),U,S,3) + (C(Lf),U,S,3).In this way we

f

: (L,U,S,3,V)

---t

have defined a functor from the category of polyadic &algebras t o the category of polyadic Boolean algebras.

4.3. Definition. Let ( L ,U,S, 3,V) be a polyadic &algebra and a E L. A subset K of U is a support of p if 3(u - K ) p = p ; a is independent of I< E u if g(I<)a = a. A polyadic &algebra is locally finite if every element has a finite support. The degree of (L,U,S,3,V) is the cardinality of U. 4.4. Lemma.

Let ( L ,U,S , 3,V) be a polyadic 6-algebra, a E L and I< 2, t h e n the following assertions are equivalent:

U. If caxd(U) 2

(i) K is a support of a; (ii) V(U - K ) a = a; (iii) a,T E

U ' ,

+ S(a)a = S ( T ) U ;

a1v-K = ~ 1 u - K

i E I , K i s a support of algebra c ( L ) .

(v) f o r every

via

in the polyadic Boolean

4.5.

In the rest of this section, by a polyadic 6-algebra w e shall m e a n a locally finite polyadic 6-algebra of infinite degree. 4.6. Example. Let L be a complete completely chrysippian &algebra,

U

an infinite set and

Monadic and polyadic Lukasiewicz-Moisil algebras

454

X a non-empty set. The set Set(X', L ) of all functions X u + L has a natural structure of &algebra. For every K G

S(a)on

U and a E Uu define three unary operations 3 ( K ) ,V(K), S e t ( X u , L ) by putting:

(4.9)

V O P ( 4 = v { ( P ( Y ) I Y E xu, Y K * 4 V ( K ) P (= ~ )A M Y ) I Y E X u , YI{*Z)

(4.10)

S W P ( 4 =P ( 4

(4.8)

X u + L , a E U u and K U. Recall that yILx iff = z l u - ~ .We can show that S e t ( X u , L ) is a polyadic d-algebra.

for any p

~Iu-K

7

:

4.7. Definition.

A polyadic hubalgebra of S e t ( X u , L ) will be called a f u n c t i o n a l L-valued polyadic d-algebra. Denote by F ( X U ,L ) t h e functional polyadic 8-algebra of all elements of Set(X', L ) having a finite support. 4.8. Remark.

F ( x ~L, ) is locally finite. 4.9. Remarks. a) The definition of a polyadic &algebra with negation can be given analo-

gously, but in this case th e universal quantifiers V(K)will be defined by

V ( K ) = N 3 ( K ) N , while axiom (4.7) will be superfluous. b) In Example 4.6, if L is a &algebra with negation, then F ( X U , L )is also with negation. 4.10. Proposition.

L e t ( L ,U , S, 3,V) be a completely chrysippian polyadic d-algebra. For e v e r y a E L, T E U' and I< C_ U t h e following equalities hold: (4.11)

S ( T ) ~ ( K= ) UV {S(a)a I ~ I ( , T }

(4.12)

S(.r)V(K)a = A {S(a)u I ~ K , T.)

Polyadic Lukasiewicz-Moisil algebras

455

Proof. By Proposition 1.4.24 we have

for every

i E I. Applying th e determination principle we get (4.11) and

similarly (4.12). 4.11. Proposition.

For any polyadic $-algebra ( L ,U,S, 3,V) there exists a 6-valued Post algebra L' such that (L,U,S,Y,V) is isomorphic t o a functional L'-valued polyadic 8-algebra. Proof. Let B be the MacNeille completion of the Boolean algebra C ( L )and

L' = B[q the &valued Post algebra associated with B . L' is complete and completely chrysippian. Consider th e map @ : L t F(U', L') defined by (4.13)

@ ( u ) ( T ) ( i )= ' p i S ( ~ ) ,a

L , T E U u and i E I. We can show t h a t @ is a morphisrn of &algebras. For example, for every a E L , T E U u and i , j E I we can write

for every a E

@(S(O)U)(T)(i)=

=

cp;S(T)S(O)U = c p ; S ( T O ) U = @(U)(TO)(i)

= S(a)@(a)(.r)(i) .

Using Proposition 4.10 for the polyadic Boolean algebra C ( L ) we obtain @(Y(I')U)(T)(i)

= cpiS(T)3(I<)U = S ( T ) j ( . f { ) c p i U

= =

=

v {S(T)'piU E uu, nK*r} = v { @ ( u ) ( r ) ( i1 )TI.*.} = IT

= ( ~ I W @ ( 4 ) ( T ) ( i )*

Monadic and polyadic Lukasiewicz-Moisilalgebras

456

If @ ( a )= cf,(b),a , b E L , then

for any

i

E

I , hence a = b.

4.12. Remark.

Let

(L,U,S, 3,V)

(4.14)

be a polyadic &algebra. Set

Eo(L)= { a E L 18 is a support of a }

.

It is easy t o prove that Eo(L)is a 6-subalgebra of L . 4.13. Proposition. Let ( L ,U,S, 3,") be a polyadic 6-algebra and F a proper 6-fiZter ofEo(L). Then there exist a non-empty s e t X and a polyadic d-morphism @ : L +

F ( X u , L y ] ) such that @ ( a )= 1, for each a E F . Proof. Consider the polyadic Boolean algebra ( C ( L ) ,U,S, 3) and denote by

E(L)the Boolean algebra of all elements of C(L)having 0 as support in C(L).It is obvious that E ( L ) = E o ( L )n C ( L ) and Fo = F n C(L)is a proper filter of the Boolean algebra E(L). By Theorem 1.4.28 there exists a non-empty set

X and

a rnorphism of

polyadic Boolean algebras Q : C ( L ) + F ( X U , L z )such that Q ( a ) = 1for each a E Fo. Define a map @ : L

for every a E

L,z E X u

of &algebras.

For every

have:

--t

F ( X U , L a )by putting

i E I. It is easy t o prove t h a t cf, is a morphism a E L, K C U ,u E U u , x E X u and i E I we

and

Polyadic Lukasiewicz-Moisil algebras

457

@ ( 3 ( K ) a ) ( z ) ( i= ) Q(cpJ(K)a)(z)= Q(3(K)cpia)(z)= = w w c p i a ) ( z )= =

v {%a)(y)

I yK*z)

=

v { ~ ( a ) ( y > ( iI>Y I ~ ~ S )= ( g ( K ) @ ( a ) ) ( z ) ( i.)

Analogously, we can show that @ commutes with V(Ic), for every K U. It follows that @ is a polyadic 19-morphism. If a E F then cpia E Fo,therefore Q(cp,a)= 1for each

i E I. Thus

@ ( u ) ( z ) ( i= ) Q ( v i a ) ( S )= 1

,

for every z E X u and

i E I.0

4.14. Definition.

A polyadic 29-morphism of the form @ : L Moisil representation of L.

-+

F ( X u , Lhq) will be called a

4.15. Example.

Let X be a non-empty set, U an infinite set, J * the chain Lhq with t h e dual order and 19* th e order type of J'. Consider t h e d*-algebra PJ.(X') of all fuzzy J*-subsets of X u ; cf. Ponasse [1978] (see also J. Coulon and J.L. CouIon [1989]). For every T E U u consider the mapping S(T) : PJ.(Xu) -+

P J * ( X ~given ) by S ( ~ ) p ( z=)p ( n )

,

for every p E P p ( X u ) and s E X u

.

If K C U then define the mappings:

3 ( K ) : PJ.(XU) + P J . ( P ) , V(K) : P J . ( P ) -+ PJ.(XU) by putting, for each p E P p ( X u ) and z E X u :

3(IOPW

=

v M Y ) I Yl<*z) ,

It is straighforward t o show t h a t ( P p ( X u ) ,U,S, 3,") is a polyadic 19*algebra. This polyadic algebra is not always locally finite, but, as in Example 4.6, we can take its elements of finite support,

458

Monadic and polyadic Lukasiewicz-Moisil algebras

4.16. Proposition. For a n y polyadic 'I(l-algebra( L ,U,S, 3, V) and f o r a n y proper filter F of EO(L) there ezists X # 0 and a polyadic representation of the f o r m @ : L + P p ( X U )such that @ ( u )= 1 f o r every u E F . Proof. Similar to the proof of

4.13.

0

459

CHAPTER 9 LUKASIEWICZ LOGICS

The theory of Lukasiewicz-Moisil algebras originated in the need for an algebraic counterpart of the many-valued logics introduced by Lukasiewicz. In this chapter we present the three-valued Lukasiewicz logic in the Wajsberg axiomatization, the n-valued Lukasiewicz logics in the Cignoli axiomatization and a logic whose theorems are the propositions true for all than a fixed

k E I . The

i E I greater

Lindenbaum-Tarski algebras of these logics are the

3-valued Moisil algebras, a subclass of n-valued Moisil algebras and t h e dvalued LM-algebras, respectively. The latter logic is studied in more detail, including predicate calculus.

31. The Wajsberg axiomatization of the three-valued Lukasiewicz logic The first system of three-valued logic was constructed by J. Lukasiewicz in 1920 in connection with his investigations in modal logic (see Lukasiewicz [1920]). His idea was t o consider a third truth-value

between 0 (false-

hood) and 1(truth). In this way th e sentences of the three-valued logic are interpreted in

L3 =

{O,i,l}.

Lukasiewicz defined in L3 a unary operation

N (negation) and a binary operation Nz=l-z

--f

(implication):

and x + y = m i n ( l , l - ~ + y ) .

The other connectives were defined by Lukasiewicz in terms of N and +: z V y = (z+ y) + y

(disjunction)

z A y = N ( N x V Ny) (conjunction) . The first axiomatization of the three-valued logic was given by Wajsberg [1931] using axioms (Al)-(A4) presented in Definition 1.1below. This section is concerned with th e Wajsberg axiomatization of the three-valued

Lukasiewicz logics

460

Lukasiewicz logic. T h e proof of some syntactic properties of the three-valued propositional calculus

W is taken from Becchio [1972] and t h e proof of the

strong completeness theorem f r o m Goldberg, H. Leblanc and Weaver [1974]. Axioms (Al)-(A4)

induce o n the Lindenbaum-Tarski algebra of W a ca-

nonical structure of Wajsberg algebra. But t h e three-valued Moisil algebras and t he Wajsberg algebras are equivalent structures (Becchio [1978d]). Thus we can assert th a t th e three-valued Moisil algebras are algebraic models for the three-valued Lukasiewicz logic.

1.1. Defi nition. The sentences of the three-valued propositional calculus W are obtained from a countable set V of proposn'tional variables and t h e logical connecti-

ves N and +, according to th e following rules:

(i)

t he propositional variables are sentences;

(ii) if p , q are sentences then N p and p

---t

g are sentences;

(iii) every sentence is obtained by the above rules (i) and (ii)applied finitely many times.

E , is the (adjacent set of the) Peano algebra of type (2,l) generated by V . In other words, t h e set of sentences denoted by

W e use t he notation

for any sentence p E E and a fixed sentence p o E E . T h e axioms of W are th e sentences of t h e following forms:

W has modus ponens (m.p.) as rule of inference:

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic P,P

+

461

4

4

A proof o f a sentence p is a finite sequence p l , ...,p, = p o f sentences such that for any i 5 n we have one of the following possibilities: (a) p; is an axiom;

(b) there exists j , k

< i such that pk

is the sentence pj

-+

pi.

A sentence p is provable (I- p ) if there is a proof of it. If p E E and S C E then a proof of p from S is a finite sequence p1, ...,pn = p of sentences such that for any i 5 n we have one of the following possibilities: (a) pi is an axiom or a member of S;

(b) there exist j , k

< i such that pk

is the sentence

In this case, we say that p is provable f r o m

0 I- p

pj

+ pi.

S (S I- p ) .

is t he same as I- p . If the sentences s, p satisfy {s}

In particular

I- p we write

simply s I- p .

A set S of sentences if syntactically consistent if there is no p in W such that S I- p and S t- N p ; if not, S is syntactically inconsistent. A syntactically consistent set S is maximal consistent if S I- p for any sentence p such that S U { p } is syntactically consistent. Now we shall give, following Becchio [1972], some syntactic properties of

w.

1.2. Lemma.

The following properties hold an W :

462

(1.5) (1.6) (1.7) (1.8) (1.9) (1.10) (1.11) (1.12) (1.13) (1.14)

(1.15) (1.16) (1.17) (1.18) (1.19) (1.20) (1.21) (1.22) (1.23) (1.24) (1.25) (1.26)

Lukasiewicz logics

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic

463

Proof. (1.1) and (1.2) follow by (Al), (A2) and m.p., while (1.3) follows by m.p. from (1.2). To obtain (1.4) use ( A l ) for p := r t p , q := ¶ and also for p := p , q := r , then apply (1.3). Now (1.5) follows from

(1.4). To obtain (1.6) use (A2), then again (A2) but for p := q 3 r , q := p --f r , r := s; finally apply (1.3). Now (1.7) follows by m.p. from (1.6), then (1.8) by m.p. from (1.7). To obtain (1.9) apply (1.1)for p := p , q := q --f N q , then (1.1)for p := ( q -+ N q ) --t p , q := p --t q, then (A3) for p := q , then (1.3) for p := p

+ q,

q := ( q + N q )

-+

q,

r := q; finally apply m.p. Further (1.10) follows by m.p. from (1.9) for p := p , q := r and (1.3) for p := p , q := p -+ r , r := r . Now

if t is a provable sentence then using ( A l ) for p := p , q := t , (1.6) for p := t , q := p , r := q , s := q , (1.3) for p := p , q := t -+ p , r := ( ( t t q ) + q ) t ( ( p t q ) --t q ) , (1.9) for p := t , q := q and (1.10) for p := (t + q ) t q , q := p , r := ( p q ) + q, we obtain (1.11).Further from (1.1)for p := q , q := r we obtain

via (1.2) for p

( ( q + r ) -+

r ) , r := p --t r ; then from (A2) for p := p , q := q --t r , r := r and (1.27) we get (1.12) via (1.3). Now (1.12) implies (1.13), then (A2) and (1.13) imply (1.14), while (1.14) implies (1.15). To obtain (1.16) apply (1.14) for p := r , q := p , r := q and (1.3). To obtain (1.17) apply (1.11) for p := q , q := r , then (1.15) for p := p , q := q , r := ( q -+ r ) -+ r. To obtain (1.18) apply (1.17) for p := p -+ q, q ._ .- q , r := r , then (1.13). To obtain := q , q := q

-+

(1.19) apply (1.3) t o ( A l ) for p := N p , q := N q and (A4) for p := q , q := p . Further (1.19) implies (1.20). To obtain (1.21) apply (1.3) t o (1.19) for p := N p , q := N q and (A4). Then (1.22) follows from (1.19) and (1.2) for p := N p , q := p + q, r := q. To obtain (1.23) apply (1.3) t o (1.21) for p := p , q := p + N p and (A3). Now (1.24) follows by m.p. from (1.23) for p := N p and (A4) for p := N N p , q := p . Further from (1.23) and (1.2) for p := N N p , q := p , r := q we get

Lukasiewicz logics

464

while from (1.24) for p T := N N q we obtain: (1.29)

:= q and (1.25) for p

:= N N p , q

:= q,

I- ( N N p -+ q ) -+ ( N N p -+ N N q ) ;

then (1.3) applied t o (1.28) and (1.29) yields

which together with (A4) for p := N p , q := N q , yields (1.25), again by 13 (1.3). Finally (1.26) follows from (1.24) and (1.23) by (1.3). 1.3. Lemma. T h e relation (1.31)

-

p-q

defined by iff

kp+q

and I - q - i p

is a n equivalence o n the set E of sentences of W . Proof. The relation is reflexive by (1.26), symmetric by definition and transitive by (1.3).

0

1.4. Theorem. Define the following operations

+, N ,

1 o n the quotient set E l -:

(1.32)

fi -+ 4 = p 3 q ,

(1.33)

Nfi = N^p

(1.34)

1 is the set of all provable sentences .

Then (E/

-,

,

N , 1) is a Wajsberg algebra, i.e. it satisfies conditions

3,

(W1)-(W 6 ) in Theorem 3.3.8. Proof. If p

-

-

p' and q q' then the sentences p + p', p' 4 p , q -+ q' and q' -+ q are provable. Then (1.15) and (1.2) imply t- ( p -+ q ) 4 ( p --+ q') and I- ( p -+ q') + (p' -+ q'), respectively. It follows by (1.3) that I- ( p -+ g ) -+ (p' -+ q') and similarly I- (p' 4 q') -+ ( p --+ q ) , therefore p + q p' -+ q'. This shows that the operation -+ is well defined. One proves similarly, using (1.25), that N is well defined. Further if t is provable and t t' then t' is

-

-

465

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic provable by m.p., while if t and t' are provable then t

-

t' by (1.1). This

shows that the set of provable sentences is actually an equivalence class. Finally axioms (Wl)-(W4)

are fulfilled in view of (Al)-(A4),

vely, (W5) reduces to the above remarked fact that I- t and

respecti-

I- t + t'

imply

I- t', while (W6) is straightforward from (1.31). 1.5. Remark. In accordance to Theorem 3.3.8, the Lindenbaurn-Tarski algebra E /

0

-

of

the three-valued propositional calculus is a three-valued Moisil algebra. This is a precise meaning of the assertion that the structure of three-valued Moisil

algebra is the algebraic counterpart of the Lukasiewicz three-valued logic. 1.6. Lemma.

The following properties hold in W : (1.35) (1.36)

(1.37) (1.38) (1.39) (1.40) (1.41) Proof. From Theorem 1.4 and Lemmas 3.3.9 and 3.3.10. 1.7. Lemma (Goldberg, Leblanc and Weaver [1974]).

Let S C E and p , q E E . (a)

If S I- p , then S' I- p for any superset S' of S .

(b) If S I- p, then there is a finite subset (c)

If S I- p and S I- p

+ q,

then S I- Q.

S'

of S such that S' I- p .

466

Lukasiewicz logics

(d) If S U { p } (e)

t- q t h e n S t- p

--t

( p + q).

If S i s syntactically inconsistent, t h e n S !- r foT a n y sentence r in E .

(f) S i s syntactically inconsistent iff S I- f. (g) If S U { p } is syntactically inconsistent, t h e n S t-

fi.

(h) If S U {fi} i s syntactically inconsistent, t h e n S t- p . Proof. (a)-(c):

Immediate from the definitions.

(d) Suppose p l , . . . , p n is a proof of q from S U { p } . We shall show, by induction on i, that S t- p t ( p t p i ) , 1 5 i 5 n. We shall distinguish the following cases:

(i) p; is an axiom or a member of S, hence S t- p i . Using twice (1.1) we get S I- p --t p i , then S I- p --f ( p + p i ) . (ii) p ; is p . Apply (Al).

(iii) pk and pk

+ pi

appear in the list p l , . . . , p i - l . Then

(1.42)

by t h e inductive hypothesis. Now apply twice (c) using (1.43) and

(1.35) for p := p , 4 := pk, (e) If S

t- p

and S I- N p for some p E

by(1.19), we get S

t- r

T

E

:= pi.

then since S

I- N p

+ ( p -+

r)

via (c).

(f) If S is syntactically inconsistent then S I- f by (e). Conversely, if S t- f then S I- N(p0 + PO) while S t- (PO --t P O ) by (1.26). (g) If S u { p } is syntactically inconsistent, then S U { p } I- N p by (e), therefore

S I- p + fi by (d), hence S t- fi via (1.40).

(h) If S U {fi} is syntactically inconsistent, then S Iand (c) yield S t- p .

5 by (g), hence (1.39) 0

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic

467

Now we turn t o semantics. We are going t o use the Lukasiewicz definition of the operations + and

N in L3:

Note that L3 endowed with the above operations is a Wajsberg algebra

1.8. Definition. An interpretation or a trzlth-value assignment of W is an arbitrary function ZJ : V + L3. Any interpretation v has a unique homomorphic extension V : E + L3, i.e. V J V = w and C(p 3 q ) = V ( p ) 3 i?(q), V ( N p ) = NV(p) for any p , q E E . We will also refer t o V as an interpretation and write simply

w instead of E.

A sentence p is valid (kp ) if w(p) = 1for any interpretation w. A set S of sentences is semanticalzy consistent if there is an interpretation 21 such that w(p) = 1for each p E S. If v(p) = 1for any interpretation v such that

v ( S ) = (1) then we shall wirte S 1.9. Proposition (Goldberg,

tactically consistent set S

p.

H. Leblanc and

Weaver [1974]). Every s y n -

c E is semantically consistent.

Proof. Assume S is syntactically consistent. As the set V of propositional variables is countable, one can prove that the whole set E of senten-

E = { p o , p l , ...}. Define by induction a sequence So = S, Sl, ... of syntactically consistent sets of sentences: Si = Si-l~{pi}if ces is also countable, say

Si-lu{pi} is syntactically consistent, otherwise Si = Si-1. Let

s=

00

U

Si.

i=O

S is syntactically consistent: 3 k p and S I- N p for some p would imply that S; I- p and Si t- N p for a sufficiently large i, a contradiction. Now we shall prove that S is maximal consistent. If Using Lemma 1.7 (b), it follows that

468

Lukasiewicz logics

it is not the case that S t- pi then pi 4 S , hence Si-l U { p i } is syntactically inconsistent. By Lemma 1.7 (f), Si-, U { p i } t- f, then S U { p i } I- f by Lemma 1.7 (a), hence S U { p i } is syntactically inconsistent. Let v be the following interpretation:

v(p) =

for any p E

1

I,

ifst-p

0,

i f SI- N p

1

otherwise

,

V . As a matter of fact we prove by induction that for any

sentence p :

(i) S I- p implies v ( p ) = I; (ii) S I- N p implies v(p) = 0; (iii) if neither

S I- p

nor

S I- N p , then v(p) = 3.

If p E V this holds by the previous definition. The inductive step comprises two cases. Case 1. p is N g , where g satisfies the inductive hypothesis.

(i)

If S I- N q then v(g) = 0, hence v ( p ) = N v ( g ) = 1.

(ii) If S I- N N q then S I- g by (1.23), hence v(g) = 1, therefore v(p) = N v ( q ) = 0. (iii) Assume neither S I- N q nor S I- N N g . But S t- q --t N N q by (1.24), hence it is not the case that S I- g. Since neither S t- g nor S I- N g we have v(g) = therefore v ( p ) = N v ( q ) =

i,

i.

Case 2. p is g -+ r , where q and r satisfy the inductive hypothesis.

(i) Suppose S I- g + r . Then the following cases are possible: S I- g 3 S I- r =+ v(r) = 1 + v(q + r ) = v(g) + w(r) = v(g) 1 = 1;

--t

The Wajsberg axiomatization of the 3-valued Lukasiewicz logic

s I- N q * v(q) = 0 * v(q s I- r * v(r) = 1 =+ v(q

469

r ) = 0 -+ v(r) = 1; -+ r ) = 1; S I- N r (using S I- q --t r and (1.25)) S I- N q +-v(q + r ) = 1; none of th e sentences q, N q , r and N r is provable from S + v ( q ) = v(r)= v(q -+ r ) = -12 4 = 1. 4

*

3*

3

(ii) Suppose S I- N ( q -t r ) . Then using (1.36) and (1.37) we obtain S I- q and S I- N r , respectively, therefore v(q -+ r ) = v ( q ) + v(r) = 1+0=0.

(iii) Suppose neither S I- q -+ r nor S t- N ( q --t r ) . Notice that v(q) = 0 would imply S I- N q hence S I- q 4 r by (1.19), while v ( r ) = 1 would imply S I- r hence S I- q + T by ( A l ) . Thus v(q) # 0 and

# 1. S t- q. Then v ( r ) # 0 because v(r) = 0 would I- N r , hence S t- N ( q + r ) by (1.38). Thus w(r) = therefore imply v(q + r ) = 1-+ f = 12 ’ Case 1. v(q) = 1, hence

$, hence it is not the case that S I-

S is maximal consistent it follows that S U { q } is syntactically inconsistent, therefore S I- d by Lemma 1.7 (g). Then v(r) # f because v ( r ) = f would imply S Iby the above argument, hence S I- q -+ r by (1.41). Thus v(r) = 0 therefore w ( q -+ r ) = f + 0 = 1. 2 Case 2. v ( q ) =

It follows from (i) and S

q. Since

S that v ( p ) = 1for every p E S.

0

Proposition 1.9 has two important corollaries.

1.10. Theorem (The Strong Completeness Theorem).

I f S C E andpEE, thenSI-piffSbp. Proof. If S

+ p then SU{$} is semantically inconsistent because v ( S ) = (1)

implies v ( p ) = 1 therefore v(@) = v ( p ) -+ N v ( p ) = 1 + 0 = 0. It follows by Proposition 1.9 that SU {i;} is syntactically inconsistent, therefore S I- p

470

Lukasiewicz logics

by Lemma 1.7 (b). Conversely, if

S t- p

and o is an interpretation such that

v ( S ) = {l},take a proof pl, ...,pn= p of p from S and prove by induction that u(p;) = 1 for all i. 1.11. Corollary (The Wajsberg Completeness Theorem). I f p E E then I- p $7 p . 1.12. Remark. An algebraic proof of the Wajsberg Completeness Theorem can be found in Becchio

[1978d].

The Cignoli axiomatization of the n-valued Lukasiewicz logic

471

$2. The Cignoli axiomatization of the n-valued Lukasiewicz logic The Lukasiewicz n-valued logics ( n 2. 3) were introduced in 1922 (see the historical note of Malinowski in W6jcicki and Malinowski [1977]). Lukasiewicz defined th e following operations on the set L, = ( 0 ' 5 ,

+ y) ;

(2.1)

x + y = min(1,l-

(2.2)

NX = 1- X ;

(2.3)

x V y = m a x ( ~y) , = (X

(2.4)

2

2

..., 5, l}:

4

y)

4

y ;

A y = min(s, y) = N ( N z V Ny)

.

Recall t h a t L, has a canonical structure of n-valued Moisil algebra, namely (L,, A , V , N , O , l,cpl, ...,( 5

V y = max(z,y),

i +j 2

N(&)

~ ~ - (PI, 1 , ...,(Pn-

vi(&)

= -,

I),

where x A y = m i n ( ~ , y ) , is 0 if

i+j <

n and 1 if

n, (Pi = 1 - 'pi, cf. Example 3.1.20, where another notation has

been used.

It is easy t o see that LL = { 0 ,

&,s,

l} is a Moisil subalgebra of L,.

We remark that, for n 2. 5,

This example, given by A. Rose (see the introduction t o Cignoli [1969]) shows that, for n 2 5, the Lukasiewicz implication (2.1) cannot be expressed in an n-valued Moisil algebra in terms of t h e operators: V, A, N , cpl,...,cpn-l.

Consequently, for rz

2 5, the

n-valued Moisil algebras are

not adequate algebraic structures for the Lukasiewicz logics. For n = 3 and n = 4 the Lukasiewicz implication can be expressed in the language of Moisil

algebras (see (2.18) and (2.19)). A natural problem is whether the Lukasiewicz implication can be defined in a particular class of Moisil algebras.

The problem was solved by Cignoli in 1982. He defined the proper n-valued Moisil algebras and gave an axiomatization of the n-valued Lukasiewicz logic. The Cignoli axiomatization uses the characterization of Moisil algebras as Heyting algebras with some unary operators (see L. lturrioz [1977c]).

Lukasiewicz logics

472

The results presented in this section are taken from Cignoli [1980], [1982] and [1984]. 2.1. Notation. In any n-valued Moisil algebra set ~ o = x 0 and 9," = 1 for any x E L and define the unary operators J;, i = 0,1, ...,n - 1: (2.5)

J i ( x ) = cp,-;(x) A N v , - ; - ~ ( x ),

for any

2

EL

.

In particular note that in L,

(2.6)

Ji(-)

j

n-1

= 1 if j

= i and

J i ( 2 )

n-1

= O if j # i.

2.2. Lemma. I n any n-valued M o i s d algebra we have:

Proof. By induction on i. 2.3. Notation. We introduce the following sets:

S n = {(i,j)E N 213 5 i 5 n -2, 15 j 5 n - 4 , j < i} , if n 2 5 and S, = 0, if n < 5 ;

2.4. Definition (Cignoli [1980], [1982]).

A proper n-valued Moisil algebra is a structure (L, {F;j}fi,j)Esn), where L is an n-valued Moisil algebra and Fij,( i , j ) E S, are binary operations on L such that

The Cignoli axiomatization of the n-valued Lukasiewicz logic

(2.8)

Y) =

(PkF,j(%,

1

473

ifksi-j

O' Ji(x) A

foranyx,yEL,(i,j)ES,and

Jj(y)

k=1,

,

if k

>i - j

...,n - 1 .

2.5. Remark. For 2

5 n 5 4,S,

= 0, therefore in this case any n-valued Moisil algebra is

proper. 2.6. Remark. For any n

2 4 we

can extend the definition of Fij for any ( i , j ) E T n :

It is easy t o see that Fij satisfy conditions (2.8) for any ( i ,j ) E Tn. 2.7. Example. If x = 5,y = J n-1 are in L,, then we denote for ( i , j ) E S, (2.11)

F , j ( z , y)

=

n-1-i+j n-1

Fij(x,y) = 0 otherwise

if (r,s)= ( i , j ) and

.

We can prove t h a t F,, introduced by (2.11) verify (2.8). For example, if k > i - j and ( r , s ) = ( i , j ) then cpkF,.,(z,y) = (Pk (n-:1;+3) = 1, since k + ( n - 1 - r s) 2 n and Jr(x) A Jr(y) = 1 . In this way, L, has a

+

canonical structure of proper Moisil algebra. 2.8. Lemma.

Let ( L , {Ej}(i,j)E~n) be a proper n-valued Moisil algebra, x, y E L and a , b E C(L). Then the following properties hold

474

Lukasiewicz logics

Proof. (2.12): W e remark that

k > i - j iff k > ( n - 1 - j ) - (n - 1- i).

For

k > i - j we have:

For k

5 i-j, the previous equality is clear, hence, using the determination

principle, we get (2.12).

(2.13): W e have

and, similarly, J j ( y A b ) = Jj(y) A b. For le

> i - j , we can write

T h e Cignoli axiomatization of the n-valued Lukasiewicz logic

475

By the determination principle, we obtain (2.13). (2.14): Follows from (2.12) and (2.13). (2.15): By (2.13), F , j ( x , O ) = F , j ( x A 1,O A 0) = f i j ( ~ , O ) A 1 A 0 = 0, therefore, by (2.14), Fij(z, b) = F;j(x V 0,O V b) = Fij(z, 0) A N b = 0. 0

2.9. Lemma. Let ( L , { F ; j } ) ,(L’,{F;j}) be two proper n-valued h : L + L‘ is a morphism of Moisil algebras then (2.16)

h(F;j(z,y)) = F i j ( h ( ~ ) , h ( y ) ) for any (z,Y) E L and ( 4 j )E

Proof. If

Moisil algebras. If

T,,.

i - j < k 5 n - 1 then cpkh(F,j(s,y)) = h(cpkFij(s,y)) =

h(J;(z)A Jj(y)) = Ji(h(x)) A Jj (h(y)) = p&(h(x), h(y)). It is obvious t h a t this equality holds for any k = 1, ...,n - 1 and ( i , j ) E T,,hence (2.16) follows by the determination principle.

0

2.10. Remark. The precedent lemma shows that the category o f proper n-valued Moisil algebras is a full subcategory of th e category of n-valued Moisil algebras. 2.10’. Remark.

Let ( L , { F ; j } )be a proper n-valued Moisil algebra, F an n-filter of L and p t he congruence associated with F ,i.e. if (x,x’)E p and (y,y‘) E p then there exist a, b E F n C ( L ) such that x A a = x’A a and y A b = y’ A b. By (2.13) we have

F;j(x,y) A a A b = Fij(~A U , y A b ) = F;j(d A a, y’ A b ) = = F’~(z’,y’) A a A b for any ( i , j ) E

T,,.Since a A b E F

we get ( ~ j j ( z , y ) , ~ , j ( z ’ , y ’ ) )E p. It

follows that L / F = L / p has a canonical structure of proper Moisil algebra.

Lukasiewicz logics

476

2.11. Proposition. Any proper n-valued Moisil algebra is isomorphic to a subdirect product 5 f a family of szlbalgebras of L,. Proof. This is a consequence of Corollary 6.1.9 and Remark 2.10.

0

2.12. Notation. Let ( L ,{ F i j } ) be a proper n-valued Moisil algebra. For any z, y E L we set (2.17)

x

-+

y = (Z

+ y) V NXV

V

F,j(~,y)

( i , j ) E Tn

where

j

denotes the residuation defined and studied in $4.3.

2.13. Remark. It will be shown in Proposition 2.15 below that (2.17) is consistent with the notation (2.1) in L,. For a moment note that if n = 3 then from T3 = 0 we obtain (2.18)

z

-+

while if n = 4,

y = (z

+ y) V NX ,

T4= {(2,1)} implies

These relations show that for n = 3,4, the implication (2.17) can be defined in terms of operations of Moisil algebras. 2.14. Proposition. In every proper n-valued Moisil algebra ( L , {Fij}) the following properties hold:

(i)

'Pl@

(ii)

2

+

Y) = 'P&

+ y = ql(z -+

* Y); y) v y;

(iii) If a E C(L)then z + a = N z V a ;

477

The Cignoli axiomatization of the n-valued Lukasiewicz logic (iv)

If b E C ( L ) then b -+ x

(v)

1

(vi) x

--f

=Nbv x;

x =x;

-,y = 1 iff x 5 y.

Proof. For every ( k , j ) E

(i)

T, we

have 1

5 i -j ,

hence ' p l & , ( x , y ) = 0, by

(2.8). But x A q 1 N x = 0 5 y, therefore we get cpl.Nz 5 y 1 ( x + y) by Proposition 4.3.20(i), (ii). Consequently 'p1(x y) = 'pl(z +Y) v V l N X v v V l F i j ( X , Y> = ' p d " =+ Y>. -+

(&j)ETn

(ii) By (i),(4.3.3)and (4.3.4). (iii) In accordance t o (2.15) and Proposition 4.3.6 (ii) we have z a= u ) V N x = N'p,-lx V a V N x = c p l N x V N x V a = N x V a. (x --f

(iv) is obtained from

(2.15)and Proposition 4.3.6(i),while (v)

is a parti-

cular case of (iv). (vi) By (i) we have the equivalences x

cpl(x

j

y = 1 iff ' p ~ ( z-+ y) = 1 iff x + y = 1 iff x 5 y. -+

2.15. Proposition. In the proper MoisiE algebra L,, the operator

y) = 1 ifF 0

defined by (2.17)coincides with the Lukasiewicz implication introduced b y (2.1).

Proof. We shall prove that z

-+

y = min(1,l

x -+ y = 1 by Proposition 2.14 (vi) and 1 and let z =

-+

-x

5 1 -x

+ y). + y.

If x

5

y, then

Suppose y

<2

5, y = 5.Then F,j(x,y) = 0 if (i,j) # ( p , q ) and + y. We have also x + y = y. We shall distinguish two

Fpq(z,y) = 1-x cases.

E T, then by (2.17)and the previous remarks we obtain x -+ y = y v N x v (1 - x + y ) = max(y,l- x , 1 - x + y ) = 1 - x + y 5 1 .

(a) If ( p , q )

478

Lukasiewicz logics

(b) If ( p , g)

4 T, then z -+

y =y V

Nz = max(y, 1- z).

Assuming q = 0,

we have y = 0, therefore max(y, 1 - z) = 1 - z = 1- z consider t h e case g

# 0.

Since (p,q) @ Tn and g

< p, by the definition

15 g 5 n-3 cannot be simultaneously true. If it is not the case t h a t 2 5 p 5 n - 2 then it follows that p = n - 1; if 1 5 q 5 n - 3 do not hold, then we have q = n - 2, hence p = n - 1. In both cases p = n - 1, therefore 0 z = 1 and we obtain z + y = max(y,O) = y = 1 - z y _< 1.

of

T, it follows

that the inequalities 2

5p 5

+ y 5 1. Let us

n - 2,

+

In accordance with Proposition 2.15 t h e operator

-, defined by (2.17)

will be called the Lulcasiewicz implication. Now we shall give the syntactic construction of an n-valued propositional calculus Luk;.

2.16. Definition. Let V be a countable set of elementsp, q, T , ... called propositional variables. The set E of sentences of th e propositional calculus Luki is obtained as the adjacent set of th e Peano algebra:

( E ,A, V, =+,N , 9 1 , ...,Vn-1,

{Ej}(i,jleT,,)

of type ( 2 , 2 , 2 , 1 , { 1 } i ~ { i , . . . , n -(2}(i,j)E~,) ~~, freely generated by V .

We shall use the following abbreviations: A u B for ( A + B ) A ( B + A) and JiA for cp,-iA A NVn-j-lA, 1 5 i 5 n - 2. 2.17. Definition. The axioms of Luk: are the sentences of t h e following forms:

(Al)

WI

*

* wi)

( ~ 2

9

The Cignoli axiomatization of the n-valued Lukasiewicz logic

479

m

where

A w:

means (...((wi A w;) A w;>

k=1

The concept of proof in

A wk)

,

Luk: is defined in terms of the above axioms

and two rules of inference, modus ponens (m.p.) and (m.p.) W 1 , W =2 wz W

w2

...

;

(Tn)

(T,):

-Y1W

2.18. Definition. Let S E and w E E . By a proof of w from the hypotheses S is meant a finite sequence of sentences wl, wz, ...,wm such that w, = w and for every i E (1, ...,m} one of the following situations hold: (i)

wi is an axiom or w; E S;

Lukasiewicz logics

480 (ii) there exist j , k < i such that (iii) there exists j

Wj

= wk

+ Wi;

< i such that wi = v l w j .

We say that the proof wl,

...,w,

is o f length m and we denote its exis-

(S implies syntactically w). In particular if 0 t- w we write simply I- w and refer t o w as a theorem o f Luk',. tence by S I- w

2.19. Lemma.

Let S (i)

E and w, w', w1, w:, w2, wi E E . If S I- w1 M w{ and S I- w2 H wt then S I- wt. V 202 # wi V w:, S I- w1 A w2 w w{ A w i and S I- (w1 =$ w2) (j (w: + wi).

(ii) If S I- w w w' then S I- N w (iii) If S t w

e w' then S t-

cpiw

* Nw' 'piw' for i = 1, ...,n - 1.

(iv)

If S t 'piw + 9;w' for i = 1, ...,n - 1 then S I- w + w'.

(v)

If S I- N w then S t- w

(vi)

If S I- w (j w' then S I- Jiw

(vii) If S

+ w' for every sentence w' E E .

* Jiw' for i = 1,...,n - 2.

t- wl H w: and S I- w2

-S w;

then S t- 4j(w1,w2)

#

Ftj(wi, wi) for every ( i , j ) E Tn). Proof. We remark that S I- w

w w' iff S I- w

+ w' and S I- w' + w (this

follows i n one direction from (A6) and (A7) and i n the opposite direction

from ( A l ) and (A8)).

The proofs o f (i) and (ii) are straightforward: for example, (ii) follows from (AlO), (A14) and ( v n ) .

(iii) Assuming S I- w

w' we have S I- cpl(w

+ w')

by ( v n ) ,therefore

n- 1

S I-

A

(piw

=$

'p;w') in accordance t o (A12).

i=l

S I- cpiw + cpiw' for every i = 1, ...,n - 1.

It follows that

The Cignoli axiomatization of the n-valued Luhiewicz logic

(iv)

If S I- 'p;w

+ (piw', 1 5 i 5 n - 1 then S I-

n-1

A ('piw

i= 1

481

+ 'piw'),

therefore S I- 'pl(w =+ w') by (A12) and, by applying (A14), we get

s I- w j . w'. If S I- N w then S I- Nw'

(v)

+ Nw

by ( A l ) and map., hence S I-

'pl(Nw' + N w ) by (rn). Using (AlO), S t- 'pl(w 3 w'), then, by (A14), we obtain S I- w + w'.

(vi)

By (ii) and (iii).

(vii) Let us suppose that S t- w1

If ( i , j ) E T, and i - j

* w;, S I- w2 H w; and 1 5 k 5 n - 1.

< k In - 1, then,

in accordance with (A18)

and (vi) we have:

S I- cp&j(wl, w2) e 'pk&j(W;, w:). This property 1 5 k 5 i - j (see (A17) and (v)). By applying (iv)

By m.p. we get

is also true for

it follows that S I- &(w17 w2) -8 F,j(w;, wk).

0

2.20. Definition. Let be the equivalence relation on E defined by w w' iff I- w e w'. Denote t he equivalence class of a sentence w E E by 6.Let 1 be the class of all theorems. The quotient algebra

-

where 0 = N1 and

-

p; = N'pi, (i = 1,...,n - 1) is called the Lindenbaum-

Tarskd algebra of Luk:.

Lukasiewicz 1ogics

482 2.21. Proposition.

The Lindenbaum-Tarski algebra of Luki is a proper n-valued Moisil algebra with residuation +. Proof. The relation

N

is clearly an equivalence; moreover, it is a congruence

by Lemma 2.19, therefore the quotient algebra exists. If w1 and

-

wzare the-

w~ by (Al), while if w1 is a theorem and w1 w2 then since I- w1 + w1 by ( A l ) it follows from (A8) that I- wz A w l , hence 202 is a theorem by (A6). Therefore the class 1consisting of all theorems does exist. The next step is t o check axioms (pl)-(plo) of relatively pseudocomplemented lattices, given in Rasiowa [1974],Ch. 2, 52 and Ch. 4, $1. But orems then w~

(PI),

N

(pz) and (p5)-(p10) are the translations of axioms (Al)-(A8)

into the

til + G2 = tiz +- til = 1 then I- w1 + w2 and k w2 + wl, hence w1 w z or til = GZ,i.e. (p3) holds. Taking a theorem w' we get I- w + w' by (Al), i.e. ti + 1 = 1, which is (p4). Note quotient-algebra language. If

-

further t h a t (2.21)

(w1 =+ w2) I- Nw2

+ Nu71

+ wz} implies syntactically w1 + 202, then.in turn 'pl(wl + WZ), 'pl(Nwz =+ N w l ) and Nwz + Null by ( r n ) , (A10) and (A14), rebecause {wl

spectively. In particular if obtain N1

5 NNG

t- w2

then k N w Z

+Nwl

= G by (A9). Therefore

El

complemented lattice with 0 = N 1 as zero, i.e.

-

El

and for w1 = N w we

-

is a relatively pseudois a Heyting algebra.

To prove

+ N(w1 A wz)and similarly k N W Z3 N(w1 A WZ),hence I- ( N w l V N w z ) + N(w1 A W Z )by (A5). For the converse implication we start from I- Nwl + ( N w l V N w z ) , which is (A3), and obtain I- N ( N w l V N w z ) + w1 by (2.21) and (A9); similarly I- N ( N w 1 V Nw2) + W Z . Applying (A8) we obtain I- N(Nw1 V N w Z )+ w1 A W Z ,hence I- N(wl A w2) + Nwl V N w z by (2.21). we notice that (A6) and (2.21) yield I- Nwl

It follows from (A9), (2.22) and Proposition 1.1.31 that N is an invois also a De Morgan lutive dual endomorphism on E l -, therefore E l N

483

The Cignofi axiomatization of the n-valued Lukasiewicz logic

algebra. Moreover, properties (4.3.7), (4.3.14) and (4.3.16)- (4.3.19) hold by (A12), (A13), ( A l l ) and (A14)-(A16),

respectively. Therefore

El

-

is

an n-valued Moisil algebra in view of the lturrioz theorem quoted in Ch. 4,

$3. The algebra is proper by (A17) and (A18).

0

2.22. Definition. An interpretation of LukR in a proper n-valued Moisil algebra L is an arbitrary mapping v : V + L. For any interpretation v : V ---t L there exists a unique mapping V : E --t L such that Vlv = v and V preserves

N , (pi for i E (1,...,n - 1) and f i j for ( i ,j) E T,. A sentence w is valid in Luk; (+:L&:, v) if V(w) = 1for any interpretation v : V L,. A, V,

=$,

2.23. Proposition (The Completeness Theorem). For every sentence w of Luk; the following assertions are equivalent:

(i)

w is provable in Luk; (I- w);

(ii) Q = 1 in

El

-;

(iii) for every proper n-valued Maid algebra L and every interpretation ) 1; v : V ---t L we have ~ ( w = (iv) w i s valid

in Luk; (bLd:,w).

Proof. The implications (i) =+ (ii) =+ (iii) =+ (iv) are obvious and for (iv)

(i) we notice first t h a t (iv)

+-

+ (iii) by Proposition 2.11,

by taking t h e interpretation v : V -+

El

(iii) =+ (ii) follows -, v ( p ) = fi, while (ii) + (i) is

already proved.

0

2.24. Remark.

Jo, J1,..., JnV1are defined (in L,) usingonly the operations N and +. The unary operators H,, H I , ...,HnVl of L, are introduced by induction: In Rosser and Turquette [1952] th e operators

(2.23)

H o ( x )= N X ;

H k + l ( ~= ) x

+H

~ ( x.)

Lukasiewicz logics

484 Now we shall define Jo,

J,-l

J1,

...,Jn-1 using Ho, HI, ...,H,-1.

We start with

and Jo:

For 1 5

k 5 n -2

(n - 1)Hi(k)

let

i ( k ) be the greatest integer < -and r(k) =

(5) *

The operators Jn-i,i = 2 , ...,n - 1 are defined by recurrence: if n r ( n - i), then (2.25) and if n (2.26)

Jn-i(X)

= Jn-l(Hi(n-i)(z) V 2)

- i < r ( n - i), Jn-i(z)

+

-i

=

(Hi(n-i)(x) A x)

then

= JT(n-i)

(Hi(n-i)(z))

*

One can prove (see Rosser and Turquette [1952], pp.

18-22) that

Jo, J17 ...,Jn-l coincide with the operators introduced by (2.5). By (2.7) it follows that (pl,...,(P,-~ can be defined in L, using only the operations N and +. In L, is also verified the equality:

By Proposition 2.14 (ii) we have in L,:

Consequently, by (2,3), (2,4) and by the previous remarks, the operations V,

A, *,91,...,(p,-l,

4jr( i , j ) E T,,can be expressed

in

L, in terms of N

and +. In Suchori

N and

[1974],th e operators (P~,...,(P,,-~ of L, are given in terms of

-+ with no use of

Jo,J1,...,J,-l.

2.25. Remarks.

V of propositional variables and using only the logical connectives -+ and N we can construct canonically the n- valued Lukasiewicz propositional calculus Luk,. An interpretation of Luk, is an arbitrary mapping v : V + L,. If E’ is the set of all sentences of Luk, Starting with a countable set

The Cignoli axiomatization of the n-valued Lukasiewicz logic

485

Luk, can be extended t o a unique mapping V : E' 4 L, t h a t preserves N and --1. A sentence w is valid in Luk, ( k ~ d w) , if V(w) = 1for every interpretation v of Luk,.

then every interpretation v of

If w is a sentence in Luk, we shall denote by w* the sentence of Luk: obtained by replacing the occurrences of + in w by the expression corresponding t o the relation (2.17). Conversely, with every sentence 20 of Luk: we can associate a sentence w o of Luk, by replacing t h e occurrences of V, A, +, cpi and Fij in w by the expressions corresponding to the relations (2.3), (2.4), (2.28), (2.7) and (2.27). By induction on the length of sentences one can prove for every sentence

w in Luk, that w is valid in Luk, iff W * is valid in Luk: and for every sentence w in Lukz that w is valid in Luk: iff wo is valid ain Luk,. This shows that the axioms (Al)-(A18) of Luk; give indeed an axiomatization o f the Lukasiewicz n-valued logic

Luk,.

2.26. Remark. Another axiomatization of the n-valued Lukasiewicz logic was given by Grigolia [1977] using the MV,-algebras

as algebraic models.

2.27. Remark.

The axioms (Al)-(A16) and the rules of inference m.p. and (r,) define an n-valued propositional calculus having as algebraic models the n-valued Moisil algebras. In accordance with Cignoli [1982], this logical system will

be called the n-valued Moisil propositional calculus. 2.28. Remark. Let us consider an n-valued Post algebra

( L , A , V, N,cpl, ..., ~ n - 1 j 0 , I r c 1 ,...,c,-2); cf. Ch. 4, $1,especially Corollary 4.1.9 and also Ch. 4, $2. Recall that cpicj = 1 if i + j

i

+j

< n. For every (i,j) E S,

(2.29)

Fij(Z, y)

and z , y E

= J ~ ( z A) Jj(y) A

An easy computation shows that

L

Cn-l-i+j

2n

and 9;cj = 0 if

let us define

-

Ej defined by (2.29)

verify (2.8) therefore

Lukasiewicz logics

486

every n-valued Post algebra has a canonical structure of proper Moisil algebra. If we replace th e axioms (A17) and ( A N ) by the axioms corresponding to the definition of the constants cl,

...,c,-~

we obtain an axiomatization of

the n-valued Post-logic (see Rasiowa [1974]). 2.29. Remark. An analysis of the predicate calculus for the n-valued Lukasiewicz logic can be found in Cignoli [1984]. The main tool in the proof of the completeness theorem for this logic is Cignoli’s theorem 4.5.12 which asserts that every nvalued Moisil algebra is completely chrysippian. Consequently the n-valued Moisil algebras provide a common algebraic framework for the treatment of

the n-valued logics of Lukasiewicz and Post. The above remarks 2.25, 2.27 and 2.28 establish the exact relations between the n-valued Moisil logic, the n-valued Lukasiewicz logic and the n-valued Post logic. See also Surma

[19751.

487

The 9-valued propositional calculus

53. The 29-valued propositional calculus In this section we introduce and examine a d-va,Jed propositional calculus with t he property that i t s theorems are the propositions true for all greater than a fixed

k E I. The

iEI

axiomatization of this calculus uses the

system of axioms of 6-valued calculus introduced by Boicescu (1973bl.The results of this section are taken from Filipoiu

3.1. Definition. Let V be an infinite

(19811.

set, whose elements will be called propositional varia-

bles. The proposation algebra of the d-valued propositional calculus on the set V is the free algebra Prop(V) on V in t h e class of all algebras of type = (2727 { l } i E ~ ,{ l } i E I ) .

The type T will be fixed throughout this section. The operations of any Talgebra will be denoted, without danger of confusion by A, V, cp,, @, (i E I).

3.2. Remark.

L Every LMd-algebra may be viewed as a .r-algebra. In particular '

is a

T-algebra.

3.3. Definition. If V is the set of propositional variables and A is a 7-algebra, then every mapping h : V + A will be called an interpretation. Now we introduce the concept of truth in the d-valued propositional calculus.

Let k E

I be fixed

in the sequel.

3.4. Definition. A valuation of Prop(V) is a .r-homomorphism w : Prop(V) -+ Lh'. We say that p E Prop(V) is k-true with respect t o w if w ( p ) ( k ) = 1 and k-false with respect t o v if w ( p ) ( k ) = 0.

Lukasiewicz logics

488 3.5. Remarks.

a) As Prop(V) is t h e free .r-algebra on V, there exists a bijection

Hom(Prop(V),

Lb4)

--t

Hornset(V, LL'), t o the effect that every valua-

tion is uniquely determined by its restriction t o the set of propositional variables and every mapping from V t o

Lh4 can be uniquely extended t o

a valuation.

b) If j E I, k

5 j , and p

j-true with respect t o

E Prop(V) is k-true with respect t o w , then p is

w.

3.6. Definition.

C Prop(V) and q E Prop(V). We say t h a t q is a k-consequence of F (or t h a t F semantically k-implies q ) if v ( q ) ( k )= 1for every valuation w such t h a t v ( p ) ( k ) = 1for all p E F . We shall write this F q.

a) Let F

+ k

b) We say t h a t p E Prop(V) is k-valid (or a k-tautology) and write /= p k

if

0

+ p , that is v(p)(lc)= 1for every valuation v. k

3.7. Remarks. a) For

F C Prop(V)

we denote by C o n k ( F ) = { p E Prop(V) 1 F

+ p}. k

Then Conk is a closure operation on Prop(V) (cf. Definition 1.1.14).

b) If j E

I, k 5 j

p then

and k

p. j

The &valued propositional calculus

489

'pk'pip) A ( P k ' p i p V (Pk'pjQiP). The computations use the fact that

homomorphism and the structure of

j

5k

@ ,

v is a

(cf. Example 3.1.3); thus e.g. if

then

Further we study the concept of proof in k-propositional calculus.

3.9. Definition.

-

For every p , 4 E Prop(V) we introduce the notation

P

q='PkPVvkq

k

(3.0)

P-

p - 4 =

(

k

--

4) A (4

k

k

P)

and call k - a x i o m , the propositions of the following forms:

(3.1)

P

(3.3)

PA4

(3.4)

PA4

k

(4

-

k

P

7

4

,

k

k

P)

7

r-

490

(3.8)

(P

-- -- q)

k

((r

k

k

Q)

k

(Pvr

k

Lukasiewicz logics

4))

,

(3.9) (3.10)

* CpjP A Cpjq ,

Cpj(p V q )

for every j E

I

k

(3.11) (3.12)

'PjP

* VicPjP

7

for every j , i E

I

,

'PjP

* CpiQjP

7

for every j , i E

I

,

Q j P e--t 'PiQjP

7

for every j 7 i E I

k

k

(3.13)

k

(3.14)

CpjP

t--t

k

(3.15)

ViP

+

CPiVjP > 'PjP

7

for every j , i E

I,

for every i , j E

I, i 5j

k

We denote t h e set o f Ic-axioms by

k-Axm.

In t h e next definition we introduce t h e logical system based o n axioms

(3.1)-(3.15)

and modus ponens as rule of inference:

3.10. Definition. Prop(V) and q E Prop(V). W e say t h a t q is a Ic-deduction f r o m F and write F t- q, if there exists a k-proof of q f r o m t h e assumption

a) Let F

k

F , i.e.,

E Prop(V) such t h a t pn = q and for each a E Gn, either p , E k-Axm U F or p , = pb + pa a finite sequence pl,p2, ...7pn of elements p ,

k

for some b, c

< a.

The &valued propositional calculus

491

b) We say that p E Prop(V) is a k-theorem and write I- p , provided k

0 I- p . We also use the

notation

k

for the property "if I-

p l and

... and

k

I- p , then I- q". k

k

3.11. Remark. The k-axioms (3.1)-(3.8) form a set of axioms of positive logic with connectors V , A, -+; it follows that all of the theorems of positive logic are k-theorems; see Rasiowa [1974]. 3.12. Example. We shall write down some &theorems: (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) (3.22)

492 (3.23)

Lukasiewicz logics

[

((p

+p k q ) k

4

pkp),

where q is a k-theorem ,

k

(3.24)

(3.25)

(3.26) (3.27)

k k

( ( p k p + p k s ) +p ) , k k

where s is a k-theorem .

Let us check e.g. (3.19) (3.21) (3.26) and (3.27). But (3.19) means k ( p k p v y k y k p ) A ( p k v k p V V k p ) , which holds because the two factors k

of the conjunction are k-theorems and in positive logic any conjunction of

theorems is a theorem. For (3.21) we use in turn the equivalence theorem for positive logic in Rasiowa [1974], (3.9) (3.13), (3.12), (3.10):

The &valued propositional calculus Formula (3.26) means I- p k p

493

v ( P k ( ( P k ( P k p v (Pkpkq) and

using again the

k

equivalence theorem it reduces t o I-

(Pkpv(Pkpvpkq;t h e latter k-theorem

k

follows by applying modus ponens t o (3.16) and

k p k p v (Pkp -+ p k p v k

k

(pkp V pkq. Finally (3.27) can be written in the following equivalent forms:

the first factor of the conjunction is k-theorem (3.16), while the second is a k-theorem because (Pks. k

3.13. Notation. For every

F E Prop(V)

we set Dedk(F) = { p E Prop(V) I F I-

p}.

k

3.14. Corollary.

Dedk i s a closure operator o n Prop(V), such that for every F E Prop(V) and every p E Dedk(F) there exists F’ F , F’ finite, such that p E Dedk( F’). In other words, Dedk is an algebraic closure operator on Prop(V). The proof is the same as in the bivalent case.

3.15. Proposition. L e t &, Vz be a n y two sets of propositional var&zbbs and f : Prop(&) + Prop(&) a .r-homomorphism. For a n y F C Prop(Vl) and p E Prop(V1) w e have

494

Lukasiewicz logics

Proof. a) Let pl,p2, . . . , p , be a k-proof of p from

F ; if pa E F

-

then f(pa) E f ( F ) .

Since f is a .r-homomorphism, if pa is a k-axiom in Prop(Vl), then f(pa) is a k-axiom in Prop(&); if p , = pb

<

p a , b,c

a, then f ( p , ) =

k

from

f(F).

b) Let v : Prop(V2) --t LLq be a valuation of Prop(V2) such that v ( f ( q ) ) ( k ) = 1 for each q E F . Then t h e composite mapping v

f

: Prop(V1) + Lh'

o

is a valuation of Prop(&), such t h a t ( v o f ) ( q ) ( k )=

1 for each q E F . Since F

p , we have (v o f ) ( p ) ( k ) = 1, i.e. k

Next we deal with consistency and completeness.

3.16. Lemma. Let F Prop(V) and p E Prop(V).

If F t- p then F I= p . k

k

v : Prop(V) -+ L'\ be a valuation such t h a t v ( q ) ( k )= 1 for every q E F. Let further p l , ...,pn = p be a k-proof of p from F . We will prove that v(p,)(k) = 1, a E 1 , . If p , E F U k-Azm then v(p,)(k) = 1, since for every q E k-Azm it is easy t o prove that q. If for b , c < a Proof. Let

+ k

495

The &valued propositional calculus

we have p , = pb

+p a k

and v ( p c ) ( k )= v ( p b ) ( k ) = 1 then it follows from

(3.0) that

3.17. Proposition (The Deduction Theorem).

If F

E Prop(V), p , q E Prop(V) then F I- ( p k

+ q ) if and only

if

k

Proof. By Remark 3.11 and th e well-known fact that t h e Deduction Theorem holds in positive logic; as a matter of fact axioms (3.1), (3.2) and (3.16) suffice (cf. e.g. Barnes and Mack [1975]).

0

3.18. Definition. Let F Prop(V). We say that F is k-consistent if p k p !$ D e d k ( F ) , for every k-theorem p ; otherwise F is said t o be k-inconsistent. F is called a mazimal k-consistent subset if it is k-consistent and maximal for inclusion

c

with this property.

3.19. Proposition. A set F Prop(V) is k-inconsistent if and only if there exists p E PrOp(V) such that p , qkp E D e d k ( F ) . Proof. If p , p k p E D e d k ( F ) then applying twice modus ponens t o (3.26) we obtain P k Q E D e d k ( F ) for every q. Conversely, if (Pkq E D e d k ( F ) for some k-theorem q, note that q E Dedk(0)

c Dedk(F).

0

3.20. Corollary.

The empty set Proof.

8 is k-consistent.

Otherwise p , p k p E Dedk(0) for some p E Prop(V), therefore

Lukasiewicz logics

496

p,Cpkp E Conk(@)by Lemma 3.16. This is a contradiction because v(Cpkp)(k) = Cpkv(p)(k) = v ( p ) ( k ) for every valuation v.

0

3.21. Lemma. T h e subset F C Prop(V) i s maximal k-consistent if and only if

# F , for

(3.28)

pkp

(3.29)

Dedk(F) = F ;

(3.30)

for every p E Prop(V), eather p E F

every k-theorem p ;

OT

pkp E F .

Proof. Let F be maximal k-consistent. For every k-theorem p , p k p # D e d k ( F ) I> F , therefore p k p # F . Since Dedk(Dedk(F)) = D e d k ( F ) , D e d k ( F ) is k- consistent and as F C D e d k ( F ) , we have F = D e d k ( F ) . Finally let p E Prop(V). If p

# F

then F U { p }

exists a k-theorem Q such that F U { p } l-

pkq, or

2

F l-

k

-

F , hence there (p

k

pkq) by

k

Proposition 3.17; using (3.23) we have p k ( p ) E D e d k ( F ) = F . If p E F then p E D e d k ( F ) hence p k p # D e d k ( F ) = F by (3.29) and Proposition 3.19. Now suppose F has properties (3.28)-(3.30). Then for every k-theorem p, pkp

#

F = D e d k ( F ) , i.e. F is k-consistent. If F1

’

#

F there exists

p E Fl such that p 6 F , then p k p E F . Thus p , p k p E I;; so that FI is not k-consistent by Proposition 3.19. 0

3.22. Remark. for some k(3.29) & (3.30) + (3.28), since if we suppose that p k p E theorem p , then p $ F = D e d k ( F ) ; but p E Dedk(0) C D e d k ( F ) .

3.23. Lemma. If Dedk(8’) i s maximal k-consistent t h e n f o r every p, q E Prop(V)

The &valued propositional calculus (3.32)

p A q E Dedk(F)

497

++p E Dedk(F)

and q E Dedk(F) ;

Proof. (3.31) and (3.34) follow from (3.19) and (3.15) respectively. Suppose p , q E Dedk(F). Then q -+

p E Dedk(F) by (3.1), while k

(q

4

q) by (3.16) and

k

I-

k

((q

k

P)

7

((q

k

q)

7

(q

k

(PQ))))

by (3.5) therefore p A g E Dedk(F). The converse implication of (3.32) follows from (3.4) and (3.5).

If p V q E Dedk(F) and p , q # Dedk(F) then @ k P , @ k q E Dedk(F) by Lemma 3.21, therefore pkp A pkq E Dedk(F) by (3.32), i.e. Cpk(p V q ) E Dedk(F) by the k-theorem dual to (3.21); this contradicts Proposition 3.19. The converse implication of (3.33) follows from (3.6) and (3.7). 0 3.24. Corollary. If F is maximal k-consistent then for every p , q E Prop(V): (3.35)

p EF

(3.36)

p A q E F H p E F and q E F ;

(3.37)

~ v ~ E F H o~ r .Eq EFF ;

(3.38)

(pip E

e (pkp E F

F

;

+ (pjp E F

for

Proof. From Lemmas 3.23 and 3.21.

i, j E I, i 5 j .

n

3.25. Proposition.

(i) Every k-consistent set is included in a maximal k-consistent set.

Lukasiewicz logics

498

(ii) FOTevery k-consistent set F there is a valuation w such that w ( p ) ( k )= 1 ~ O each T p E F. (iii) FOT every mazimal k-consistent set F there is a valuation w such that (3.39)

v(P)(~)= 1 u p i p E F

(Vp E Prop(V), Vi E I )

.

Proof.

(i) By a well-known argument using the Zorn lemma. (iii) Define f : V + Li4 by

(3.40)

1

if pix E F

0

if pix

f ( x > ( i )=

# F.

The mapping is well defined because (3.38) in Corollary 3.25 ensures that f ( x ) ( i ) 5 f ( x ) ( j ) for i 5 j . Let w : Prop(V) + Li' be the valuation extending f. We prove (3.39) by induction on the length of p . For p E V , (3.39) reduces to (3.40). The inductive step is based on Lemmas 3.21 and 3.23. For instance if p satisfies (3.39), so does p i p because taking also into account (3.14) and (3.13) we have

(ii) We use (i) and (iii): Let F*1 F be a maximal k-consistent set and w the valuation satisfying (3.39) with respect to F*. If p E F then p E F* = Dedk(F*) hence (pkp E Dedk(F') = F* by (3.31) in Lemma 0 3.23, therefore w ( p ) ( k )= 1.

499

The 29-valued propositional calculus 3.26. Theorem (completeness).

Let F E P r o p ( V ) , p E P r o p ( V ) . Then F

+ p if and only if F k

I-

p.

k

Comment. In other words Conk = Dedk. Proof. If F

t- p

then

i=

F

k

p by Lemma 3.16.

Suppose

F

k

+

p.

k

We prove that F U { P k p } is k-inconsistent: if F U {pkp} is k-consistent, iL such that by Proposition 3.25 there is a valuation v : P r o p ( V ) 4 ' v ( q ) ( k ) = 1 for every q E F U { P k p } , i.e. v ( q ) ( k ) = 1 for every Q E F and 1 = v ( @ k p ) ( k )= ( P k V ( p ) ) ( k ) = v ( p ) ( k ) , which contradicts F

p. k

Therefore F U { p k p } is not k-consistent, thus there exists a k-theorem s such that FLJ { p k p }

(PkS,

k

hence F

t- ( P k p

-+ p k S ) by t h e Deduck

k

tion Theorem 3.17 and using (3.27) we obtain F

t- p .

0

k

3.27. Corollary.

FOTp E P r o p ( V ) we have

+ k

p if and only if I-

p.

k

In the sequel we construct the Lindenbaum-Tarski algebra of our &valued calculus. 3.28. Lemma.

If v

LLq is a valuation then the set Vk = { p E P r o p ( V ) I v ( p ) ( k ) = 1) is maximal k-consistent and every : Prop(V)

4

maximal k-consistent set i s of this form for a unique valuation. Proof. It is easy t o prove t h a t v k satisfies conditions (3.28)-(3.30) in Lemma

3.21 (for (3.29) use Theorem 3.26). Therefore v k is maximal k-consistent. If F P r o p ( V ) is maximal k-consistent then the valuation of Proposition

500

Lukasiewicz logics

Now let

be the set of k-theorems on the set Prop(V) and define the following relation: (3.42)

p

-

q ++ (pip

* cpiq E Tk ,

for every

iEI

k

This relation is an equivalence relation because the syllogism rule is valid in positive logic, hence in our calculus as well. 3.29. Lemma. T h e following assertions are equivalent:

(ii) w(p) = w ( q ) , for every valuation w.

3.30. Theorem. T h e relation is a congruence on Prop(V) and the r-afgebra Prop(V)/ is an LMd-algebra (called the Lindenbaum-Taraki algebra of the considered 19-calculus and denoted by P(V)).

-

-

The 6-valued propositional calculus Proof. The compatibility of prove: for example p

+

N

g

-

"N"

501

with the operations of

Prop(V)

+ w(p) = w(g), Vv valuation + .(pip)

is easy t o

= v(piq),

-

pig. It follows that P ( V ) = Prop(V)/ is a .r-algebra with operations defined by 01[2] = [ o ~ x ] , 0 2 [y] = [X 0 2 y], for

Vv valuation

pip

[XI

each unary operation o1 and binary operation

02.

Since all theorems of posi-

tive logic are k-theorems it follows that ( P ( V ) A, , V) is a distributive lattice (see e.g. Rasiowa [1974]). Moreover,

P ( V ) is bounded with 0 = [pkp] and

1 = [Cpkp],where p E Tk. Finally P(v)satisfies axioms (3.1.2)-(3.1.5) in Definition 3.1.1 of an LMd-algebra; for example

e

v(cp;p) = w(cpiq) Vi E I Vv valuation H

H cp;v(p) = cpiv(q) Vi E I Vv valuation

++

* cp;v(p)(j)= cpiw(q)(j)V i , j E I Vw valuation ++ e v ( p ) ( i )= w( q) ( i ) Vi E I Vw valuation

++

v(p) = v(q) Vv valuation

H

++ [p] = [q] .

0

Let L be an LM6-algebra. We introduce the operations + and k

as follows:

z * y = ( z + y ) A ( y - + ~ ) . k

k

k

3.31. Lemma. In every LMd-algebra L the following relations hold:

t+

k

Lukasiewicz logics

502

(3.46)

2

+(y k

--t

z ) = (x + y ) + ( x + z ) ; k

k

x + (y + Z ) = ( X A y )

(3.48)

k

k

k

-

k

z;

k

Proof. Straightforward .

0

3.32. Definition. Let L be an LMO-algebra. A subset S C L will be called a k-filter of L if s is a proper filter of the lattice L and x E s iff (Pkx E s. 3.33. Lemma. If S C_ L , t h e n the following assertions are equivalent: (i)

S

a k-filter of L ;

23

Proof.

(i)

+ (ii):

If x E S and x

+y E

S then (Pkx E S and (Pkx V (Pky E S

k

hence ( ~ kAx y k y = ' p k x A ( ( P k x V p k y )E S, which implies 'pky E S , therefore y E s. (ii) (i): If x , y E S then y --t (x --t y ) = 1 E S , z --t y E S;

+

since x

k

k

(z A y ) = x

-+

k

k

y it follows that x A y E

k

S. If x

E S , y E L,

503

The d-valued propositional calculus since x

--t

2

V y = 1, we have x V y

E S. Finally:

k

3.34. Remark. The set of k-filters of an LMd-algebra L is in bijective correspondence with the set of filters of the Boolean algebra

C(L).

3.35. Definition.

A proper subset F

#

Prop(V) will be called a k-deductive system of

formulas if

3.36. Proposition. The set of k-deductive systems of Prop(V) i s in bijective correspondence with the s e t of k-filters of the LMd-algebra P(V).

Prop(V)/ N= P ( V ) be the canonical map which is a homomorphism of .r-algebras. Let F C Prop(X) be a k-deductive system; we shall prove that g(F) is a k-filter of P(V). Since F it folh!s that 1 = g((pkp) E g(F), where p E Tk;if pk(Tk) c Tk g ( p ) , g ( p ) + g ( q ) E g ( F ) , then there exists pl,ql E F such that g(p) = Proof.

Let g

k

:

Prop(V)

--t

504

Lukasiewicz logics

p + q E F , hence q E F and finally g ( q ) E g ( F ) . k

q E g-'(g). Finally g(g-'(g)) = g for every k-filter

a of P(V)

since g is

& Prop(V); indeed if p E g-'(g(F)), then g ( p ) E g ( F ) , or g ( p ) = g(p'), p' E F , or p p', p' E F , hence p E F because F is a k-deductive system. surjective, and q-l ( g ( F ) ) = F for every k-deductive system F

N

3.37. Remark. 1) A subset F & Prop(V) is a k-deductive system iff F

# Prop(V)

and

F = Dedk(F). 2) Every k-deductive system is a k-consistent set. 3) A subset F k-consistent .

Prop(V) is a maximal k-deductive system iff F is maximal

Andytic tableaux for the &valued propositional calculus

505

54. Analytic tableaux for the &valued propositional calculus In this section we introduce the method of analytic tableaux for the

6-

valued propositional calculus and obtain a completeness theorem using this method. For the classical calculus see Smullyan [1968]. The results of this section are taken from Filipoiu [1978].

Let v : P r o p ( V ) --t ' \L

be a valuation of P r o p ( V ) and set

then

4.1. Definition.

A family {vkl k E I } , where v k E P r o p ( V ) (Vk E (or truth-set) if conditions (4.1)-(4.5) hold.

I),will be called saturated

4.2. Remark.

If v : P r o p ( V ) -+ LLq is a map and v k ( k E I ) are defined by (4.0) then v is a valuation of P r o p ( V ) iff { v k I k E I } is a saturated family. Conversely, if { v k I k E I } , vk E P r o p ( V ) and we define v : P r o p ( V ) --t Li' by v ( p ) ( k ) =: 1 iff p E vk, then { v k 1 k E I } is a saturated family iff is a valuation of P r o p ( V ) in

L\'.

We consider the language of the 0-valued propositional calculus and for each k E I we introduce the symbols T k and T k .

Lukasiewicz logics

506 4.3. Definition.

An expression of the form T k p (or T k p ) , where p E Prop(V), will be called a signed proposition, or signed formula, of prefiz

T k(ofprefix

Tk).

Now we write down the following Gentzen-style (meta) formulas:

(4.9)

T k qj p

Tkpjp Tjp

Tjp

*

Formulas (4.9)-(4.9) suggest properties like "if p v q then p or q", "if not

( p V q ) then not p and not q" etc. This suggests further a classification of signed formulas into two classes according as they have "and" consequences or "or" consequences. Let us follow this suggestion in a formal definition. 4.4. Notation.

We denote by R any signed formula having one of the following forms: T k ( pv q ) 3 T k ( pA q ) 7 T k v j p7 T k p j p

(A)

and we denote by

T k q j p7 T k q j p 7

any signed formula having one of the following forms:

Thus formulas (4.6)-(4.9)

can be succintly lumped into the following

two: (4.10)

sz

A :-,

B:-

521

r rl I r 2

522

where in the case of signed formulas of type (A) of the form Tktpjp, Tktpjp,

T k + j p , T k p j p we consider Ri = R2 = T j p .

R1 = R2, for

example if

R

= T k p j p we set

Analytic tableaux for the 19-valued propositional calculus

507

4.5. Lemma.

Let S be a set of signed formulas and S; = { p E Prop(V) 1 T k p E S}, Si = { p E Prop(V) I T k p E S } . Suppose for every j , k E I, j 5 k we have Sr3 -C S; and Sj' 2 3;. Then {S; I k E I } is a saturated family i.f the following conditions hold:

(i) for every k E I and every p E Prop(V) ezactly one of T k p , T k p belongs to S; (ii) 52 E S ($ R1 E S and (iii) r E S ($ rl E S

OT

R2 E S; E S.

Proof. If we define v : Prop(V) + L '\

(4.11)

V ( P ) ( k )=

{

by

1,

if T k p E S ,

0,

if T k p E S

,

v is well defined and v is a valuation iff conditions (i)-(iii) hold. Now apply Remark 4.2.

0

4.6. Definition. A set S of signed formulas is said t o be saturated if it satisfies conditions (i)-(iii) of Lemma 4.5.

We now describe the concept of tree. 4.7. Definition. An unordered tree is a triple A = (A,lev,s) where:

(i) A is a nonempty set, whose elements are called points.

(ii) lev : A + IN is a mapping, lev(x) is called the level of the point x E A. (iii) s E A x A such that if ( x l , y ) E s , (z2,y) E s then z1 = 5 2 . If (x,y) E s we say y is a direct successo~of x and x is a direct predecessor of y. More generally, if (z,y)E sn (i.e., there is a sequence

508

Lukasiewicz logics

...,(x,+~,y) E s),

(5,zl),(XI,zz),

while y is a successor of

we say t h a t z is a predecessor of y

x.

(iv) there exists a unique element 0 E A , such t h a t lev(0) = 1 and 0 has no predecessor.

(v)

if ( x , y ) E s then lev(y) = lev(z)

+ 1.

If x E A has no successor we say th a t z is an end point; if 2 E A has only one direct successor we say that x is a simple point, while if it has several direct successors we say th a t x is a branch point. A finite sequence (z1,22, ...,zn)E A” is called a path t o x, in the tree A if x1 = 0 and ( x 1 , ~ 2 ) , ( ~ 2 ,,..., ~ ) (zn-l,zn) E s. B y a mazimal p a t h or a brunch we mean a path

(21,

...,2,)

such th a t zn is an end point.

4.8. Definition. a) A tree

A is said to be finitely generated if each point has a finite number

of direct successors.

b) Let A = ( A ,lev, s) and A’ = (A’,lev’, 3’) be tree; A is called a subtree of

A’ if A G A’, lev

c) A tree is said t o be

= lev’lA, 0 = 0‘ and

S’IA~A

= s.

dyadic if each point has a t most two direct successors.

4.9. Definition. An analytic tableau for a signed formula x is a dyadic tree 7, whose points are signed formulas and there exists a finite sequence trees such t ha t

z,7z,..., 7, of dyadic

is th e tree w i th the single point z, 7, = 7 and for

E 1,n - 1, 7~~~ is a direct eztension of TA,t h a t is 7~is a subtree of 7 ~ + ~ and 7xtl is obtained fro m 7~by the application of one of the following t w o rules: a) choose an end point y of 7~and on th e path to y choose an $2, then adjoin either ill, or $2, as th e sole successor of y;

b) choose an end point y of 5 and on the p a t h to y choose a I?, then adjoin

rl and rzas the direct successors of y.

509

Analytic tableaux for the &valued propositional calculus 4.10. Definition.

a) A branch of an analytic tableau is called c h e d if it contains two points

of the form Tkp and T i p , for some k , j E

I , k 5 j and p E Prop(V). A

tableau is closed if all of its branches are closed.

b) A branch R of an analytic tableau is called complete, if for every point

R

which occurs in

which occurs in

R

R,both Q1 and n2occur in R a t least one of

called completed if every branch of

rl, I'2

and for every point

occurs in

R. A

tableau

I'

7 is

7 is either closed or complete.

For example an analaytic tableau for the signed formula X = rfi'[(ql(PkXV y3j(pkx)

A

('pl'pjrpkx V ( P k X ) ]

is given below:

This tableau i s closed. 4.11. Definition. a) Let

'u

: Prop(V) -+ Li4 be a valuation. A signed formula Tkp(or T k p )

is said t o be t r u e under

v if v(p)(lc)= 1(if v(p)(lc)= 0).

Lukasiewicz logics

510

b) A branch of an analytic tableau is said to be true under v , if all o f its points are true under v ; an analytic tableau is called true u n d e r v if it has at least a branch which is true under v. c) A set of signed formulas (a branch) is called satisfiable if there exists a valuation under which all the elements of the set (all t h e points of the branch) are true.

4.12. Definition. By a k-analytic proof of a formula p E Prop(V) is meant a closed tableau for T k p . 4.13. Proposition.

(i) If T k p has a closed analytic tableau, then p is a k-tautology. (ii) If Tkp has a closed analytic tableau, t h e n p is n o t k-satisfiable. Proof.

(i) If 7'is a direct extension of the analytic tableau 7, then 7' is true under every valuation v w i th the property t h a t 7 is true under v. Hence using Definition 4.1, it follows by induction t h a t every analytic tableau is true under a valuation v if i t s origin is true under this valuation. If 7

7 cannot be true under any valuation, because branch of 7 contains some Tkq,T j q , k , j E I , k 5 j and if such

is a closed tableau, then every

a branch is true under a valuation v , then v ( q ) ( k ) = 1, v ( q ) ( j )= 0, a contradiction. This implies th a t if T k p has a closed analytic tableau (i.e. p E Prop(V) has a k-analytic proof), then for every valuation v we have v ( p ) ( k ) = 1.

(ii) If T k p has a closed analytic tableau then v ( p ) ( k ) = 0 for every valuation 0 and so p is not k-satisfiable. 4.14. Remark. T he method of analytic tableaux is consistent, i.e. it is n o t possible t h a t

Analytic tableaux for the &valued propositional calculus

511

both T k pand T k p have closed analytic tableaux, where p E Prop(V). 4.15. Corollarv.

If a signed f o r m u l a y h a s a closed analytic tableau t h e n y i s n o t satisfiable. 4.16. Definition. By a Hintikka s e t we mean a set S of signed formulas which satisfies the following conditions:

(i) for every k , j E I , k 5 j, x E V it is not the case that Tka:E S and Tjx E S, where V is the set of propositional variables;

(ii) if R E S then

R1 E S and Q2

(iii) if r E S then

rl E S or rZ E S.

E S;

4.17. Lemma.

E v e r y Hintikka s e t i s satisfiable. Proof. Let vo : V -+

Lbq be the

interpretation of propositional variables

given by

{

~ o ( ~ ) (=i )

(4.12)

1,

0

,

if T k x E S for some k 5 i otherwise

,

.

It follows from Definition 4.16 (i) that vo(z) E ' \L

and v o ( z ) ( i ) = 0 whe-

PI E S. Furthermore there exists a valuation v : Prop(V) + L2 such that vlV = vo. It follows from (4.12) that every signed variable of S is true under v. As a matter of fact every element of S is true under v ; this is never

px

established easily by induction on the length of the formula, using conditions

(ii) and (iii) in Definition 4.16.

13

4.18. Corollary.

E v e r y complete n o t closed branch of a n y analytic tableau is satisfiable. Proof. If R is a complete not closed branch of the analytic tableau

7, then

the set of its points is a Hintikka set, hence it is satisfiable by Lemma 4.17. 0

Lukasiewicz logics

512

4.19. Theorern (corn pleteness). If p E Prop(V) is a k-tautology then every completed analytic tableau starting with T k p is closed. Proof. Let

7 be

a completed analytic tableau starting with T'p.

not closed then there is a complete branch Corollary 4.18 there is a valuation v : point of

R. It follows that v

be a k-tautology.

R

of

Prop(V)

If 7 is

7 which is not closed. By

--f

which satisfies every

satisfies T k p , i.e. v ( p ) ( k ) = 0, hence p cannot

The d-valued predicate calculus

513

$5. The d-valued predicate calculus In this section we introduce a &valued predicate calculus, a system of axioms for this calculus and we prove a completeness theorem. The construction is due t o Filipoiu [1981] and follows closely the model of Barnes and Mack [1975] for the classical calculus. 5.1. Definition. Let X be an infinite set whose elements are called individual variables, R a set whose elements are called predicate symbols and a map ar :

R

-+

N

called arity function. We will work with algebras of type T

= (2,2, { l } , ~ r ,{ l } i E ~ { , l } z E{ ~l, } z E their~ operations ); will be denoted

by A, V, {Cpi}iEz, {&};El, { V Z } ~ € Xand { ~ X } ~ respectively. ~ X , By the f u l l first order algebra on (X,R) we mean the Peano algebra Prede(X,R) of type T freely generated by the set {r(zl,..., z,)Iz1, ...,z, E X , r E R,, n E N } , of atomic formulas, where r ( q ,...,z,) stands for (r,zl, ...,zn) and R, = { r E R I a r ( r ) = n } for n E N . The elements of Preds(X,R) are referred to as well-formed expressions or formulas.

5.2. Definition. The set of variables involved in w E Pred$(X, R ) is

(5.1)

X ( w ) = n { Y I Y C X , w E Preds(Y,R)}

5.3. Lemma.

FOTevery zl, ...,z,

z E X, r E

&, n E N ,

i E I

(5.5)

X(Vzw) = X ( 3 X W ) = X(w)

u {z} .

WI,2 0 2 , w

E Preds(X, R),

Lukasiewicz logics

514 Proof. Left t o the reader.

5.4. Corollary.

X ( w ) is finite for every w E Preds(X, R). Proof. Apply Lemma 5.3 t o a formative construction of w.

5.5. Definition. The depth of quantification of w E Prede(X, R ) is the number d(w) E

JV

defined recursively as follows:

(i)

d(r(x1, ...,x.,)

= 0,

(ii) d(wl A w2) = d(wl v w2>= max(d(wl),d(w2)), (iii) d(cpiw) = d(cpiw) = d(w), (iv) d(Vxw) = d(32w) = d(w)

+ 1.

5.6. Notation.

If w E Preds(X, R ) and x,y E X, we denote by w(x/y) the result of substituting y for x a t every occurrence of x in w, if any. In other words, w(z/y) is the image of w by the unique endomorphism h of Preds(X, R ) such that h ( x ) = y and h ( z ) = z for z E X - {x}. 5.7. Remarks. (i)

w(x/y) E Preds(X, R).

(ii) If 2 (iii) x

# X(w) then w(x/g)

= w.

4 x(w(x/y)).

The idea behind th e next definition is t h a t formulas Vxw and Vyw(x/y) express “the same thing” provided y g! X ( w ) - {x} (cf. Lemma 5.13). 5.8. Definition. w2 E Predfl(X,R), w1 M w2 i f For every wl,

515

The d-valued predicate calculus

(A) d(w1) = d(w2) = 0 and w1 = w2; or ( 6 ) d(w1) = d(w2)

> 0 and one of the following situations

hold:

(i)

Wh

= w i A w! ( h = 1,2) and w: x w; and wy x ws, or

(ii)

Wh

= wk V w; ( h = 1,2) and w i x w; and wy x w;, or

(iii) W h = cp;(wi) (i E I , h = 1,2) and w: (iv)

Wh

= ~p;(wL)( i E I , h = 1,2) and

M

w;, or

W: M

w:; or

(C) w1 = Vxw: and w2 = Vyw: and one of the following situations holds: (i) x = y and w: x w;, or

(ii) w{ x w and wk x w(z/y) where y

# X ( w ) ; or

(D) similar t o (C) but with 3 instead of V. 5.9. Lemma. (a) Suppose wl, w2 and w satasfy condition (C.ii) in Definition 5.8 and

take z E X , z # X(w1) U X(w2). T h e n there is w' E Preds(X,R) such that z # X(W') and ( C i ) hold3 f o r wl, w2 and w'. (b)

If w1 x w2 t h e n d(wl)

(c)

~f wl x w2 and y # X(w1) u X(w2) t h e n wl(x/y)

(d) T h e relation

XI

= d(w2) M

w2(x/y).

is transitive.

Proof. Check simultaneously (a)-(d)

by induction on the depth of quantifi-

cation.

5.10. Proposition.

T h e relation x is a congruence of the algebra Preds(X, R). Proof. Reflexivity and consistency with the operations of Preds(X,R ) are immediate, while transitivity is Lemma 5.9 (d). The only delicate point in establishing symmetry is case (C.ii) of Definition 5.8. But w = w(z/y)(y/x) z

# X(w(z/y))

by Remark 5.7 (iii).

0

Lukasiewicz logics

516

5.11. Definition. The set F X ( w ) of free variab2es of a formula w E Prede(X,R) is defined recursively as follows:

(i)

~ ~ ( r ( z..., 1 2. ,))

= (51,...,G,},

(ii) FX(w1 A wz) = FX(w1V

WZ)

= FX(w1)U FX(w2),

(iii) FX(cpiw)= FX(cpiw) = FX(w), (iv)

FX(Vxw) = FX(3xw)= FX(w)- {x}.

The elements of the set

(5.6)

BX(w) = X(W)- F X ( w )

are referred t o as the bound varaables of w .

5.12. Lemma. If w1 M w 2 t h e n FX(wl)= FX(w2). Proof. By induction; left t o th e reader. 5.13. Lemma.

Let w E Preds(X, R) and x, y E X.

(i) If y

# X ( w ) - {x} t h e n Vxw

(ii) Ifx,y

# X(w)

M

Vyw(x/y).

t h e n Vxw M Vyw.

Proof.

(i) If y = x then Vyw(x/y)= Vxw. If y # X ( w ) the desired conclusion follows from case ( C i ) in Definition 5.8 with w{ = w and wi = w(x/y). (ii) From (i) and Remark 5.7 (ii).

0

5.14. Notation. Let Pd(X,R) or simply

Pd

stand for th e quotient algebra Preds(X,R)/

M.

The equivalence class containing w E Preds(X,R) will be denoted by

The 8-valued predicate calculus

517

[w]= E P $ ( X , R )or simply by [w] whenever there is no danger of confusion.

5.15. Remark. Using the simplified notation 5.14, condition (C) in Definition 5.8 reads: VXWl

=Vyw2

*

(i) x = y and w1 = w 2 ,or (ii) w2 = wl(s/y) where y $?! X(wl). 5.16. Lemma. Every element of Ps(X, R ) can be represented in t h e f o r m [w] where w E Preds(X, R) satisfies

(i) n o variable x E X appears in w more t h a n once in a quantifier 'dx OT 3 x and (ii) F X ( w ) n B X ( w ) = 0. Proof in two steps.

I) Let

be a formative construction (cf. Definition 1.5.b) of w E Preds(X, R).

6

Take a variable y X ( w ) and set wI, = Vyw*(x/y); then w i M Wh by Lemma 5.13 (i). Let further wy, ,..,w:-~, wI, be a formative construction o f w;L and let w;+~,...,w; be obtained from wl, ...,Wh-l, w i in the same way as W h + l , ..., wn were obtained from wl, ...,wh-1, Wh in (5.7). Then

is a formative construction of wk and since follows that w:M wt

(t = h+l,

M

is a congruence it also

...,n). Thus w; M w and the occurrences

of s in Vxw* have been removed from w;.

Lukasiewicz logics

518

II) Since X is infinite the above construction can be applied as many times as necessary, i.e. until we obtain a formula w' M w fulfilling properties (i) and (ii). 0 The next definition introduces an appropriate concept of interpretation. 5.17. Definition.

U be a non-empty set. By a 8-valued interpretation (or simply an interpretation) in U we mean a couple (f,G), where f : X --f U and Let

(5.9)

G : R

is such that

G(r)

-+

:

{gig : U"

U"'(')

--f

nEN}

--f

' i L

for every r E

R.

We will think of z E X as a name for the object f(z)E as a name for the &relation G ( r ) .

U

and of r

ER

5.18. Lemma. For every 8-interpretation (f,G)in U there is a unique f u n c t i o n v : Po 4 Lh' such that f o r every r(z1,...,I,) E Ps (r E R, 21,...,z, E X , n =

ar(r)), w1,w2,w,wo E

P79,; , j E 1:

(a) v(r(z~,*-*,zn))(j) = 1 @

G(r)(f(z~), * - . i f ( z n ) ) ( j )= 1,

(b) v(w1 A w ~ ) ( j )= 1 H w(wl)(j) = 1 and v(wz)(j) = 1, (c) v(w1

v w2)(j) = 1 @ v(w1)(j)

= 1 or v(wz)(j) = 1,

(f) condition (fn) holds f o r every n E N , (g) condition (g,) holds f o r every n E N ,

where conditions (fo), (go) are vacuously fulfilled, while f o r n > 0, (fn) if 20 = Vzwo and d(w) = n t h e n [v(w)(j) = 1 f o r every X' = X U { t } with t # X , every extension f' : X' + U o f f and every

The &valued predicate calculus (5.10)

V‘

I

519

: {w‘ E Pa(X‘,R) d(w’)<

that fu&l (a)-(e)

and (fk) f o r all k

TI} +

L2[Jl

< n, it follows that v’(w0(z/t))(j)=

117

(gn) similar to (fn) but with 3x instead of Vx and “there as a n extension”.

f‘ quantified by

Proof. Every element of the Peano algebra Preda(X,R) is uniquely represented in one and only one of the forms r(xl, ...,xn), w 1 A

cpiw,

Cpiw,

Vxw or 3x20; therefore conditions (a)-(g)

202,

w1 V w2,

determine uniquely

o(w) for each w E Preda(X,R). It remains t o show that if w1 M w2 then v ( w l ) = v ( w 2 ) E Liq. Suppose first d(wl)= 0. Then w1 = w 2 by Definition 5.8 (A). Clearly (a) implies that w(r(xl, ...,xn)) E LLq and it follows by induction via (a)-(e) t h a t v(wl) E LLq whenever d(wl) = 0. Next proceed by induction for d(wl) > 0. If w1 and w 2 satisfy Definition 5.8 (B) t he desired conclusion follows immediately from (b)-(e). Now suppose w1 = Vxw{ and

w2

= Vyw: satisfy Definition 5.8 (C). Then Lemma

5.9 (c) implies that in case (i) w{(x/t)M w i ( x / t ) = w t ( y / t ) , while in case (ii) w i ( x / t ) M w ( x / t ) and wk(y/t)M w ( s / y ) ( y / t ) = w ( x / t ) , therefore in both cases w : ( x / t )

w;(y/t),hence in condition (fn) we do have d ( w { ( x / t ) ) = d(w;(y/t)), therefore v(w1) = ~ ( 2 ~ 2 ) Condition . (fn) implies also

V(wl)

E

M

LL~.

0

5.19. Defi nitio n.

A quadruple (U, f,G , V ) satisfying the conditions of Lemma 5.18 will be called a d - d u e d valuation of Pa(X,R ) in the domain U .We also say simply that w is a valuation of Pa. As was done in 53, in the sequel we consider a fixed element k E

I.

5.20. Definition. Let H Pa and w E Pa. We say that H semantically k-implies w ,and we write

H

+ k

w ,if for every valuation v of Pa such that w(w’)(k) = 1for

520

Lukasiewicz logics

all w‘ E H it follows t h a t v ( w ) ( k ) = 1. We also use the notation (5.11)

Dedk(H) =

{w E J‘s I H

k

w1

k

In particular if

0

20

we say that

w is a k-tautology and write simply

k

A “good” construction of a predicate calculus should “include” the corresponding propositional calculus. This is actually the case of the $-valued theory, as shown by Proposition 5.21 below, which is stated in semantical terms but is also valid for the corresponding syntactical concepts due t o the Completeness Theorems 3.26 and 5.36.

Let us extend th e notation Prop(V) introduced in Definition 3.1 t o the case of finite sets V. Preds(X,

If w’ E Prop({zl

R), we write simply w’(w1,

,...,5,))

...,w, E ...,z,/wn).

and w1,

...,w,)instead of w’(z1/w1,

5.21. Proposition.

Let n E and

IN - {0}, H U {w}

Prop({zl,

...,z,}),

w1,...,w, E Prede(X, R )

(i) If H semantically k-implies w i n the d-valued propositional calculus then H(w1, ...,w,) w(w1, ...,w,).

+ k

(ii) If w is a k-tautology in the d-valued propositional calculus then

I= k

w(w1,

...,%).

521

T h e &valued predicate calculus Proof.

(i) Let v be a valuation of Pd such that v(w")(k) = 1 for every w" E H(wl, ...,w,). Let h : Prop({z1,...,zn}) + Li' be the valuation of Prop((z1, ...,2,)) such t h a t h(xj) = v(wj) ( j = 1, ...,n). Then for every w' E Prop( {q, ...,z,}) it follows that

=

= w'(h(z,),...,h(z,)) =

...,z,))

h(w') = h(w'(z1,

...)v(wn))

W+(Wl),

In particular if 20' E H then w'(w1,

= v(w'(wl, ...,w,)) .

...,w,,)E H(w1, ...,w,) and

h ( w ' ) ( k )= v(w'(wl,..., wn))(k) = 1. This implies v(w(wl,

...,w,))(k) = h ( w ) ( k )= 1.

(ii) Immediate from (i).

0

The above elements of semantics will be related t o syntax, i.e. to the concept of proof which we introduce now. 5.22. Definition. The set k-Axm of axioms of the &valued predicate calculus is the set of those elements of P s ( X ,R)that have representatives of the following forms:

(5.13)

w'(w1,

...,wn), where ~ ' ( 2 1 ..., , z,)

is a k-axiom

of the &valued propositional calculus and

w1, ...,w, E Preds(X, R) (5.14')

Vx(w1

(5.15')

Vxw

(5.15")

w(x/y)

-+ w2) + k k

(wl

, + Vxw2) k

,

+ w(x/y) k

,

y

# BX(Vzw) ,

3zw

,

y

# BX(3xw) ,

--t

k

x $? F X ( w , )

,

Lukasiewicz logics

522

~~(VZW * ) V X C ~ ~, W i E

(5.16“)

I,

k

(5.17’)

Pi(3sw)

H

Vx~piw,

iEI,

k

where + and k

*

are defined by (3.0).

k

We define by induction the concept of k-proof. 5.23. Definition. Let H C PS and w E Pa. A k-proof of length n of w from the hypotheses H is a sequence w1,w2, ...,w, of elements of Pd such that w, = w and wl, ...,w,-~ is a k-proof of length n - 1of w,-~ and (a) w,

E k - A x m U H , or

(b) wt = w,

--f

w, for some t,s

< n, or

k

t l , ...,t, E (1, ...,n } such that wtl, ...,wtm subset Ho H such that

(c) w, = Vxw’ and there exist is a k-proof of wt from a

x $! FX(H0) $ ! lJ{FX(w’)/w‘ E Ho} We denote by H I- w th e existence of a k-proof of w from H and we k

also say that

F syntactically k-implies w. In particular if 0 Ik

t h a t w is a k-theorem and write simply Ik

w.

w we say

The &valued predicate calculus

523

Note that by removing formulas involving cp; or

pi from axioms in Defi-

nition 5.22 we get the axioms of the bivalent propositional calculus. Also, the inference rules are the same as in the bivalent case: modus ponens and

generalization: W -

vxw

(6. Definition 5.23 (b) and (c), respectively). 5.24. Definition.

Let Pi = Po(Xl, R') and Pj = Pg(X2,R 2 ) . By a semi-morphism from to

Pj we

mean a couple (f,g) where

f

:

Pj

+

P$', g : XI

+

Pj

X2 and

the following conditions are fulfilled: (a)

g(X1) is infinite;

(b) f is a morphism with respect t o A, V, pi and pi (i f I);

Proof.

(i) As for the bivalent case; cf. Lemma 4.2 in Barnes and Mack [1975]. (ii) By induction. The only non-trivial case is w = Vzwo (or similarly w = 3zwo), where wg satisfies (ii). Using t h e infiniteness of g(X') and Lemma 5.13 (i) we may suppose t h a t g(z) # g(x), g(y) and z $!

X'(wo).

Then

Lukasiewicz logics

524

5.26. Theorem (The Substitution Theorem). L e t ( f , g ) be a semi-morphism from

Pi

Pj

to

and H G

Pi, w E Pi. If

w1,...,w, is a k-proof of w f r o m H , then f(wl), ,..,f(w,) is a k-proof of f(w)f r o m

f(W

Proof. Induction over n. For n = 1 note that f(k-Azm' U H ) = k-Asm2U

f ( H ) because one sees readily that the axioms of the forms (5.13), (5.16). (5.17) in Definition 5.22 are transformed into the corresponding axioms for

Pi,while for

the axioms of the forms (5.14) and (5.15) the same result is

obtained via Lemma 5.25; for (5.15) we have t o choose a representative of

f(w)such that g(y) # BX2(Vg(z)f(w))and similarly for 3. The inductive step amounts t o proving that f preserves modus ponens and generalization. The former is clear. To prove the latter suppose t h a t wn = Vxw and there is a k-proof w t l..., , wt, of w = w t mfrom a subset

HO

H such that x

# FX'(Ho).Then

by the inductive hypothesis

f(wt,), ...,f(w t m )is a k-proof o f f(w) from f ( H o ) . For each w' E Ho, from x # FXl(w') we infer g(x) @ FX2(f(w'))by Lemma 5.25 (i), therefore

is a k-proof from f ( H o ) .

0

5.27. Lemma.

For every H

5 PS and w

E Ps,af

H I- w

then H

k

Proof. Let w l ,...,w, = w be a k-proof o f such that v(w')(lc) = 1 for all w' E

w. k

w from H and v

H. We will

a d-valuation

prove by induction over n

that v(w)(lc) = 1. For n = 1 we take w E k-Azm, say of the form (5.14') in Definition 5.22. Note first that

525

The &valued predicate calculus

w w' + w")(k)= 1

(5.18)

(

k

or w(w")(k)= 1 , therefore

2)(vx(w1

- w2)

k

(wl + Vxw2))(k)= 1 H

k

k

w(wl)(k) = 0 or w(Vxwz)(k)= 1 -S

f' : X U { t } + U of f such that w'(w1(z/t))(k) = 1and w'(w2(x/t))(k)= 0) or w(wl)(k) = 0 or (for every extension f' : X U

H (there is an extension

{t}

-+

u, w'(wz(./t))(k)

= 1)

and the latter statement is clearly true. Similar proofs hold for the other k-axioms. For the inductive step suppose first that

w, = w t

-

wn, where

k

w(wd)(k) = w(wt)(k)= 1 by the inductive hypothesis; then (5.18) implies

w(w,)(k) = 1.

w = Vxwo and there exist t l , ...,t , E (1, ...,n} such that wtl, ...,wt, = w o is a k-proof of w ofrom HO C H , where x # F X ( H o ) . Take X' = X U { t } and w' as in condition (f,) from Lemma 5.18; the task Now suppose that

is t o prove w'(wo(x/t))(k) = 1. Reasoning as in Lemma 5.18, w' can be

extended t o a 6-valuation w" of Pa(X', R). On the other hand we claim that the following pair

(f,g)

is a semi-morphism:

f

: Pd(X, R) -+

Ps(X', R)

X' defined by g(x) = t and g(y) = y for y # z. Conditions (a)-(c) in Definition 5.24 are easily checked, while (d) becomes r(y/z)(z/t) = r(x/t)(g(y)/g(z)) and is also easily verified by considering the cases y # z # z, y # x = z and y = x # z . According t o Theorem 5.26, w t , ( x / t ) ,..., wt,(x/t) is a k-proof of wo(x/t) defined by f ( w ) = w ( x / t ) and g

:

X

--t

Lukasiewicz logics

526 from f ( H 0 ) . But f(H0) = HObecause z hypothesis implies

Ho

4 F X ( H o ) ,therefore the inductive

wo(z/t), hence t h e extension d' of v' satisfies k

v " ( w O ( z / t ) ) ( k= ) 1, which implies d(wo(x/t))(k)= 1.

0

5.28. Corollary.

~f I- w t h e n - ( Ik

(PkW).

k

Proof. As for Corollary 3.20.

D

5.29. Theorem (The Deduction Theorem).

If H

C_

P,q

and

w,w' E P,q then H

I-

(w

H U { w } I-

-

if and only i j

w')

k

k

w'.

k

Proof. Similar to that of Proposition 3.17, except that in the inductive step of the "if" part we must add the case when w is obtained by generalization: w' = Vzwo and

Ho I- wo, where Ho C H

U {w} and

k

z $! F X ( H 0 ) . If

w

4 HO then HO E H

followed by the k-axiom w' + (w

HO- {w} I- w k

z

# FX(w).

-

w') yields a b-proof of w

k

k

from H . If w E Ho, from

--f

and a k-proof of w' from k

k

wo,therefore Ho - {w} I- Vz(w

Applying (5.14') we get HO -

k

k

--t

-

wo), where

k

{w} Ik

k

+ w'

Ho I- w o and the inductive hypothesis we obtain

k

H I- (w 4 w')

Ho

w

Vxwo, hence

k

0

527

The $-valued predicate c d c u l u s

5.30. Definition. we say that H Pd is k-consistent if (PkW @ Dedk(H) for every k-theorem w ;otherwise H is said to be k-inconsistent. H is called a maximal kconsistent subset if it is k-consistent and maximal for inclusion with this property. 5.31. Remark. Proposition 3.19, Lemmas 3.21 and 3.23, Corollaries 3.20 and 3.24 and Remark 3.22 are extended to the &valued predicate calculus with the same proofs.

5.32. Lemma. Let H C Pd be k-consistent. If 3xw E H and t { w ( s / t ) } is k-consistent.

#

F X ( H ) then H U

Proof. Suppose there is a k-theorem w' such that H U { w ( z / t ) } I-

(PAW'.

k

Then H I- w ( z / t ) + (pkw' by Theorem 5.29, therefore k

k

H I- (pk(w(./t)) by (3.23), hence H k

we get

v@k(W(Z/t))

and using (5.17')

k

H I-

(pk3tW(T/t),

which yields H I-

k

(pk3SW

by Lemma 5.13 (i).

k

On the other hand from 3 s w E H and (3.19) we deduce H

I- cpk3sw, k

therefore H is k-inconsistent by Remark 5.31. 5.33. Lemma. F o r every k-consistent set H

P a ( X , R ) there ezist X' 2 X and H' c

PG(X*,R) such that (5.19)

H

(5.20)

H' is masimal Ic-consistent in Pd(X',R) ;

H' ;

528

Lukasiewicz logics

if 3 x w E H* t h e n w ( s / t ) E H' f o r s o m e t E X *

(5.21)

.

Proof. We are going to define four increasing families of sets X i , Hi, Hi, Pi such that Pj = P8(Xi,R ) and each Hi is maximal k- consistent in Pi (i E N ) . First we put X o = X and Ho = H ,then

where tL are new variables. We prove that if Hi is k-consistent, so is Hi+l. If not, there is a k-theorem wo such that @kw0 has a k-proof from H,!+l;let

A = {zf?;(x/t$,), ...,w~(s/t$h)} be the set of all elements of - Hi occurring in that k-proof. Then H j U A I- PkWO, which contradicts Lemma 5.32. Thus Hi+l is k-consistent k

and we take a maximal k-consistent set Hi+l 2 Hit1, which exists by the Zorn lemma. Now we take X * = U Xi and H* = U Hi. Then (5.19) holds and we prove (5.20) using Lemma 3.21 via Remark 5.31. First if (PkW E H* for some k-theorem w then (PkW E Hi for some i and this contradicts Lemma 3.21 for Hi; hence (Pkw $! H* for every k-theorem w. Further if w E Dedk(H*) take a k-proof w1, ...,wn-lr w, = w of w from H* in Pfl(X*,R);then this is also a k-proof of w from Hi in Pfl(Xj,R) for a sufficiently large i, hence w E Dedk(Hi) = Hi by Lemma 3.21 for Hi,therefore w E H'. This proves that Dedk(H*) = H*. One proves similarly (3.30) for H* by using (3.30) for

Hi for some i. Finally if w = 3sw' E H* take i such that w E Hi; then w'(z/tL) E

Hitl C H*.

0

5.34. Proposition. For every k-consistent set H c Pg(X, R ) there is v ( w ) ( k )= 1f o r every w E H . Proof. If H*

2H

satisfies conditions (5.19)-(5.21)

0

valuation v such that

and v* is a valuation

The &valued predicate calculus

529

of P8(X*,R)such that v*(w)(k) = 1 for every w E H*, then the restriction v = v*

I Pd(X,R ) is obviously a valuation

of P . ( X , R ) which satisfies

v(w)(k) = 1 for every w E H. Therefore we may suppose without loss of generality that (5.20) and (5.21) hold for

H . The desired valuation will

be a quadruple, as required in Definition 5.19, namely ( X , i d x , G , v ) , where --f {g I g : X" --t J$} is defined by

G : R

and v is obtained from the &interpretation (idx, G) in X by the construction in Lemma 5.18. The first point is the correctness of this construction: to prove that (idx, G) is actually an interpretation in X it remains to check that G(r)(xl,...,xn) E $. But if i , j E I,i I j and cp;r E H , then H I- cpjr k

by (3.15), therefore cpjr E H by Lemma 3.21 (3.29) via Remark 5.31. Now our theorem will be established if we succeed to prove that (5.23)

v ( w ) ( ~=) 1 ($ cpjw E H

(jE

I)

because if w E H then by applying Lemma 3.21 we obtain in turn (pkW $! H , (pkqkw E H and (PkW E H via (3.12), therefore v(w)(k) = 1by (5.23). Thus it remains to prove (5.23) by induction. For w = ~ ( $ 1 ,...,z,,),(5.23) follows from Lemma 5.18 (a) and (5.22). For the inductive step we use Properties (a)-(g) in Lemma 5.18 (6. Definition 5.19) and again Lemma 3.21.

If 201,

satisfy (5.23), so do wl A w2 and w1 V w2; e.g. for w1 A w2 we also use (3.36) and get 202

Now we suppose w satisfies (5.23) and prove that so do cpiw, p;w, Vxw

and 3xw. But

Lukasiewicz logics

530

by (3.11) and (3.29), while (3.30), (3,14), (3,13) and (3.29) imply

For Vxw suppose first yjVxw # H . Then cpjVxw E H as was just seen in (5.24), hence 3xcpjw E H by (5.17"), therefore (5.21) implies (cpjw)(x/t) E H for some t E X, hence (cpjcpjw)(z/t) E H by (3.13); but since cpjw satisfies also (5.23) we get v((cpjw)(x/t))(j) = 1, i.e. v(w(x/t))(j) = 0, therefore v(Vxw)(j) = 0. Conversely, suppose vjVxw E H. Then pjVxw # H by (5.24), hence 3xCpjw # H by (5.17"). Then it follows from (5.15") that for every t # BX(3xcpjw) = B X ( 3 x w ) we have (cpjw)(x/t) # H , hence (cpjw)(x/t) E H by (5.24), therefore v(w(x/t))(j) = 1. As t is arbitrary, the l a t t e r equality is equivalent to v(Vxw)(j) = 1. 0 The similar proof for 3 z w is left to the reader. 5.35. Theorem (The Completeness Theorem). Let H Pd(X,R) and w E Pd(X,R). T h e n H

w if and only if k

H t w. k

Proof. As for Theorem 3.26 except that Lemma 3.16, Proposition 3.25 and Theorem 3.17 are replaced by Lemma 5.27, Proposition 5.34 and Theorem 5.29, respectively. 0 5.36. Corollary. For w E Pd(X, R ) we have

+ k

w if and only if I- w. k

Kripke-style semantics for $-valued predicate logics

531

$6. Kripke-style semantics for 8-valued predicate logics In this section we introduce the semantic 29-models for the 8-valued predicate calculus and establish the relationship with the algebraic models for

the case when 8 is an ordinal number. A similar semantics for the Post logic is defined in Maksimova and Vakarelov [1974].

6.1. Definition. Consider again the &valued predicate calculus Ps = P s ( X ,R ) . An algebraic 8-model is a triple M = ( L , U , S ) , where L is a completely chrysippian 8algebra, U a non-empty set and S : x U x --$ L is a map satisfying the following conditions:

w E Ps,z E X and for any interpretation f E U x exist in L : V{ws(f;) I c E U } , A {ws(f:) [ c E U},

for any there

where ws(f) = S ( w , f ) and

fay)

=

[

f(Y) c ,

7

if Y

#.

ify=x

6.2. Remark. Property (6.2) holds for any formula w .

,

for any y E X ;

532

Lukasiewicz logics

6.3. Definitiom. For every

H E

P8 and

w E

P8 let

H

b

w mean that for any algebraic

k

hf = ( L , U , s ) , if Yk(W$(f)) = 1 for any w' E H and f E U x then V k ( W s ( f ) ) = 1. In particular, if 8 w then we say 20 is algebraically $model

k

k-valid and write simply

w. k

6.4. Proposition.

Po and w E Ps then the following assertions are equivalent:

If H (a)

H t-

w;

k

Proof.

Let w1,w2,...,w , be a k-proof of w from H and M an algebraic &model such t h a t ( P k ( W $ ( f ) ) = 1 for any w' E H . We are .j (b):

(a)

going t o prove by induction on n that Y k ( w s ( f ) ) = 1. If n = 1 then

w E k - h m U H , therefore ( P k ( W S ( f ) ) = 1, because for any k-axiom s we have b s. Thus e.g. for the axioms (5.14') note first t h a t if z $ F X ( w ) k

then w s ( f ) = ws(f,")for every c E

U

by Remark 6.2, therefore

533

Kripke-style semantics for 19-valued predicate logics

The inductive step is performed as follows:

(1) if cpk(ws(f)) = 1 and (P~((w+ ~ ' ) ~ ( f=) 1 ) then k

(2) if wl,...,w, is a k-proof of w from Ho (Pk(wS(f)) (Pk(WS(f))

(b)

+ (c):

= 1 then

EH

# F X ( H o ) and {wS(ff) 1 c E u } =

where z

( P k ( ( v z W ) S ( f ) )= (Pk

A

= 1. Let us take

U"'(')

L = 'iL

and G :

R

+ {g

Ig

:

U" + Lh'}

Li'. We can consider the algebraic &model M = (Li4,U,S), where S : P8 x U x + LL4 is given by S(w,f) = v(f, G), where v(f, G) is the d-valuation associated with the interpretation (f, G). For this algebraic 19-model we have ( ~ k ( w ~ ( f ) = ) 1 iff v(f,g)(w)(k) = 1, such t h a t G ( r ) :

4

therefore we get the desired implication. (c)

+ (a): By the Completeness Theorem 5.35.

0

Lukasiewicz logics

534

Now we shall define the concept of semantic &model using the concepts of 6-structure and &space introduced in Definition

S

6.4.1and 6.4.6.If

( A , { A f } t E ~ , ; Eis ~ ) a &structure then we shall work below with a relation \I- C A x U x x Pd and we shall write x It- w instead of =

f ( X , f , W )E

It-.

6.5. Definition. A semantic 6-model (s.6-m.) is a triple 0 = (S,U,\I-) with S,

U ,IF as

above and satisfying the following conditions:

(6.7)

If x IF w and x 5 y then y [I-

(6.8)

For any atomic formula T and for any interpretations fi, f2 such that fl(ui) = f2(u:),i = 1, ...,rn we have

(6.9)

x It w v w ' e x [I- w or x It- w';

f

f

(6.10)

f

f

x [I- WAW'U x [I- w and x \I- w'; f

f

f

(6.11)

w;

f

x /I- 3zw

e there is c E U

such that

x It- w ; ff

(6.12)

x It- Vzw u for any

CE

U , x It- w

(6.13)

x

It-

;

ff

f

pjw

there is t E

T such that x E At

f and for any y E A:, y

It- w ; f

(6.14)

x It- cpjw u there

is t

E T such that z E A'

f and there is y E A; such that y [If w f

.

535

Kripke-style semantics for 9-valued predicate logics

6.6. Remark. Property (6.8) holds for any formula w. 6.7. Definition. Let us consider a triple

0 = ( Y ,U,I I-),

where

Y

=

(Y, I , {di}iEi)

is a

&space. Such a triple is called a special semantic 9-model (s.s.9-m.) if it verifies axioms (6.7)-(6.12)

x IF Cpjw

(6.16)

in Definition 6.5 and

there is y

2 djx

such that y

f

Iy w . f

6.8. Remark. Using (6.7) we get the equivalence: d j x

Iy

w

e there

is y

2 djs

such

f

Iy

that y

w.

f

6.9. Lemma. Any 8.8.8-m. is a 3.9-m. Proof. Since djx E A?] and y

2 djx for any y E A?], the following asserti-

ons are equivalent:

(i)

djx IF w;

(ii) y

1-

w, for any y E A?];

f

(iii) there is t E Y/ M , z E St and y

Itf

(iv)

2

IF 'pjw. f

w for any y E A;;

536

Lukasiewicz logics

6.10. Definition.

If 0 = (S, U,IF) is a s.6-m. we shall say that w E Pg is k-valid in 0 if for any

f E U x and for

any x E

A , x Il-

(PkW.

A formula w is semantically

f

k-valid if it is k-valid in any s.6-m. If H G Ps and w E Po let H k

mean t h a t for any s.9-m.0, if every in

(s)w

w' E H is k-valid in 0 then w is k-valid

0. The following results establish the relationship between algebraic 9-models

and semantic 6-models. 6.11. Theorem.

Let S = ( A ,{ A f } t E ~ , jbe E ~a )9-structure and M ( S ) = ( B ( A ) ,n, U,8,A, {(pi}iEl, { @ i } i E ~ )the corresponding Moisilfield. If 0 = (S,U, I )-l as a semantic 6 - m o d e l define S by ws(f) = {t E

A

12

Il-

w}f o r a n y w E Ps and f E U x . Then:

f

(1) M = ( M ( S ) , U , S )i s a n algebraic 6-model; ( 2 ) For any formula w,w is algebraically k-valid i n M iff w is semantically k-valid i n 0.

It is obvious that ws(f) E B ( A ) , for any w and f. The triple ( M ( S ) U, , S) satisfies conditions (6.1)-(6.6). We shall prove only (6.4)(6.6). In the 9-algebra M ( S ) we have, for w E Ps,f E U x ,i E I and any x E A: Proof.

Kripke-s tyle semm tics for &valued predicate logics

Iy

(for cp;w a similar proof with

537

and $Z),

f 5

E (3uw)s(f)

($5

II- 3uw H f

u

z

IF w for some c E U f,"

++ z E

U ws(f,"), CEU

(for Vuw a similar proof with "for all c"). For any formula w the following equivalences hold:

($for all 5 E

A,

5

\I-

QkW

.

0

f

6.12. Theorem.

Let L be a completely chrysippian 8-algebra with card(I) _< X,, M = ( L , U , S ) a n algebraic 8-model and Q the countable set of the elements of L having the f o r m V ws(f:) OT A w~(f:), where u is a variable, CEU

CEU

w a formula and f a valuation. T h e n we can construct the 19-space S = (SpecL(Q),&,{d;};cl) and define the relation [I-c SpecL(Q) x U x x P8 by M It- w -Hws(f) E M , such that: f (a)

0 = (S,U, It) is a special semantic d-model;

(b) FOTa n y formula w, w is k-valid in 0 ifl w is k-valid in M . Proof. The triple satisfies the conditions in the definition of a s.s.8-m.:

538

Lukasiewicz logics

* ws(f:)

E M for some c E U u M II- w for some c E U ; f:

(Pkw fs ( ( P k w ) S ( f ) E

f

M

*

(Pk(wS(f))

E

M

*

T h e verification of the remaining conditions is easy. From the construction of 0 we obtain the equivalence of the following properties:

(i)

M II-

(ii)

((Pkw)S(f)

(iii)

((Pkw)S(f)=

(Pkw, for any

M E SpecL(Q)

and

f

E

Ux;

f

E M , for any M E SpecL(0) and f E U x ;

1.

0

6.13. Corollary. If 6 ds an ordinal number and card(I) 5 Xo f o r any w E F'd and H C Pa, then the following conditions are equivalent: (a)

H I- w ; k

539

APPENDIX APPLICATIONS TO SWITCHING THEORY

The first written evidence of th e idea t h a t t h e algebra of logic can be applied t o t he study of electrical networks is due t o the physicist P. Ehrenfest, in a review of the Russian translation of L. Couturat's book Algibre de la logique; the review was published in the Journal of the Russian PhysicalChemical Society, Physics section, 42, section II, no. 10, p. 382, 1910. Later on, in two papers published in December 1936 and February 1937 in

the Journal of the Institute of Engineering and Electro-Communication of Japan (Japanese, with English abstracts in the Japanese periodical Nippon Electrical Communication Engineering, 1938- 1941), A. Nakashima and M. Hanzawa pointed out several algebraic laws of series and parallel connections which in fact were identical t o th e laws of Boolean algebra; but t h e authors were not aware of the latter concept. Independently of Ehrenfest, Nakashima and Hanzawa and independently of each other, C.E. Shannon in Transactions

A.I.E.E. 1938, no. 57, p. 713 and V.I. Shestakov in his doctoral thesis a t the University of Moscow, also in 1938, applied Boolean algebra t o t h e study of switching circuits. Their works were followed up, especially from 1950 on, when the algebraic theory of switching circuits developed vertiginously. Nowadays the starting point of the theory seems quite simple. Associate with each contact of a circuit, a variable x which takes the value 1when the contact is closed and the value 0 when the contact is open. Further let

t

be

a variable which takes the value 1if a current flows through the circuit and

the value 0 if no current flows through the circuit. If the circuit consists of

x, y connected in series then t = x A y, while if the contacts x, y are connected in parallel then t = x V y. Also, two contacts may be mounted on an armature, as in Fig. A l . When the armature is in position a

two contacts

the upper contact is closed and th e lower contact is open, while in position b the upper contact is open and the lower contact is closed. That is why

540

Appendix

a Fig. A l .

we also refer t o the upper and the lower contacts as opening contacts and closing contacts, respectively. Let us associate with the armature a binary variable x defined as follows: x = 0 in position a ,

2

= 1 in position b;

then x is also th e variable associated with the opening contact, while the variable associated with the closing contact is 3. Therefore the two-element Boolean algebra Lz can be applied t o the study of contact circuits. More generally, Boolean algebra is a suitable tool for the study of networks made up of binary devices, which means that each device has two possible states. However it turns out that this theory uses Boolean algebras distinct from

Lz

as well. On t he other hand the study of networks involving multi-positional devices and of the so-called hazard and race phenomena have imposed the use of other algebraic tools, namely Galois fields, Lukasiewicz-Moisil algebras

and t he theory of discrete function; see e.g. Moisil [1959c], the reference given a t t he end of this appendix and Davio, Thayse and Deschamps [1978], respectively. In this appendix we illustrate by a few typical examples the adequacy of Lukasiewicz-Moisil algebras as a tool for the study of circuits, then we indicate appropriate references for the main topics dealt with in the theory.

It has been remarked that the description depicted in Fig. A 1 neglects the fact t h a t the armature has also an intermediate position, when both contacts are open; cf. Fig. A2. Therefore it is more appropriate t o use a three-valued variable x in order to describe the position of the armature:

541

Applications to switching theory

Fig. A2.

z = 0 in position a , Fig. A l ; z = 1/2 in the intermediate position depicted in Fig. A2; z = 1 in position b, Fig. A l . On the other hand if in the construction of the Moisil algebra L, given in Example 3.1.20 we change the notation i t o for i = 0, ..., n - 1 , we obtain L, = 0, ..., 2,l} and in particular the algebra L3 = {0,1/2,1}, whose operations are recalled in Table A l . Thus the variable z associated with the armature takes values in

5

{ 5,

0

Table A l .

0

1

Table A2.

L3 and if we denote by xo and z1the variables associated with the opening contact and the closing contact we see that zo = Ncpp and z1 = q l z ; cf. Table A2. The above type of contacts is known as break-before-make; there is also another type of contacts, called make-before-break, for which the associated variables fulfil zos= pzz and zlS = Npls.

As will be illustrated hereinafter, the study of the sequential behaviour of circuits requires the knowledge of the structure of functions with arguments and valued in L,. Consider the following functions Ak : L, ---t L,(k E

L,) A1

:

=

A0

91.

= N Q , - ~ , A;/(,+q = p,-i A Nv,,-iv1 (i = 1 ,..., n - 2) and It foIIows easily from N = n-j-1 n-1 and cp; = 0 or 1

Appendix

542 as j

< n -i

or j

2 n -i

-&= 0 or 1 as j

function

f

:

L,

-+

L,

Table 3.1.1 in the new notation) that i or j = i. This implies further that any

(cf.

#

can be written in the form

which will be referred t o as the interpolation formula for each k E L, the right side reduces t o

f(k)

A Xk(k)

f. P r o o f for = f(k). As a matter

of fact the interpolation formula can be generalized t o functions of several arguments in L, and with values in L,. In particular for n = 3 we get

hence the interpolation formula becomes

The study of the sequential behaviour of circuits is based on the hypothesis (made explicit by C.P. Popovici) that the axis of time can be divided into sufficiently small intervals, so that within each of them the position of each device of the circuit remains unchanged. Therefore the time values are discrete: 0,1,2,...,n,... , both in the Boolean switching theory and in its generalizations. Notations like a,

...,z, ... mean the values of the variables

... a t the n-th time interval.

a, ...,5,

B y way of illustration let us study the operation of a relay. This device consists of an electro-magnet and an armature with contacts. When the current flows through the coil of the electro-magnet the armature is attracted and remains attracted; when there is no current in the flow the armature comes back and remains in the rest position. More exactly, this operation is described i n Table A3, where ( is the variable associated t o the current and z is the variable associated to the armature. Thus z,+~ = f(&,z,), where

f : L2x L3+ L 3 is the function depicted in Table A3. But we have the identity

543

Applications to switching theory

1

0

1 0

1/2

Table A 5

which is checked by taking in turn

In = 1 and tn= 0.

Further ( A l ) yields,

via Table A3.

hence we obtain th e formula

which describes the operation of the relay and is called the characteristic

equation of the relay. Beside relay contacts there exist key contacts, for which the armature is controlled from outside the automaton. Let a,

...,c,

...,z

5,

be the varia-

bles associated t o the armatures of the keys and of the relays, respectively,

I, ...,C

and

t he variables associated t o the current in the relays. The syn-

thesis problem consists in constructing an automaton having a prescribed operation. This behaviour can be described by a system of equations of the form

(A51 where

{

xn+1 = F(an,

cn, xn,

***y

Zn)

7

2,)

7

............... zn+1 = H(an,

F,..., H

cn, xn,

are functions with arguments and values in

LJ.But

Appendix

544

= f(tn9 5,) ...............

5n+1

zn+1 = f(5nr zn)

where

9

f stands for the right side of the characteristic equation of the relay,

therefore

(A71

9

{

F(an, ---, cn, zn, * - * , zn) = f ( t n ,

5,)

7

...............

.

~ ( u n , . . . , c n , 5 n , . * - , z ~ )= f(Cnn,xL’n)

If we succeed t o obtain

{<

t,..., 5 as functions

[ = q u , ...,c,5,

”., z ) ,

............... = x(u, ...,c, 2, ...,2) ,

(A81

then in view o f the interpolation formula generalized t o functions o f several variables, t he functions

...,x

(A9)

Q1

:

L3 x

... x L3 + Lz C LJ

can be expressed in terms of LM-operations and this will enable us t o construct the circuit. But it follows from the above discussion that @, satisfy

(A101

1

F(u, ”‘1

c,z,

...,2) = f ( @ ( U , ....c , 5,.... 2),5)

............... H ( u , ....c, 5,’”,2) = f ( x ( u ,..., c , 5 ,...,z),2)

...,x must

. .

so that what we must do is to solve the latter system of functional equations

in the unknowns

a, .... x.

Consider the following example. Construct a series-parallel circuit with a key

A and two relays X , Y ,all of the contacts being break-before-make,

and such that:

I)

i n t h e rest position both the key and relays are unoperated;

It) at the setting o f the key A relay X is operated, then both relays remain operated;

Y is operated and

545

Applications to switching theory

I l l ) by putting the key off relay X becomes unoperated then Y becomes unoperated.

As a matter of fact there are several possible exact interpretations of conditions 1-111 in terms of positions 0, 1/2 and 1; one of them is given in Table

where

F

and

G

a

zn

Yn

0

0

0

1/2 1 1 1

0

0

0

0

1/2 1

0 0

1 1 1/2

1 1 1

1/2 1/2 1/2

1/2

0 0

%+1

Yn+l

0 0

0 0

1/2 1 1

0

1/21 1 1 1 1 1/2 0 1 0 1

I

I I

1/2

0

0 1/2 1 1 1 1

1/2

0

are the functions described in Table A4. On -he other hand

we have seen that for each key contact or relay contact z , the variables zo and z1 associated with the opening contact and the closing contact satisfy zo

= N p z = XOZ and z1 = cplz = Xlz; cf. (A2). It follows easily by

induction on t h e (obvious!)

definition of series- parallel circuits that the

546

Appendix

functions 9 and Q can be expressed in terms of the operations A , V, XO and

Al.

Using distributivity, this implies further that

V (a9 A XOU A XIS)

(A12.2)

Q(u, Z,y) = b1 V

9 and Q are of the form

V

... V (b27 A Xla A Xiz A Xiy)

and we are going t o use ( A l l ) and the lines o f Table A4 in order t o determine the coefficients a l ,

..., b27 in (A12).

From line 1 and ( A l l . l ) we get 0 = F(O,O,O) =

@(O,O,O)

A 1/2, i.e.

Q ( O , O , 0) = 0, or equivalently a1 V a2 V a4 V a6 V a8 V a10 V a16 V a23 = 0 by

(A12.1), i.e. (A13)

a1 = a2 = a4 = a6 = a8 = a10 = a16 = a23 = 0

Similarly, using line 10 we obtain 0 = 9(1/2,0,1)A1/2, or equivalently

while from line 8 we deduce

i.e. @(1/2,0,1) = 0,

547

Applications to switching theory

1/2 = @(1/2,1,1) V (G(1/2,1,1) A 1/2)

,

i.e. @(1/2, 1,1) = 0, or equivalently

In t he same way we obtain from lines 2, 9 and 11 that @(1/2,0,0) = @(1/2,1/2,1) = @(1/2,0,1/2) = 0, but these conditions are ensured by (A13)-(A15). Further line 4 is equivalent t o 1 = @(1,1/2, l),which reduces to

(A16)

a3

V

a14

=1

.

Similarly, lines 3, 5, 6 and 7 imply @(1,0,0) = @ 1,O) ( =I @(l, , 1,1/2) =

@(1,1,1)= 1, but these conditions are ensured by (A14). We have thus proved that formulas (A13)-(A16) describe all functions @ satisfying ( A l l . l ) and we can obtain similarly all the solutions

9 of the functional equation

(A11.2). In particular taking a3 = 1and all the remaining coefficients equal t o 0 we obtain the particular solution @ ( a ,2,y) = Xla and we obtain similarly the particular solution Q ( a ,2,y) = Xlz.The latter pair (a, Q) yields the circuit in Fig. A3.

Fig. A3.

The above example is typical for the way in which t h e applications of Moisil algebras to switching circuits are worked out. The theory investigates circuits involving devices such as polarized relays with unstable neutral,

Appendix

548

ordinary relays under low self-maintaining current, valves, resistances; both break-before-make and make-before-break contacts are taken into consideration, as well as multi-positional relays. For each case t h e characteristic equation o f the device is established, then the corresponding analysis problem and synthesis problem are studied. The analysis problem is converse of the synthesis problem: it consists in describing the operation of a given circuit. In the case o f series-parallel circuits, the bijection between the basic algebraic operations and their physical realizations enables us t o translate immediately the structure of t h e circuit into its working function, i.e. the function which yields the value of the variable associated with the current through the circuit from the values of the variables associated with th e devices of the circuit. The operation of the automaton is then obtained from the working function and the characteristic equations of the devices o f the circuit. As concerns the synthesis problem there are many examples worked out and a few theorems. First one discusses the way i n which the wording of a program can be translated into a table in

terms of 0, 1/2, 1and expressing the recurrence equations (A5); for instance

it is proved that in the above example Table A4 is the only consistent specification of the previous conditions I, ll and 111. Then an operation program is called consistent if the table which expresses it contains no contradiction, i.e. nopairF(ao, a!

# p.

...,co,so,...,z O ) =a!and

F(ao,...,c0,zo, ...,)'2

=pwhere

Unlike what happens in the Boolean theory of switching circuits,

there exist consistent programs which cannot be realized by series-parallel circuits. However another type of circuits has been pointed out, called P - I

circuits, such that any consistent program can be realized by a P - I circuit. Another theorem states that a working function W ( a ,...,c, z, y, ...) can be realized only with break-before-make contacts of the relay X if and only if

W ( a ,...,c, 1/2, y, ...) 5 W ( a ,...)c, 0, y, ...) 5 W ( a ,'",c , 1, y, ...) and a dual condition for make-before-break contacts. O n the other hand every function with arguments in L3 and values in Lz is the working function of a certain series-parallel circuit with break-before-make and make-beforebreak contacts.

Applications to switching theory

549

The equivalence of automats and th e simplification problem are topics somewhat complementary t o the synthesis problem. Roughly speakaing, two circuits are said t o be equivalent if their operation is the same and we are looking for a simple (possibly the simplest) circuit in a given class of equivalent circuits. Other topics include bridge two-terminals, multi-terminals, applications of the Moisil algebra

L5 etc.

The first works in this field were published as a series of short communications (Moisil [1956], [1958], [1959a]; the series begins with the invited paper Greniewski (19561). The first survey of the field is Moisil [1957]; for another variant of it see Moisil [1972], pp. 104-148. The book Moisil [1967] deals both with the above mentioned types of circuits and with electronic circuits, using both Boolean algebras and Moisil algebras. See also Moisil [1959b], [1961], [1963b], [1964a], [1964b], [1965b] and lvanescu (Hammer) and Rudeanu (19641. These references are not exhaustive.

This Page Intentionally Left Blank

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1945. Post algebras and rings. Duke Math. J., 12, 389-395. WAJSBERG, M.: 1931. Axiomatization of the three-valued propositional calculus (Polish). C.R. Seances SOC.Sci. Lattres Varsovie, CI. Ill, 24, 126-145. W6JCICK1, R.: 1975. A theorem on the finiteness of the degree of maximality of the n-valued Lukasiewicz logic. Proc. Intern. Symp. Multiple-Valued Logic, Indiana, 244-251 = Polish Acad. Sci. Inst. Philos. Sociol. Bull. Sect. Logic, 4, 19-25. W6JCICK1, R. and MALINOWSKI, G. (editors): 1977. Selected Papers on Lukasiewicz sentential calculi. Ossolineaum, Wroclaw and Warsaw.

574

References

YAQUB, F.M.: 1962. Free extensions of Boolean algebras. Doctoral dissertation, Purdue Univ., cf. Pacific J. Math., 13,761-771 (1963).

575

AUTHOR INDEX

Abad, M. 184,188,192,196,200 Adams, M.E. 275 Balbes, R. 7,75,176,203,209, 213,218,219,221,223,257, 275,283,295,311,359,364, 365,377,388,391,392,396, 411,415 Bsdele, I. 371 Banaschewski, B. 371,415 Barnes, L. 495,513,523 Beazer, R. 222,257-259 Becchio, D . 132,138,141,220, 310,311,316,460,461,470 Berman, J. 258 Beznea, L. 101,106,365,397 Bialynicki-Birula, A. 92 Birkhoff, G. 60,61,66,287,289 Boicescu, V. 106,113,125,155, 179,184,189,198,199,202, 224,239,250,251,253,258, 259,275,279,288,292,296, 300,305,306,310,311,317, 349,366,369,371,379,396, 487 Brezuleanu, A. 83 Bruns, G. 371,415 Burris, S. 54

Chang, C.C. 282,283 Cherciu, M. 248,267 Cignoli, R. x,113,122,124,125, 132,176,196,201,203,204, 211,212,229,233,247,248, 261,265-267,269,276,283, 288,289,292,296,300,301, 317,332,337,350,371,380,

382,384-386,397,407,411, 412,415,471,472,485,486 Cohn, P.M. 54 Coppola, L.G. 349,441,444 Cornish, W.H. 102,397 Coulon, J. 457 Coulon, J.L. 457 Daigneault, A. 43

Davio, M. 540

Day, G. 237,415 De Gallego, M.S. 211,212,300, 301 Deschamps, J.-P. 540 Dummet, M. 208 Dwinger, Ph. 7,75,176,203,209, 213,218,219,221,223,257, 283,295,311,359,364,365, 377,385,388,391,392,396, 407,411,415

576 Ehrenfest, P.S. 539 Epstein, G. 165,176,209,232,277, 279,290,308-310,312

Author index Leblanc, L. 47 Lemmon, E.J. 430 tukasiewicz, J. ix,459,471

Ferentinou-Nicolacopoulou, J. 22 Georgescu, G. 106,169,176,229,

232,266,268,271,332,340, 371,378,389,415,417,452 Goldberg, H. 460,465,467 Gratzer, G. 54,213 Greniewski, M. 549 Grigolia, R. 485 Halmos, P.R. 43,47,50,51,240,

371,395,436,440 Hammer, P.L. (Ivhescu, P.) 549 Hanzawa, M. 539 Hashimoto, J. 295 Hatvany, Cs. 247 Hecht, T. 310 Horn, A. 208,209,282,308-310, 312 Iorgulescu, A. 41,106,110,111,

116,241,250,251,254-256, 278,299,317,324,337,353, 421,436,441,443,444 Iturrioz, L. 201,204,207,220, 311-316,437,471,483 Kalman, J. 22

Katrindk, T. 176,218,223,310 Leblanc, H. 460,465,467

Mack, J. 495,513,523 Mac Lane, S. 81 Makinson, D. 278 Maksimova, J. 326,531 Marek, W. 110,116 Marrona, R. 23 McKinsey, J.C.C. 434 Mitschke, A. 176 Moisil, Gr.C. ix,x,23,111,116, 119,126,131,132,139,176, 177,187,199,201,204,207, 211,214,215,248,285,289, 292,302,305,314,385,540, 549 Monteiro, A. ix,xi,132,136,138, 199,204,207-209,222,229, 248,261,267,273,275,276, 314,315,349,350,385,386, 441,444 Monteiro, L. 135,136,182,184, 188,192,196,200,207,208, 232,234,310,311,314,316, 349,371,385,417,436,441, 444 Mugkavdin, V. 92 Nachbin, L. 37,438 Nadiu, Gh. 441 Nakasima, A, 539

Author index Onicescu,

0. x

Petcu, A. 136,138

Pierce, R.S. 54,61,77,235,258, 289,359,366,410,411 Ponasse, D. 111,457 Popescu, N. 81 Popovici, C.P. 542 Potthoff, K. 51 Post, E. ix Priestley, H.A. 95 Radu, A. 81 Radu, E. x Rasiowa, H . 42,52,92,412,486, 491,501 Rose, A. x,471 Rosenbloom, P. x,176 Rosser, J.B. 483,484 Rousseau, G. 203

577 Thayse, A. 540 Thomas, J. 310 Tolosa, J. 131 Traczyk, T. 110,116,165,168, 176,277,279,283 Turquette, A. 483,484 Vakarelov, D. 326,531 Varlet, J. 206,214,215,218,220, 223,227,257,259,260 Voiculescu, I. (Petrescu, 1.) 92 Vraciu, C. 106,169,176,266,340, 360,371,378,389 Wade, L. 165 Wajsberg, M. 459,470 Weaver, G. 460,465,467 Wcjjcicki, R. 471 Yaqub, F.M. 410

Rudeanu, S. 124,155,549 Zadeh, Sade, A. 571 Sankappanavar, H.P. 54 Sestakov, V.I. 539 Shannon, C.E. 539 Sholander, M. 20 Sicoe, C. 113,127,128,278,303 Sikorski, R. 42,52,235,371,410, 412 Smullyan, R.M. 505 Suchori, W, 115,121,484 Surma, S.J. 486 Tarski, A. 434

L.

ix

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579

SUBJECT INDEX

Algebra

B - 308 B L - 308 Boolean - 25,43

De Morgan - 22 free - 61 Heyting - s. Heyting Heyting-Brouwer - 220 Kleene - 113

L

-

207 Lindenbaum-Tarski - 481

m - 241 monadic- 43 P - 209 Peano- 61 Post - 165,168,169 propositional - 457 quotient - 57 regular - 243 relative K - 223 semisimple - 40,45,425 simple - 40,45,268 Stone- 213 subdirectly irreducible - 60 weakly projective - 391 amalgamation property 363 ascending basis 183 atom 235

automorphisrn 12,59 axiom 124,460,478,489,521 axis 177 Boolean algebra 25,43,47,49,429

- - 314 topological - - 429

symmetric

Boolean interval 196 Boolean spectrum 321 bounded poset (lattice) 3 Category 73 algebraic - 74 dual - 74 equational - 74 centre 169 chain 3 chain of constants ascending - - - 166 descending - - - 168 closure operator 7,428,431 congruence 3757 maximal - 268 proper - 267

m - 256 congruence extension property 258 consequence 263 consistent set 461,467,495 constant 46

580

Subject index

coproduct 76

-

complete

completely prime

Decreasing set 95 deductive system 261 degree 47,453

maximal

direct product 2,58,76 double Stone algebra 213 Element

strong

-

248

- 41 m-maximal - 41 m-complete m-prime-

(m,d)

greatest

-

join irreducible

free

-

293

- 3 least - 3 maximal - 3 minimal - 3 regular - 216 last

strictly chrysippian - 302 embedding 11 endomorphism 12,13,59

-

41 256

-

256

formative construction 55

3

independent - 453

(P;

-

++-closed

first - 3

31

248

complemented - 23 216

35,41

-

chrysippian - 23

-

106

C extension

410

free mD regular extension function antitone

-

8

-8 increasing - 8

decreasing

isotone - 8 strongly continuous

-

83

functor 77 (left,right) adjoint covariant

-

contravariant faithful - 78

equivalent algebras 69

fully faithful

equivalent categories 78

identity

-

-

-

77 78

77

extension 75 Generalization 523

g.1.b. 3,243

-

77

equational class 65

essential - 75

1

free variable 516

epimorphism 74

F i l t e r 31

3

proper - 31,41 Stone-

direct sum 76

-

-

prime - 33,41 principal

diamond 20

dense

41

79

Subject index Heyting algebra 202 linear--

207

symmetric - - 207,313 homomorphism 9,27,58,121,175 Ideal 31 maximal - 35 monadic- 44 prime - 33 principal - 31 proper - 3 1 increasing set 95 individual variable 513 infimum, inf 3 injective huli 82

injective object 81 interior operator 428,431,483 interpretation 467,483,484,487, 518 involution 12,22 external - 110 internal - 110 isomorphism 8,59,75,78 dual - 8 Join 4,6

K k-axiom 489

k-deducibility 490 k-deductive system 503 k-filter 502 k-implication 488,519,522 k-consistent set 495,527

581 k-proof 490,510,522 k-theorem 491,522 k-valid sentence 488,522,532 Lattice 4 atomic- 235 complemented - 23 complete - 6,228 m-complete - 6,228 distributive

-

20

dual - 5 fully normal - 208 normal - 273 pseudocomplemented - 213 relatively complemented - 213 relatively pseudocomplemented 202 strongly atomic - 238 superatomic - 237 lexicographic product 2 lower bound 3 1.u.b.

3

Lukasiewicz implication 478 Lukasiewict-Moisil = LM LMn-algebra 106,116 axled - - 177 LMd-algebra 106,116 closure - - 432 completely chrysippian - - 229 m-completely chrysippian -

-

241

-

dual modal - 432 fully complete - - 241 generalized - - 119

582

Subject index

- - 432 modal - - 432 d-simple - - 287

-- 2 strict - - 1

interior

LM space 332,335

- with negation

dual

pentagone 20 polyadic 9-algebra 452

337

MacNeille completion by cuts 11,15 meet 4,6 modal operator 428,431 modal structure 429

functional

exactly n-valued - - 159 irredundant

--

155

proper - - 471,472 symmetric - - 314 Moisil field 328 Moisil pre-n-algebra 116

Moisil representation 457 monadic &algebra 418,419 functional - - 420 monomorphism 74 monotone representation 166,179 Moore family 7 morphism 43,47,58,73,418,452 canonical - 59,67 functorial - 78 Natural t ra nsformation 78 negation 22,110 Ordinal sum 2

polynomial 67 poset 1 predecessor 3,112 Priestley space 95,341

- - with involution 102 - - with negation 345 projective object 81 proof from hypotheses 461,479,490 property R 260 propositional variable 460,478,487

Q Q-filter 437 Q-prime- 437 Q-semisimple &algebra 439 quantifier existential - 43,45,417 universal - 44,418 Reflector 80 regular set 242 retract 75 Saturated set 507 segment 28 semantic implication 467,488,519 semilattice

Partial order 1

454

locally finite - - 453

modus ponens 460,479,490,523 Moisil n-algebra 116,131

--

join

-

4

Subject index meet

-

583

4

length of a

semimorphism 513

-

segment of a

61

-

61

sentence 460 signed formula 506

19-congruence 250,267,268

Stone space 83,86

&filter

- - with involution

92

247

maximal - 265

subalgebra 28,44,55,123,322

monadic- 423

subcategory 74

prime- 265 proper

full - 74

-

265

&ideal 247

reflective - 80 subdirect decomposition 60

monadic- 423

subdirect product 60

maximal monadic - 424 8-model

sublattice 14,28 subobject 75

algebraic - 531

subsemilattice 14

semantic - 534,535

successor 3,112

d-space 328

support 47,453

&structure 326

supremum, sup 3

7-algebra 54

syntactic implication 461,479,490, 522 Toset

3

tree 507 type of algebra 54 Ultrafilter 35 ultraproduct 72 upper bound 2 Valid sentence 467,483,485 valuation 487,519 variety 65 Weak implication 262 word 61

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