Central European Journal of Mathematics

CEJM 3(1) 2005 1–13

Some alternating sums of Lucas numbers ∗ ˇ Zvonko Cerin

Department of Mathematics, University of Zagreb, Kopernikova 7, 10010 Zagreb, Croatia

Received 8 August 2004; accepted 12 October 2004 Abstract: We consider alternating sums of squares of odd and even terms of the Lucas sequence and alternating sums of their products. These alternating sums have nice representations as products of appropriate Fibonacci and Lucas numbers. c Central European Science Journals. All rights reserved.

Keywords: Fibonacci numbers, Lucas numbers, integer sequences, alternating sums MSC (2000): 11B39, 11Y55

1

Introduction

The Fibonacci and Lucas sequences Fn and Ln are defined by the recurrence relations F1 = 1,

F2 = 1,

Fn = Fn−1 + Fn−2 for n > 3,

L1 = 1,

L2 = 3,

Ln = Ln−1 + Ln−2 for n > 3.

and Let uk = F2k−1 , sk = L2k−1 , vk = F2k , tk = L2k , Uk = u2k , Sk = s2k , Vk = vk2 and Tk = t2k denote odd and even terms of the Fibonacci and Lucas sequences and their squares. In the P P P note [1] we presented formulas for the sums ji=0 Sk+i , ji=0 Tk+i and ji=0 sk+i tk+i of Lucas numbers. The purpose of this paper is to establish similar results for the alternating P P P sums ji=0 (−1)i Sk+i , ji=0 (−1)i Tk+i and ji=0 (−1)i sk+i tk+i . Therefore we prove three theorems covering each of these alternating sums. The proofs are by induction. For each theorem we establish three relations among Fibonacci and Lucas numbers that are either initial steps of the induction or accomplish the inductive step in the proofs. ∗

E-mail: [email protected]

ˇ Z. Cerin / Central European Journal of Mathematics 3(1) 2005 1–13

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In each theorem we must treat odd and even summations separately. The results are similar in form and the method of proof. They could be viewed as examples of situations where the following integer sequence appears. Let the sequence ak be defined by the recurrence relation a0 = 1, a1 = 6, and . This sequence is listed as A049685 in the ak = 7 ak−1 − ak−2 , for k > 2. Then ak = t2k+1 3 data bank of Sloane [4]. P For k > 0, let bk = ki=0 ai . Since ak = bk − bk−1 we see that the terms bk satisfy the recurrence relations b0√= 1, b1 = 7, b2 = 48, and bk = 8 bk−1 − 8 bk−2 + bk−3 for k > 3. Let √ 1+ 5 α = 2 and β = 1−2 5 . Note that β = − α1 . Looking at the roots of the characteristic equation we find that bk =

4 α + 11 β 11 α + 4 β (5 α + 2 β)k + (2 α + 5 β)k , 15 15

for every k > 0. Also, bk = F34 k . This sequence is listed as A004178 in [4]. The following three theorems are our main results: Theorem 1.1. a) For every k > 1 we have 1 + Tk = 5 uk uk+1 . b) For every m > 0 and k > 1 the following equality holds: D) δ2 m+1 +

2X m+1 i=0

(−1)i Tk+i = −5 bm uk+m+2 sk+m ,

c) For every m > 1 and k > 1 the following equality holds: E) δ2m +

2m X

(−1)i Tk+i = 5 am uk+m+1 uk+m ,

i=0

where the sequence δm is defined as follows:   if m = 0, 1;  1, 15, δm = Sn+1 , if m = 2 n and n > 1;   δ + 5 t , if m = 2 n + 1 and n > 1. m−2

m

Theorem 1.2. a) For every k > 2 we have 9 + Sk = 5 uk−1 uk+1 . b) For every m > 0 and k > 1 the following equality holds: K) θ2m+1 +

2m+1 X i=0

(−1)i Sk+i = −5 bm uk+m+1 sk+m

c) For every m > 1 and k > 1 the following equality holds: L) θ2m +

2m X i=0

(−1)i Sk+i = 5 am Uk+m ,

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where the sequence θm is defined as follows:   if m = 0, 1, 2;  9, 5, 14 θm =

θm−2 + 5 an , if m = 2n + 1 and n > 1;   2 T − θ n m−2 , if m = 2n and n > 2.

Theorem 1.3. a) For every m > 0 and k > 1 the following equality holds: P ) ξ2m +

2m X

(−1)i sk+i tk+i = 5 am uk+m vk+m ,

i=0

b) For every m > 0 and k > 1 the following equality holds: Q) ξ2m+1 +

2m+1 X i=0

(−1)i sk+i tk+i = −5 bm vk+m sk+m+1 ,

where the sequence ξm is defined as follows:   if m = 0, 1;  2, 5 ξm = 1 + an , if m = 2n and n > 1;   5 b , if m = 2n + 1 and n > 1. n

In the last section we shall describe how one can discover these results with the help from computer in the package Maple V.

2

Preliminaries for Theorem 1.1

For the initial step in an inductive proof of the part b) of our first theorem we shall use the following lemma. Lemma 2.1. For every k > 1 the following equality holds: A) 15 + Tk − Tk+1 = −5 uk+2 sk . Proof. In terms of the Fibonacci and Lucas sequences the relation A) becomes A∗) 15 + L22 k − L22 k+2 + 5 F2 k+3 L2 k−1 = 0. k

k

−β By the Binet formula Fk is equal to αα−β and Lk is equal to αk + β k (see [2] and [3]). It follows that the left hand side of A*) is equal to α3 (αM2 +1) , where M denotes the expression

15 α5 + 15 α3 + α4 k+5 + α4 k+3 + α−4 k+5 + α−4 k+3 − α4 k+9 −

α4 k+7 − α−4 k+1 − α−4 k−1 + 5 α4 k+6 − 5 α8 + 5 − 5 α2−4 k .

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The fourteen terms of M can be considered as one group of four terms and two groups of five terms that have similar exponents. The first group is


15 α5 + 15 α3 − 5 α8 + 5 = −5 α2 − α − 1 α2 + 1 α4 + α3 + α2 − α + 1 = 0 because α is the root of the equation x2 − x − 1 = 0. One can see similarly that the other two groups are zero. Hence, M = 0 and the proof is complete.  For the induction step in the proof of the part b) of our first theorem we shall use the following lemma. Lemma 2.2. For every r > 0 and k > 1 the following equality holds: B) 5 t2r+3 + Tk+2r+2 − Tk+2r+3 + 5 br+1 uk+r+3 sk+r+1 = 5 br uk+r+2 sk+r . Proof. In terms of the Fibonacci and Lucas sequences the relation B) becomes B∗) 5 L4 r+6 + L22 k+4 r+4 − L22 k+4 r+6 +

5 [br+1 F2 k+2 r+5 L2 k+2 r+1 − br F2 k+2 r+3 L2 k+2 r−1 ] = 0.

It follows that the left hand side of B*) is equal to δ(M1 + M2 + M3 ), where δ = √ M1 = 2 (2 + 5)(A−k−r B r − A−k C −r ), √ M2 = (65 + 25 5)(B r − A−r ), √ M3 = (64079 + 28657 5)(Ak C r − Ak+r Ar ), √

√

1165−521 4

√

5

,

√

5 A =B = A2 . Hence, the left hand side of A = 7+32 5 , B = 7−32 5 = A1 , and C = 47+21 2 B*) is zero because M1 = M2 = M3 = 0 and the proof is complete. 

For the induction step in the proof of the part c) of our first theorem we shall use the following lemma. Lemma 2.3. For every r > 0 and k > 1 the following equality holds: C) 3 [Sr+2 − Sr+1 + Tk+2 r+2 − Tk+2 r+1 ] = 5 uk+r+1 [t2 r+3 uk+r+2 − t2 r+1 uk+r ]. Proof. In terms of the Fibonacci and Lucas sequences the relation C) is equivalent to the relation C∗) 3 [L22 r+3 − L22 r+1 + L22 k+4 r+4 − L22 k+4 r+2 ]−

5 F2 k+2 r+1 [L4r+6 F2 k+2 r+3 − L4 r+2 F2 k+2 r−1 ] = 0.

It follows that the left hand side of C*) is equal to

2 α2 M , (α2 +1)2

where M denotes the expression

α4 k+8 r+2 − α4 k+8 r+10 + 3 α−4 k−8 r−8 + α−4 r + 3 α4 k+8 r+8 −

3 α−4 r−2 − 3 α4 k+8 r+4 − 3 α−4 k−8 r−4 + α−4 k−8 r−2 − α−4 k−8 r−10 +

α4 r − α4 r+8 + 3 α−4 r−6 − 3 α4 r+2 + 3 α4 r+6 − α−4 r−8 .

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The sixteen terms of M can be considered as four groups of four terms that have similar exponents. The first group is α4 r − α4 r+8 − 3 α4 r+2 + 3 α4 r+6 = α4 r 1 − α4






α2 − α − 1 α2 + α − 1 = 0

because α is the root of the equation x2 − x − 1 = 0. One can see similarly that the other three groups are zero. Hence, M = 0 and the proof is complete. 

3

The proof of Theorem 1.1

a). For m = 2 k − 1 and n = 2 k + 1 the formula (17b) in [5] says that L4 k − (−1)2 k−1 L2 = 5 F2 k−1 F2 k+1 and since L2 = 3 we get L4 k + 3 = 5 F2 k−1 F2 k+1 . On the other hand, the formula (17c) in [5] implies the relation L4 k + 2 = L22 k so that 1 + Tk = 5 uk uk+1 . b). The proof is by induction on m. For m = 0 the relation D) is 15 + Tk − Tk+1 = −5 uk+2 sk (i. e., the relation A)) which is true by Lemma 2.1. Assume that the relation D) is true for m = r. Then 2(r+1)+1

δ2(r+1)+1 +

X

(−1)i Tk+i =

i=0

δ2 r+1 + 5 t2(r+1)+1 +

2X r+1 i=0

(−1)i Tk+i + Tk+2 r+2 − Tk+2 r+3 =

5 t2(r+1)+1 + Tk+2 r+2 − Tk+2 r+3 − 5 br uk+r+2 sk+r = −5 br+1 uk+(r+1)+2 sk+(r+1) , where the last step uses Lemma 2.2. Hence, D) is true for m = r + 1 and the proof is completed. c). The proof is also by induction on m. For m = 0 the relation E) is 1 + Tk = 5 uk+1 uk (i. e., the relation under a)) which we already proved. Assume that the relation E) is true for m = r. Then 2(r+1) X δ2(r+1) + (−1)i Tk+i = i=0

Sr+2 +

2r X i=0

(−1)i Tk+i − Tk+2 r+1 + Tk+2 r+2 =

5 ar uk+r+1 uk+r + Sr+2 − Sr+1 − Tk+2 r+1 + Tk+2 r+2 = 5 ar+1 uk+(r+1)+1 uk+(r+1) , where the last step uses Lemma 2.3. Hence, E) is true for m = r + 1 and the proof is completed.

ˇ Z. Cerin / Central European Journal of Mathematics 3(1) 2005 1–13

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4

Preliminaries for Theorem 1.2

For the initial step in an inductive proof of the part b) of our second theorem we shall use the following lemma. Lemma 4.1. For every k > 1 the following equality holds: F ) 5 + Sk − Sk+1 = −5 uk+1 sk . Proof. In terms of the Fibonacci and Lucas sequences the relation F) becomes F ∗) 5 + L22 k−1 − L22 k+1 + 5 F2 k+1 L2 k−1 = 0. It follows that the left hand side of F*) is equal to

M , α (α2 +1)

where M denotes the expression

5 α3 + 5 α + α4 k+1 + α4 k−1 + α−4 k+5 + α−4 k+3 − α5+4 k −

α3+4 k − α−4 k+1 − α−4 k−1 + 5 α4 k+2 − 5 α4 + 5 − 5 α−4 k+2 .

The fourteen terms of M can be considered as one group of four terms and two groups of five terms that have similar exponents. The first group is 5 α3 + 5 α − 5 α4 + 5 = −5 (α2 + 1) (α2 − α − 1) = 0 because α is the root of the equation x2 − x − 1 = 0. One can see similarly that the other two groups are zero. Hence, M = 0 and the proof is complete.  For the induction step in the proof of the part b) of our second theorem we shall use the following lemma. Lemma 4.2. For every r > 0 and k > 1 the following equality holds: G) 5 [br+1 uk+r+2 sk+r+1 − br uk+r+1 sk+r + ar+1 ] + Sk+2 r+2 − Sk+2 r+3 = 0. Proof. In terms of the Fibonacci and Lucas sequences the relation G) becomes G∗) 5 [3 br+1 F2 k+2 r+3 L2 k+2 r+1 −

3 br F2 k+2 r+1 L2 k+2 r−1 + L4 r+6 ] + 3 [L22 k+4 r+3 − L22 k+4 r+5 ] = 0.

It follows that the left hand side of G*) is equal to δ(M1 + M2 + M3 ), where δ = M1 = 6 (2 +

√

√

445−199 4

√

5

,

5)(A−k C −r − A−k−r B r ),

√ M2 = (25 + 11 5)(A−r − B r ), √ M3 = −3(9349 + 4181 5)(Ak C r − Ak+r Ar ),

√

√

5 A =B = A2 . Hence, the left hand side of A = 7+32 5 , B = 7−32 5 = A1 , and C = 47+21 2 G*) is zero because M1 = M2 = M3 = 0 and the proof is complete. 

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For the induction step in the proof of the part c) of our second theorem we shall use the following lemma. Lemma 4.3. For every r > 1 and k > 1 the following equality holds: H) 2 [Tr+1 − θ2 r ] + Sk+2 r+2 − Sk+2 r+1 = 5 [ar+1 Uk+r+1 − ar Uk+r ]. Proof. In terms of the Fibonacci and Lucas sequences the relation H) is equivalent to the relation H∗) 6 [L22 r+2 − θ2 r ] + 3 [L22 k+4 r+3 − L22 k+4 r+1 ]−

5 [L4 r+6 F22k+2 r+1 − L4 r+2 F22k+2 r−1 ] = 0.

In order to check the relation H*) we must use the fact (easily proved by induction) √  √ r √ −r √  that θ2 r = 3+3 5 7+32 5 + 3−3 5 7+32 5 + 2 for every r > 1. It follows immediately either in Maple or in Mathematica that the left hand side of H*) is equal to zero and the proof is completed with the help of the computer. 

5

The proof of Theorem 1.2

a). For m = 2 k + 1 and n = 2 k − 3 the formula (17b) in [5] says that L4 k−2 − (−1)2 k+1 L4 = 5 F2 k−3 F2 k+1 and since L4 = 7 we get L4 k−2 + 7 = 5 F2 k−3 F2 k+1 . On the other hand, the formula (17c) in [5] implies the relation L4 k−2 − 2 = L22 k−1 so that 9 + Sk = 5 uk−1 uk+1 . b). The proof is by induction on m. For m = 0 the relation K) is 5 + Sk − Sk+1 = −5 uk+1 sk (i. e., the relation F)) which is true by Lemma 4.1. Assume that the relation K) is true for m = r. Then 2(r+1)+1

θ2(r+1)+1 +

X

(−1)i Sk+i =

i=0

θ2 r+1 + 5 ar+1 +

2X r+1 i=0

(−1)i Sk+i + Sk+2 r+2 − Sk+2 r+3 =

5 ar+1 + Sk+2 r+2 − Sk+2 r+3 − 5 br uk+r+1 sk+r = −5 br+1 uk+(r+1)+1 sk+(r+1) , where the last step uses Lemma 4.2. Hence, K) is true for m = r + 1 and the proof is completed. c). The proof is also by induction on m. For m = 0 the relation L) is 9 + Sk = 5 uk+1 uk−1 (i. e., the relation under a)) which we already proved. Assume that the relation L) is true for m = r. Then 2(r+1)

θ2(r+1) +

X i=0

(−1)i Sk+i =

8

ˇ Z. Cerin / Central European Journal of Mathematics 3(1) 2005 1–13

2 Tr+1 − θ2 r +

2r X i=0

(−1)i Sk+i − Sk+2 r+1 + Sk+2 r+2 =

5 ar Uk+r + 2 Tr+1 − 2 θ2 r − Sk+2 r+1 + Sk+2 r+2 = 5 ar+1 Uk+(r+1) , where the last step uses Lemma 4.3. Hence, L) is true for m = r + 1 and the proof is completed.

6

Preliminaries for Theorem 1.3

For the initial step in an inductive proof of the part b) of our third theorem we shall use the following lemma. Lemma 6.1. For every k > 1 the following equality holds: M ) 5 + sk tk − sk+1 tk+1 = −5 vk sk+1 . Proof. In terms of the Fibonacci and Lucas sequences the relation M) becomes M ∗) 5 + L2 k−1 L2 k + L2 k+1 [5 F2 k − L2 k+2 ] = 0. It follows that the left hand side of M*) is equal to

Pα , α2 +1

where P denotes the expression

α4 k + α4 k−2 − α−4 k+2 − α−4 k + 5 α4 k+1 −

α4 k+4 − α4 k+2 + 5 α−4 k−1 + α−4 k−2 + α−4 k−4 .

The ten terms of P can be considered as two groups of five terms that have similar exponents. The first group is α4 k + α4 k−2 + 5 α4 k+1 − α4 k+4 − α4 k+2 =



− α2 − α − 1 α4 + α3 + 3 α2 − α + 1 α4 k−2 = 0

because α is the root of the equation x2 − x − 1 = 0. One can see similarly that the other group is zero. Hence, P = 0 and the proof is complete.  For the induction step in the proof of the part a) of our third theorem we shall use the following lemma. Lemma 6.2. For every r > 0 and k > 1 the following equality holds: N ) ar+1 − ar + sk+2r+2 tk+2r+2 − sk+2r+1 tk+2r+1 +

5 [ar uk+r vk+r − ar+1 uk+r+1 vk+r+1 ] = 0.

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, in terms of the Fibonacci and Lucas sequences the relation N) Proof. Since ak = t2 k+1 3 is equivalent with the relation N ∗) L4 r+6 − L4 r+2 + 3 L2 k+4 r+3 L2 k+4 r+4 − 3 L2 k+4 r+1 L2 k+4 r+2 +

5 [L4 r+2 F2 k+2 r−1 F2 k+2 r − L4 r+6 F2 k+2 r+1 F2 k+2 r+2 ] = 0.

It follows that the left hand side of N*) is equal to

Q , (α2 +1)2

where Q denotes the expression

− 5 α−4 r−3 − α−4 r+2 + α−4 r−6 + 2 α−4 k−8 r−7 + 6 α−4 k−8 r−1 − 5 α4 r+3 −

2 α−4 r − 5 α−4 r−1 − 2 α4 r+4 − 2 α1−4 k−8 r + 6 α4 k+8 r+9 − α4 r+2 +

5 α4 r+7 + 5 α−4 r−5 − 6 α−4 k−8 r−5 + 2 α4 r+8 + 5 α1−4 r − 2 α11+4 k+8 r +

α4 r+10 − 6 α4 k+8 r+5 + 2 α4 k+8 r+3 + 5 α4 r+5 + 2 α−4 r−4 − 5 α9+4 r .

The twenty four terms of Q can be considered as two groups of eight terms and two groups of four terms that have similar exponents. The first group is α−4 r−6 − α2−4 r − 2 [α−4 r − α−4 r−4 ] − 5 [α−4 r−1 − α−4 r−5 − α1−4 r + α−4 r−3 ]


= 1 − α4 α2 − α − 1 α2 − 4 α − 1 α−4 r−6 = 0 because α is the root of the equation x2 − x − 1 = 0. One can see similarly that the other three groups are zero. Hence, Q = 0 and the proof is complete.  For the induction step in the proof of the part b) of our third theorem we shall use the following lemma. Lemma 6.3. For every r > 0 and k > 1 the following equality holds: O) 5 [br+1 (1 + vk+r+1 sk+r+2 ) − br (1 + vk+r sk+r+1 )] =

sk+2r+3 tk+2r+3 − sk+2r+2 tk+2r+2 .

Proof. In terms of bk ’s and the Fibonacci and Lucas sequences the relation O) is equivalent with the relation O∗) 5 [br+1 (1 + F2 k+2 r+2 L2 k+2 r+3 ) − br (1 + F2 k+2 r L2 k+2 r+1 )]−

L2 k+4 r+5 L2 k+4 r+6 + L2 k+4 r+3 L2 k+4 r+4 = 0. √ It follows that the left hand side of O*) is equal to −360 + 161 5 times a factor with eight terms that can be divided into two groups of four terms such that both groups are evidently zero. Hence, the left hand side of O*) is zero and the proof is complete. 

7

The proof of Theorem 1.3

a). The proof is by induction on m. For m = 0 the relation P) is 2 + L2 k−1 L2 k = 5 F2 k−1 F2 k . Its proof is done as follows.

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For m = 2 k − 1 and n = 2 k the formula (17b) in [5] says that L4 k−1 − (−1)2 k−1 L1 = 5 F2 k−1 F2 k and since L1 = 1 we get L4 k−1 + 1 = 5 F2 k−1 F2 k . On the other hand, the formula (17a) in [5] implies the relation L4 k−1 − 1 = L2 k−1 L2 k so that 2 + L2 k−1 L2 k = 5 F2 k−1 F2 k . Assume that the relation P) is true for m = r. Then 2(r+1)

ξ2(r+1) +

X

(−1)i sk+i tk+i =

i=0

1 + ar+1 +

2r X i=0

(−1)i sk+i tk+i − sk+2 r+1 tk+2 r+1 + sk+2 r+2 tk+2 r+2 =

5 ar uk+r vk+r + ar+1 − ar − sk+2 r+1 tk+2 r+1 + sk+2 r+2 tk+2 r+2 = 5 ar+1 uk+(r+1) vk+(r+1) , where the last step uses Lemma 6.2. Hence, P) is true for m = r + 1 and the proof is completed. b). The proof is also by induction on m. For m = 0 the relation Q) is 5 + sk tk − sk+1 tk+1 = −5 vk sk+1 (i. e., the relation M)) which we already proved in Lemma 6.1. Assume that the relation Q) is true for m = r. Then 2(r+1)+1

ξ2(r+1)+1 +

X

(−1)i sk+i tk+i =

i=0

5 br+1 +

2X r+1 i=0

(−1)i sk+i tk+i + sk+2 r+2 tk+2 r+2 − sk+2 r+3 tk+2 r+3 =

−5 br vk+r sk+r+1 + 5 br+1 − 5 br + sk+2 r+2 tk+2 r+2 − sk+2 r+3 tk+2 r+3 = −5 br+1 vk+(r+1) sk+(r+1) , where the last step uses Lemma 6.3. Hence, Q) is true for m = r + 1 and the proof is completed.

8

Computer assisted proofs

In the rest of this note I will describe how one can discover these results and check the above proofs with the help of the computer. The presentation is for the software Maple V. The input with(combinat): calls the package that contains the function fibonacci that computes the terms of the Fibonacci sequence. We first define functions fF, fL, fu, fv, fU, fV, fs, ft, fS, fT, and fD that give terms of the sequences Fk , Lk , uk , vk , Uk , Vk , sk , tk , Sk , Tk and the left hand side of D). fF:=x->f0(x):

fL:=x->g0(x):

fu:=x->fF(2*x-1):

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fv:=x->fF(2*x): fU:=x->fu(x)^2: fV:=x->fv(x)^2: fs:=x->fL(2*x-1): ft:=x->fL(2*x): fS:=x->fs(x)^2: fT:=x->ft(x)^2: f0:=fibonacci: g0:=x->f0(2*x)/f0(x): fD:=(a,m,k)->a+sum((-1)^i*fT(k+i),i=0..2*m+1): The following procedure tests for values a and k between a0 and a1 and k0 and k1 and m, k) is an integer and prints out a, k, for given values m and n if the quotient q = f D(a, f s(k+n) and q (factored into primes). gD:=proc(a0,a1,m,k0,k1,n) local a,k,q;for a from a0 to a1 do for k from k0 to k1 do q:=fD(a,m,k)/fs(k+n): if denom(q)=1 then print([a,k,ifactor(q)]);fi:od;od;end: The input gD(1,1000,1,17,17,1); is asking to determine among first thousand positive integers the number δ3 that makes the quotient q of the left hand side of the formula D) for m = 1 by sk+1 an integer for k = 17. The output [105, 17, −(2)(5)(7)(233)(135721)] indicates that α3 = 105 is a good candidate. This is ”confirmed”when we input gD(105,105,1, 1,100,1); whose outputs all have integers as the third term in each of the hundred triples of the outputs. We can now replace fs(k+n); with fu(k+n); and make some experiments to discover that the same number 105 appears in the search gD(1,1000,1,17,17,3);. All this suggests to modify gD as follows: gD:=proc(a0,a1,m,k0,k1) local a,k,q;for a from a0 to a1 do for k from k0 to k1 do q:=fD(a,m,k)/fs(k+m)/fu(k+m+2)/(-5): if denom(q)=1 then print([a,k,ifactor(q)]);fi:od;od;end: With the new function the command gD(105,105,1,1,100); has for output the same number 7 as the third term in each of the hundred triples of the outputs. When we repeat this for values of m between 1 and 5 we discover that numbers δ2 m+1 and the third terms are 105, 720, 4935, 33825, 231840 and 7, 48, 329, 2255, 15456 (i. e., b1 , b2 , b3 , b4 , b5 ), respectively. These five values are sufficient to discover the rule by which δ2 m+1 ’s are built. All other formulas in this paper are discovered by similar procedures. It remains to explain how to invoke the help of the computer in the proof of Lemmas. We shall prove only Lemmas 2.1 and 2.2 because the proofs of other lemmas are analogous. We first define the function that expresses the Binet formula and make the assumption that α > 0 and that r and k are positive integers. a:=alpha: b:=-1/a: f:=x->(a^x-b^x)/(a-b): g:=x->a^x+b^x: assume(alpha>0,r,posint,k,posint): f0:=f: g0:=g: The numerator of the left hand side of A*) is evaluated by the the following input. M:=numer(simplify(15+fT(k)-fT(k+1)+5*fu(k+2)*fs(k))); The output is the expression: 15 α6 + 15 α4 + α6+4 k + α4+4 k + α6−4 k + α4−4 k − α4 k+10 −

α4 k+8 − α−4 k+2 − α−4 k + 5 α4 k+7 − 5 α9 + 5 α − 5 α−4 k+3 .

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We consider three groups of terms with similar exponents. factor(add(op(i,M),i in [1,2,12,13])); factor(expand(add(op(i,M),i in [3,4,7,8,11]))); factor(expand(add(op(i,M),i in [5,6,9,10,14]))); The outputs


−5 α α2 − α − 1 α2 + 1 α4 + α3 + α2 − α + 1 ,
4 2

−α4 αk α − α − 1 α4 + α3 + 3 α2 − α + 1 , (α2 − α − 1) (α4 + α3 + 3 α2 − α + 1) (αk )4

are all zero because α is the root of the equation x2 − x − 1 = 0. In order to prove Lemma 2.2 we need the function that computes the integer sequence bk . s5:=sqrt(5):fb:=x->factor((15+7*s5)/30*((7+3*s5)/2)^x+ (15-7*s5)/30*((7-3*s5)/2)^x): Since the function fb uses √explicit values, we must do the same with other functions (i. e., we must define α = 1+2 5 ). The left hand side Bs of the relation B*) is defined as follows: alpha:=(1+s5)/2:FS:=x->factor(simplify(x)): Bs:=FS(5*ft(2*r+3)+fT(k+2*r+2)-fT(k+2*r+3)+ 5*(fb(r+1)*fu(k+r+3)*fs(k+r+1)-fb(r)*fu(k+r+2)*fs(k+r))): The second part Bt of Bs is the key part for which we must prove that it is zero. We do this by making some substitutions into Bt which result in a significant simplification Bu of it. Bt:=op(2,Bs): Bq:=FS(Bs/Bt); Bu:=FS(subs({3-s5=4/(3+s5),7/2+3/2*s5=A,47/2+21/2*s5=A^2, (7/2+3/2*s5)^(-r)=A^(-r),9+4*s5=A*(3+s5)/2},Bt)); √ 5 and Bu is Note that Bq is − 233 + 521 12 60  √ √ √ −r 982090 A2 r+k + 435 52r 7 + 3 5 + 439204 A2 r+k 5−   √ −r √ −r − 143285 A2 r+1+k + 975 2r 7 + 3 5 − 975 7/2 + 3/2 5   √ √ √ −r 435 5 7/2 + 3/2 5 − 64079 A2 r+1+k 5. Now it is obvious that the second and the fourth term are opposites of the seventh and the sixth term, respectively. The other terms too add up to zero because the output of the code FS(add(op(i,Bu),i in [1,3,5,8])/A^(2*r+k))*A^(2*r+k); √ √ √
2 r+k 5 7+3 5 −7 − 3 . 5 + 2 A A and A = is − 143285+64079 2 2

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References ˇ [1] Z. Cerin: On sums of odd and even terms of the Lucas sequence, (preprint). [2] V.E. Hoggatt, Jr.: Fibonacci and Lucas numbers, The Fibonacci Association, Santa Clara, 1979. [3] R. Knott: Fibonacci numbers and the Golden Section, http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html. [4] N.J.A. Sloane: On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/ njas/sequences/. [5] S. Vajda: Fibonacci and Lucas numbers, and the Golden Section: Theory and Applications, Halsted Press, Chichester 1989.

CEJM 3(1) 2005 14–25

On two theorems for flat, affine group schemes over a discrete valuation ring Adrian Vasiu∗ Mathematics Department, University of Arizona, 617 N. Santa Rita, P.O. Box 210089, Tucson, AZ-85721, USA

Received 27 July 2004; accepted 26 September 2004 Abstract: We include short and elementary proofs of two theorems that characterize reductive group schemes over a discrete valuation ring, in a slightly more general context. c Central European Science Journals. All rights reserved. ° Keywords: Group schemes, discrete valuation rings MSC (2000): 11G10, 11G18, 14F30, 14G35, 14G40, 14K10, and 14J10

1

Introduction

Let k be a field. Let p ∈ {0} ∪ {n ∈ N|n is a prime} be the characteristic of k. Let V be a discrete valuation ring of residue field k. Let π be a uniformizer of V and let K := V [ π1 ] be the field of fractions of V . Let F = Spec(P ) and G = Spec(R) be flat, affine group schemes over V . We will assume that F is a reductive group scheme over V ; so F is smooth over V and its fibres are connected and reductive groups over fields. In this paper we present elementary and short proofs of the following two basic theorems on reductive group schemes over V . Theorem 1.1. Let f : F → G be a homomorphism such that its generic fibre fK : FK → GK is a closed embedding. Then the following four properties hold: (a) Suppose fK is an isomorphism and G is of finite type. Then F is the smoothening of G in the sense of [1, Thm. 5, p. 175] (the definition is recalled in 2.3.2). (b) The kernel Ker(fk : Fk → Gk ) is a unipotent, connected group of dimension 0. (c) The homomorphism f is finite. ∗

E-mail: [email protected]

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(d) If p = 2, then we assume that FK¯ has no normal subgroup that is an SO2n+1 group for some n ∈ N. Then f itself is a closed embedding. Theorem 1.2. Suppose GK is a connected, smooth group over K and the identity component (Gk )0red of the reduced group (Gk )red is a reductive group over k of the same dimension as GK . Then the following three properties hold: (a) The group scheme G is of finite type over V , has a connected special fibre Gk , and its generic fibre GK is a reductive group. (b) If p = 2, we assume that the group GK¯ has no normal subgroup that is isomorphic to SO2n+1 for some n ∈ N. Then G is a reductive group scheme over V .

(c) The normalization Gn of G is a finite G-scheme. Moreover, there is a faithfully flat V -algebra V˜ that is a discrete valuation ring and such that the normalization (GV˜ )n of GV˜ is a reductive group scheme over V˜ .

If V is of mixed characteristic (0, p), then 1.1 (a) and (b) were first proved in [13, proof of 3.1.2.1 c)]. For the sake of completeness, in §3 we recall the proofs of 1.1 (a) and (b) in a slightly enlarged manner that recalls elementary material. The passage from 1.1 (a) and (b) to 1.1 (d) (resp. 1.1 (c)) is a direct consequence of the classification of the ideals of the Lie algebras of adjoint and simply connected semisimple groups over k¯ (resp. is only a variant of Zariski Main Theorem). In [13, 3.1.2.1 c)] we overlooked the phenomenon of exceptional nilpotent such ideals and so the extra hypothesis of 1.1 (d) for p = 2 does not show up in [13, 3.1.2.1 c)]. Theorem 1.1 (resp. Theorem 1.2) is also proved in [12], under the extra assumption that GK is of finite type over K (resp. that Gk is of finite type over k). The methods of [12] are essentially the same as of [13] except that the proofs of [12] contain many unneeded parts. Here we follow [13] to get short and efficient proofs of 1.1 and 1.2. The importance of 1.1 (resp. 1.2) stems from its fundamental applications to integral canonical models of Shimura varieties of Hodge (resp. to dual groups), cf. [13] (resp. cf. [12]). We also mention that 1.1 (c) and (d) are powerful tools in extending results on semisimple groups over a field, from characteristic 0 to arbitrary positive characteristic. As an exemplification, in §5 we include such an application that pertains to adjoint groups. The proofs of 1.1 and 1.2 are carried on in §3 and §4 (respectively). Few notations and preliminaries needed in §3 and §4 are gathered in §2. We would like to thank U of Arizona for good conditions for the writing of this work. We would like to thank G. Prasad for pointing out that [13, 3.1.2.1 c)] omits to add the extra hypothesis of 1.1 (d) for p = 2.

2

Preliminaries

In 2.1 we list notations. Elementary properties of Lie algebras, dilatations, representation theory, and quasi-sections are recalled in 2.2 to 2.5. In 2.6 we present in an accessible way a result of the classical Bruhat–Tits theory for reductive groups over K.

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2.1 Notations Let k, V , K, π, F = Spec(P ), and G = Spec(R) be as in §1. Let V sh be the strict henselization of V . Let W be the set of finite, discrete valuation ring extensions of the completion of V sh . If H is a reductive group scheme over an affine scheme Spec(A), let Z(H), H der , H ad , and H ab denote the center, the derived group, the adjoint group, and respectively the abelianization of H. So Z ab (H) = H/H der and H ad = H/Z(H). Let H sc be the simply connected semisimple group cover of H der . For a free A-module N of finite rank, let GL(N ) be the group scheme over Spec(A) of linear automorphisms of N .

2.2 Lie algebras Let Lie(H) be the Lie algebra of an affine group H of finite type over k or over V . We view Lie(H) as a (left) H-module via the adjoint representation. Until 2.3 we assume that k = k¯ and that H is over k. For x ∈ Lie(H) we have a unique and functorial Jordan decomposition x = xs + xn such that for any monomorphism H ֒→ GLm (m ∈ N), xs ∈ Lie(GLm ) is semisimple, xn ∈ Lie(GLm ) is nilpotent, and xs and xn are polynomials in x ∈ Lie(GLm ) (cf. [2, Ch. 1, §4]). We say x is nilpotent (resp. semisimple) if x = xn (resp. x = xs ). So if H is Ga (resp. Gm ), then x is nilpotent (resp. semisimple). 2.2.1 Lemma 2.1 ¯ Let n be a non-zero ideal of Lie(H) Lemma 2.1. Let H be a reductive group over k = k. that is formed by nilpotent elements and is a simple H-module. Then p = 2 and H has a normal subgroup H0 isomorphic to SO2n+1 (n ∈ N) and such that n ⊂ Lie(H0 ). Proof. The image of n in Lie(H ab ) is trivial. So n ⊂ Lie(H der ). So we can assume that H = H der is semisimple. Let nad := Im(n → Lie(H ad )). As Lie(Z(H)) is formed by semisimple elements, the Lie homomorphism n → nad is an isomorphism. Thus nad is a Q non-zero ideal of Lie(H ad ) that is a simple H ad -module. Let H ad = i∈I Hi be the product decomposition into simple groups. As H ad is adjoint, there is no element of Lie(H ad ) fixed by H ad . So as n is a simple H ad -module, we get that there is i0 ∈ I such that nad ⊂ Lie(Hi0 ). Thus we can assume H ad is a simple group. Let T be a maximal torus of H ad and let L be the Lie type of H ad . As Lie(H ad ) has non-zero semisimple elements, we have nad 6= Lie(H ad ). So as nad is a simple H ad -module, from [11, Prop. 1.11] we get that either nad = Im(Lie(H sc ) → Lie(H ad )) or (p, L) ∈ {(2, F4 ), (3, G2 ), (2, Bn ), (2, Cn )|n ∈ N}. If nad = Im(Lie(H sc ) → Lie(H ad )), then nad has non-zero semisimple elements except when dimk (Lie(Z(H sc ))) is the rank of L, i.e. except when (p, L) = (2, A1 ) (cf. loc. cit.). If (p, L) is (2, F4 ) (resp. (3, G2 )), then nad is the unique proper ideal of Lie(H ad ) and so it is generated by the direct sum s of the eigenspaces of the adjoint action of T on Lie(H ad ) that correspond to short roots. Thus dimk (nad ) is 26 (resp. 7) and nad ⊂ s ⊕ Lie(T ), cf. [7, pp. 408–409] applied to a semisimple group over Fp whose extension to k is H sc . But L has 24 (resp. 6) short roots, cf. [3, PLATES VIII and IX]. So dimk (Lie(T ) ∩ nad )

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is 2 = 26 − 24 (resp. 1 = 7 − 6) and so nad has non-zero semisimple elements. If (p, L) = (2, Cn ) with n ≥ 3, then we similarly argue that nad is the ideal associated to short roots and that dimk (Lie(T ) ∩ nad ) = 2n2 − n − 1 − ε − 2(n2 − n) = n − 1 − ε > 1, where ε is 1 if n is even and is 0 if n is odd (cf. [7, p. 409] and [3, PLATE III]). Thus we have (p, L) = (2, Bn ), with n ∈ N. As nad is a simple H ad -submodule of Lie(H ad ), from [11, Prop. 1.11] we get that that nad is the direct sum of the eigenspaces of the adjoint action of T on Lie(H ad ) that correspond to short roots, that dimk (nad ) = 2n, and that nad ⊂ Im(Lie(H sc ) → Lie(H ad )). Let nsc be the inverse image of n in Lie(H sc ). We have dimk (nsc ) = dimk (n) + dimk (Lie(Ker(H sc → H))) = 2n + dimk (Lie(Ker(H sc → H))). But as nsc is an H sc -module, we have Lie(Z(H sc )) ⊂ nsc (cf. [11, Prop. 1.11]). Thus dimk (nsc ) = 2n + 1. Thus dimk (Lie(Ker(H sc → H))) = 2n + 1 − 2n = 1 and so Ker(H sc → H) = Z(H sc )→µ ˜ 2 . Thus H is an adjoint group and so isomorphic to SO2n+1 . ¤

2.3 Dilatations In this section we assume GK is reduced. Let S be a reduced subgroup of Gk . Let J be the ideal of R that defines S and let IR be the ideal of R that defines the identity section of G. Let R1 be the R-subalgebra of R[ π1 ] generated by πx , where x ∈ J. By the dilatation of G centered on S one means the affine scheme G1 := Spec(R1 ); it has a canonical structure of a flat, affine group scheme over V and the morphism G1 → G is a homomorphism whose special fibre factors through the closed embedding S ֒→ Gk (cf. [1, Prop. 1 and 2, pp. 63–64]). In §3 we will need the following elementary Lemma. 2.3.1 Lemma 2.2 Lemma 2.2. Suppose G is a closed subgroup of a smooth group scheme H over V of l−dim(S) relative dimension l. Then Ker(G1k → Gk ) is isomorphic to a subgroup of Ga . l−dim(S) ∼ Moreover, if G = H, then Ker(G1k → Gk ) → Ga . Proof. We can assume V is complete. As G1 is a closed subgroup of the dilatation of H ˆ centered on S (cf. [1, Prop. 2 (c) and (d), p. 64]), we can also assume that G = H. Let R and IˆR be the completions of R and IR with respect to the IR -topology. Let s := dim(S). ˆ = V ⊕ IˆR = V [[x1 , ..., xl−s , y1 , ..., ys ]], where x1 , ..., As G is smooth over V , we can write R ˆ defines the completion of S xl−s , y1 , ..., ys ∈ IˆR are such that the ideal (x1 , ..., xl−s , π) of R ˆ = V [[x1 , ..., xl−s , y1 , ..., ys ]][ x1 , ..., xl−s ]. Let δ : along its identity section. We have R1 ⊗R R π π \ ˆ→R ˆ ⊗V R ˆ be the comultiplication map of the formal Lie group of G. As S is a subgroup R P P of Gk , for i ∈ {1, ..., l−s} we have δ(xi ) = xi ⊗1+1⊗xi + j∈I a (aij ⊗bij )+ j∈I b (aij ⊗bij ), i i where Iia and Iib are finite sets, where each aij and bij belong to IˆR , and where for each j ∈ Iia (resp. j ∈ Iib ) the element aij ∈ IˆR (resp. bij ∈ IˆR ) is divisible by either some xu

or by some πyv ; here u ∈ {1, ..., l − s} and v ∈ {1, ..., s}. ˆ 1 , ..., xl−s , y1 , ..., ys ). We have Ker(G1k → Gk ) = Spec(A1 ), where A1 := R1 ⊗R R/(x

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Let x¯i be the image of xπi in A1 . So A1 is a k-algebra generated by x¯1 , ..., x¯l−s . Taking P P a b the identity δ( xπi ) = xπi ⊗ 1 + 1 ⊗ xπi + j∈I a πij ⊗ bij + j∈I b aij ⊗ πij modulo the ideal i i ˆ we get that the comultiplication map δ1 : A1 → A1 ⊗k A1 (x1 , ..., xl−s , y1 , ..., ys ) of R1 ⊗R R,

of the group Ker(G1k → Gk ) is such that δ1 (¯ xi ) = x¯i ⊗ 1 + 1 ⊗ x¯i for any i ∈ {1, ..., l − s}. As G1 is smooth over V (cf. [1, Prop. 3, p. 64]) of relative dimension l, Ker(G1k → Gk ) = Ker(G1k → S) has dimension at least l − s. So as A1 is k-generated by x¯1 , ..., x¯l−s and its dimension is at least l − s, we get that A1 = k[¯ x1 , ..., x¯l−s ] is a polynomial k-algebra. From the description of δ1 we get that Ker(G1k → Gk ) is isomorphic to Gl−s ¤ a . 2.3.2 Smoothening We assume that GK is smooth over K and that G is of finite type over V . We take S to be the Zariski closure in Gk of the special fibres of all morphisms Spec(V sh ) → G of V -schemes. We refer to G1 → G as the canonical dilatation of G; it is a morphism of finite type. There is m ∈ N and a finite sequence of canonical dilatations G′ := Gm → Gm−1 → ... → G1 → G0 := G such that G′ is uniquely determined by the following two properties (cf. [1, pp. 174–175]): (i) the affine group scheme G′ is smooth and of finite type over V ; (ii) if Y is a smooth V -scheme and if Y → G is a morphism of V -schemes, then Y → G factors uniquely through the homomorphism G′ → G.

From very definitions, we get that G′ (V ) = G(V ) and that the smoothening (GV sh )′ of GV sh is G′V sh . We also point out that the V -schemes G1 , ..., Gm = G′ are of finite type.

2.4 Representations We denote also by δ : R → R ⊗V R the comultiplication map of G. Let L0 be a finite P subset of IR . For l0 ∈ L0 we write δ(l0 ) = j∈I(l0 ) a0j ⊗ l0j , where I(l0 ) is a finite set and a0j , l0j ∈ R. Let L be the V -submodule of R generated by 1, by l0 ’s, and by l0j ’s. It is known that we have δ(L) ⊂ R ⊗V L, cf. [9, 2.13]. So L is a G-module and thus we have a homomorphism ρ(L) : G → GL(L) between flat, affine group schemes over V . Let B P be a V -basis of L contained in {1} ∪ IR . For l ∈ B ∩ IR , we write δ(l) = l′ ∈B all′ ⊗ l′ , where each all′ ∈ R. As δ(l) − l ⊗ 1 + 1 ⊗ l ∈ IR ⊗V IR , we have a1l = l. But if GL(L) = Spec(A(L)) and if q(L) : A(L) → R is the V -homomorphism that defines ρ(L), then R(L) := Im(q(L)) is the V -algebra generated by the all′ ’s. As a conclusion we have: (i) The V -subalgebra of R generated by L is contained in R(L). So if G (resp. if GK ) is of finite type over V (resp. over K) and if L0 generates the V -algebra R (resp. the K-algebra R[ π1 ]), then ρ(L) (resp. ρ(L)K ) is a closed embedding homomorphism.

2.5 Quasi-sections Let X be a reduced, flat V -scheme of finite type. Let y ∈ X(k). From [6, Cor. (17.16.2)] ˜ of K such that there is a faithfully flat, we get the existence of a finite field extension K

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˜ that is of finite type and we have a morphism z : Spec(V˜ ) → X local V -subalgebra of K whose image contains y. 2.5.1 Lemma 2.3 Lemma 2.3. (a) If V is complete, then we can assume V˜ is a discrete valuation ring. (b) Let a : Y → X be a morphism between reduced, flat V -schemes of finite type. Suppose ¯ and that for any W ∈ W the map a(W ) : Y (W ) → X(W ) that V is complete, that k = k, is onto. Then the map a(k) : Y (k) → X(k) is surjective. Proof. As V is complete, it is also a Nagata ring (cf. [10, (31.C), Cor. 2]) and thus the ˜ is a finite V -algebra. So as V˜ we can take any local ring of normalization VK˜n of V in K ˜ (it is a local ring of V ˜n ) that is a discrete valuation ring. the normalization of V˜ in K K From this (a) follows. By taking W to be V˜ , we get that y ∈ Im(a(k)). So as y was arbitrary, we get that (b) holds. ¤

2.6 Lemma 2.4 ¯ Let f : F → G be a homomorphism such Lemma 2.4. Suppose V is complete and k = k. that fK is an isomorphism. Then f (V ) : F (V ) → G(V ) is an isomorphism. If moreover G is smooth, then f is an isomorphism. Proof. Let T be a maximal split torus of F , cf. [5, Vol. III, Exp. XIX, 6.1]. We show that the assumption that F (V ) © G(V ) leads to a contradiction. We have F (K) = F (V )T (K)F (V ), cf. Cartan decomposition of [4, 4.4.3]. So as F (V ) © G(V ), there is 1 g ∈ G(V ) ∩ (T (K) \ T (V )). We write T = Gsm = Spec(V [w1 , ..., ws ][ w1 ...w ]). Let w ∈ P s 1 be such that under the V -homomorphisms P ։ V [w1 , ..., ws ][ w1 ...ws ] → K that define g ∈ T (K) 6 F (K), it is mapped into an element of the set {wi , wi−1 |i ∈ {1, ..., s}} that maps into K \ V . As fK is an isomorphism we can identify P [ π1 ] = R[ π1 ]. Let n ∈ N be such that π n w ∈ R. Under the V -homomorphism R → K that defines g n+1 , the element π n w maps into an element of K \ V . Thus g n+1 ∈ / G(V ). Contradiction. So F (V ) = G(V ). Let now G be smooth. We show that the assumption that f is not an isomorphism ¯ As R 6= P , there is w0 ∈ P \ R such leads to a contradiction. We can assume k = k. that x := πw0 ∈ R \ πR. Let g¯ : R → k be a k-homomorphism such that g¯(x) 6= 0. Let g : R → V be a V -homomorphism that lifts g¯ (as V is henselian and G is smooth). The K-homomorphism g[ π1 ] : R[ π1 ] → K maps w0 into an element of K \ V . So g defines an element of G(V ) \ F (V ). So F (V ) © G(V ). Contradiction. Thus f is an isomorphism.¤

3

Proof of Theorem 1.1

In this chapter we prove 1.1. To prove 1.1 we can assume that V is complete, that ¯ and that fK is an isomorphism. So f : F → G is defined by a V -monomorphism k = k,

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∼ P ֒→ R that induces a K-isomorphism P [ π1 ] → R[ π1 ] to be viewed as an identity. Let ρ(L) : G → GL(L) be as in 2.4, with L0 ∈ IR ⊂ R such that it generates the K-algebra P [ π1 ] = R[ π1 ]. The generic fibre of ρ(L) is a closed embedding, cf. 2.4 (i) and the choice of L0 . To prove 1.1 for f , it suffices to prove 1.1 for ρ(L) ◦ f . So we can assume ρ(L) is a closed embedding; so G is a reduced, flat, closed subgroup of GL(L) and so of finite type over V .

3.1 Proofs of 1.1 (a) and (b) Let G′ = Gm → ... → G1 → G0 = G be as in 2.3.2. As F is smooth over V and due to 2.3.2 (ii), f : F → G factors through a homomorphism f ′ : F → G′ . As G′ is smooth, f ′ is an isomorphism (cf. 2.6). So 1.1 (a) holds. For i ∈ {0, ..., m − 1}, Gi is a reduced, flat group scheme of finite type (cf. end of 2.3.2) and so a closed subgroup of some general linear group Hi (cf. 2.4 (i)). So each group Ker(Gi+1k → Gik ) is a subgroup of a product of a finite number of copies of Ga , cf. 2.3.1. As f ′ is an isomorphism, Ker(fk ) = Ker(Fk → Gk ) has a composition series whose factors are subgroups of Ker(Gi+1k → Gik ) (i ∈ {0, ..., m − 1}). Thus Ker(fk ) is a unipotent group in the sense of [5, Vol. II, Exp. XVII, 1.1]. As Ker(fk ) ⊳ Fk and as Fk has a trivial unipotent radical (being reductive), Ker(fk ) has dimension 0. But Fk is connected and so its action on (Ker(fk ))red via inner conjugation is trivial. So (Ker(fk ))red 6 Z(Fk ). Let g¯ ∈ (Ker(fk ))(k). By induction on i ∈ {0, ..., m} we show that g¯ ∈ Ker(F (k) → Gi (k)). The case i = 0 is obvious as G0 = G. For i ∈ {0, ..., m − 1} the passage from i to i + 1 goes as follows. Let Si be the reduced subgroup of Gik such that Gi+1 is the ˜ i+1 be the dilatation of Hi centered on Si ; we have dilatation of Gi centered on Si . Let H ˜ i+1 (cf. [1, Prop. 2 (c) and (d), p. 64]). a closed embedding homomorphism Gi+1 ֒→ H ˜ i+1 ֒→ GLn (with ni ∈ N, cf. 2.4 (i)). We consider a closed embedding homomorphism H i ¯ ∈ GLn (k) be the image of ˜ i+1 ֒→ GLn . Let h We have homomorphisms F → Gi+1 ֒→ H i i ¯ is a semisimple element. As Ker(Hi+1k → Hik ) is a g¯ via F → GLni . As g¯ ∈ Z(Fk )(k), h ¯ is a unipotent element. product of Ga groups (cf. 2.3.1) and as g¯ ∈ Ker(F (k) → Hi (k)), h ¯ is the identity element. So g¯ ∈ Ker(F (k) → Gi+1 (k)). This ends the induction. So h As f ′ : F → G′ = Gm is an isomorphism, we get that g¯ is the identity element of F (k). Thus (Ker(fk ))red is a trivial group. So Ker(fk ) is also connected. So 1.1 (b) holds.

3.2 Proof of 1.1 (c) As V is a Nagata ring, R is also a Nagata ring (cf. [10, (31.H)]). So the normalization Gn = Spec(Rn ) of G is a finite G-scheme. The homomorphism F → G factors through a morphism F → Gn . We have F (V ) = G(V ) = Gn (V ), cf. 2.6; this also holds if V is replaced by a W ∈ W. Thus f (k) : F (k) → Gn (k) is an epimorphism, cf. 2.5.1 (b) applied to F → Gn . So Gnk is connected and the morphism Fk → (Gnk )red is dominant. So both F and Gn have unique local rings O1 and O2 (respectively) that are faithfully

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flat V -algebras and discrete valuation rings. The natural V -homomorphism O2 → O1 is dominant and becomes an isomorphism after inverting π. So we can identify O1 = O2 . So as FK = GnK , the normal, noetherian, affine schemes F and Gn have the same set of local rings that are discrete valuation rings. Thus we have Rn = P (cf. [10, (17.H), Thm. 38]) and so F → Gn is an isomorphism. So f is a finite morphism. So 1.1 (c) holds.

3.3 Proof of 1.1 (d) As Ker(fk ) is unipotent, it is a subgroup of the unipotent radical of a Borel subgroup of some general linear group over k (cf. [5, Vol. II, Exp. XVII, 3.5]). So Lie(Ker(fk )) is formed by nilpotent elements (cf. 2.2) and is normalized by Fk . The root data of Fk and FK¯ are isomorphic (cf. [5, Vol. III, Exp. XXII, 2.8]) and determine the isomorphism classes of Fk and FK¯ (cf. [5, Vol. III, Exp. XXIII, 5.1]). So the hypothesis of 1.1 (d) implies that either p > 2 or p = 2 and Fk has no normal subgroup isomorphic to SO2n+1 (n ∈ N). So Lie(Ker(fk )) has no non-trivial simple Fk -submodule, cf. 2.2.1. Thus Lie(Ker(fk )) = 0. From this and the connectedness part of 1.1 (b), we get that Ker(fk ) is trivial. So fk is a closed embedding. As fk and fK are closed embeddings, from Nakayama’s lemma we get that the finite morphism F ×G Spec(O) → Spec(O) is a closed embedding for any local ring O of G. Thus f is a closed embedding and so an isomorphism (as fK is so). So 1.1 (d) holds. This ends the proof of 1.1.

3.4 Remarks (a) The reference in [14, proof of 4.1.2] to [13, 3.1.2.1 c)] does not always work for p = 2 (cf. 1.1 (d)). However, in [14, proof of 4.1.2] one can always choose ρB(k) and L such that ρ is a closed embedding (cf. 2.4 (i)). ¯ As Lie(Z(Fk )) (b) We continue to assume that fK is an isomorphism and that k = k. is formed by semisimple elements, the connected, unipotent group Z(Fk ) ∩ Ker(fk ) is trivial. Suppose now that p = 2 and f is not an isomorphism. So fk is not a closed embedding (see 3.3) and so (fk )red : Fk → (Gk )red is not a closed embedding. We get that Fkad → (Gk )red /fk (Z(Fk )) is an isogeny (cf. 1.1 (c)) that is a finite product Q j∈J fj : Fj → Gj of isogenies fj with Fj as a simple, adjoint group. Any fj is a purely inseparable isogeny (cf. 1.1 (b)) and there is j0 ∈ J such that fj0 is not an isomorphism. Q We have Ker(fk ) := j∈J Ker(fj ) and so Ker(fj0 ) is a non-trivial, unipotent subgroup of Fj whose Lie algebra is formed by nilpotent elements (see 3.3). From 2.2.1 and its proof we get that Fj0 is an SO2n+1 group and that Lie(Ker(fj0 )) contains the unique simple Fj0 -submodule nj0 of Lie(Fj0 ) of dimension 2n. The quotient of Fj0 by nj0 (see [2, §17]) is an Sp2n group (cf. [2, 23.6, p. 261]). So fj0 factors through a purely inseparable isogeny gj0 : Sp2n → Gj0 whose kernel is a unipotent group. So Lie(Ker(gj0 )) is formed by nilpotent elements and so it is trivial (cf. 2.2.1). Thus gj0 is an isomorphism. As a conclusion, the root data of Fk and (Gk )red are not isomorphic and so Fk and (Gk )red are not isomorphic.

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Proof of Theorem 1.2

¯ In this chapter we prove 1.2. To prove 1.2 we can assume that V is complete, that k = k, and that tr.deg.(k) ≥ 1.

4.1 The group Γ ¯ (Gk )0 is a split reductive group. Let T1 , ..., Ts be a finite number of Gm As k = k, red subgroups of (Gk )0red that generate (Gk )0red . For i ∈ {1, ..., s} let yi ∈ Ti (k) be an element of infinite order (here is the place where we need, in the case when p ∈ N, that tr.deg.(k) ≥ 1). So the Zariski closure in Ti of the subgroup of Ti (k) generated by yi , is Ti itself. Let Γ be the subgroup of G(k) generated by y1 , ..., ys . We conclude: (i) The Zariski closure of Γ in Gk is (Gk )0red .

4.2 The finite type case In this section we prove 1.2 under the assumption that G is of finite type over V . For i ∈ {1, ..., s} let Vi be a finite V -algebra that is a discrete valuation ring and such that there is wi ∈ G(Vi ) that lifts yi (cf. 2.5.1 (a)). By replacing V with its normalization in the composite field of the fields Vi [ π1 ] (i ∈ {1, ..., s}), we can assume that for i ∈ {1, ..., s} there is wi ∈ G(V ) that lifts yi . Let G′ be as in 2.3.2. As w1 , ..., ws ∈ G′ (V ) = G(V ), from 4.1 (i) we get that the Zariski closure of Im(G′k (k) → Gk (k)) in (Gk )red contains ′ (Gk )0red . So (Gk )0red 6 Im(G′k → Gk ). So if G′0 k is the identity component of Gk , then we 0 have an isogeny G′0 k → (Gk )red . From this and [2, 14.11] we get that the unipotent radical ′0 of G′0 k is trivial. So Gk is a reductive group, cf. [2, 11.21]. ′ Let G′0 be the open subgroup of G′ formed by G′0 k and by GK . Let (Gj )j∈J be a covering of G′ by open affine subschemes such that each G′jk is an open subscheme of ′ ′0 ′0 ′ ′ ′ ′0 either G′0 k or Gk \ Gk . The product scheme G ×G′ Gj is: (i) Gj if Gjk ֒→ Gk , and ′0 ′ (ii) G′jK if G′jk ֒→ G′k \ G′0 k . So G ×G′ Gj is affine for any j ∈ J. Thus the morphism G′0 → G′ is affine and so G′0 is affine. So G′0 is a smooth, affine group scheme over V whose special fibre is a reductive group and whose generic fibre is connected. This implies that G′0 is a reductive group scheme over V , cf. [5, Vol. III, Exp. XIX, 2.6 and 2.7]. So GK is a reductive group. So 1.2 (a) holds. Let y ∈ G(k). From 2.6 applied to G′0 → G′ , we get that G′0 = G′ and that G′0 (V ) = G′ (V ) = G(V ). So by replacing V with a W ∈ W, we can assume there is w ∈ G(V ) that lifts y (cf. 2.5.1 (a)). So y ∈ Im(G′0 (k) → G(k)). So we have an epimorphism G′0 k ։ (Gk )red and so Gk is connected. So 1.2 (a) holds. We check 1.2 (b). From 1.1 (d) we get that the homomorphism G′0 → G is a closed embedding and so an isomorphism (as its generic fibre is so). So 1.2 (b) holds. We check that 1.2 (c) holds. The homomorphism G′0 → Gn is an isomorphism (cf. 1.1 (c)) and so Gn is a reductive group scheme. So 1.2 (c) also holds. This ends the proof of 1.2 for the case when G is of finite type.

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4.3 The general case To end the proof of 1.2 we are left to show that G is of finite type. Let I0 be the ideal of R that defines (Gk )0red . Let L be the set of all G–modules L obtained as in 2.4. For L ∈ L, let L0 , L, R(L), and ρ(L) : G → GL(L) be as in 2.4. We can identify G(L) := Spec(R(L)) with the Zariski closure in GL(L) of the image of ρ(L). As GK is smooth over K and connected, it is also of finite type. We now choose L0 such that ρ(L)K is an isomorphism (cf. 2.4 (i)) and L0 modulo I0 generates the finite type kalgebra R/I0 . So R(L) surjects onto R/I0 (cf. 2.4 (i)) and thus ρ(L)k induces a closed embedding homomorphism (Gk )0red ֒→ (G(L)k )0red between smooth, connected groups of dimension dim(GK ). So by reasons of dimensions, we get that (G(L)k )0red is a reductive group isomorphic to (Gk )0red . As in 4.2, by replacing V with some W ∈ W we can assume that for any i ∈ {1, ..., s} there is wiL ∈ G(L)(V ) that lifts yi ∈ (G(L)k )0red (k) = (Gk )0red (k). So from 4.2 applied to G(L) (instead of G), we get that the normalization G(L)n = Spec(R(L)n ) of G(L) is a reductive group scheme over V . ˜ ∈ L be such that L ⊂ L. ˜ So we have a homomorphism ρ(L, ˜ L) : G(L) ˜ → G(L) Let L ˜ 1] = ˜ L) ◦ ρ(L). So R(L) ֒→ R(L) ˜ ֒→ R ֒→ R(L)[ 1 ] = R(L)[ such that ρ(L) = ρ(L, π π ˜ L)n : G(L) ˜ n → G(L)n be the morphism defined by ρ(L, ˜ L). Let W ∈ R[ π1 ]. Let ρ(L, ˜ ˜ k )0 (k) = (Gk )0 (k). W be such that there is wiL˜ ∈ G(L)(W ) that lifts yi ∈ (G(L) red red ˜ W )n of G(L) ˜ W is a reductive group scheme over W , cf. 4.2 The normalization (G(L) ˜ W . The morphisms (G(L) ˜ W )n → G(L) ˜ n → G(L)n → G(L)W define a applied to G(L) W W ˜ L, W )n : (G(L) ˜ W )n → G(L)n between reductive group schemes over homomorphism ρ(L, W ˜ L, W )n is an isomorphism, cf. 2.6. This W whose generic fibre is an isomorphism. So ρ(L, ˜ L)n is an isomorphism. So R(L) ֒→ R(L) ˜ ֒→ R(L)n = R(L) ˜ n ֒→ R[ 1 ]. implies that ρ(L, π ˜ we have V -monomorphisms R(L) ֒→ R ֒→ R(L)n . But R(Ln ) is So as R = ∪L∈L R(L), ˜ a finite R(L)-algebra, cf. 4.2. So as R(L) is a noetherian V -algebra, we get that R is a finite R(L)-algebra and so a finitely generated V -algebra. This ends the proof of 1.2.

5

An application

¯ We take V to be the Witt ring W (k) We assume that p ∈ N is a prime and that k = k. of k. The goal of this section is to exemplify how one can use 1.1 (d) to extend results on semisimple groups over a field of characteristic 0 to results on semisimple groups over k. Let Hk be an absolutely simple, adjoint group over k. Let Qk be a parabolic subgroup of Hk different from Hk . Let Uk be the unipotent radical of Qk and let Lk be a Levi subgroup of Qk .

5.1 Proposition Proposition 5.1. Let ρk : Lk → GL(Lie(Uk )) be the representation of the inner conjugation action of Lk on Lie(Uk ). Then ρk is a closed embedding.

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Proof. Let H be the adjoint group scheme over V = W (k) that lifts Hk . Let Q be a parabolic subgroup of H that lifts Qk . Let U be the unipotent radical of Q and let L be a Levi subgroup of Q that lifts Lk . Let ρ : L → GL(Lie(U )) be the representation of the inner conjugation action of L on Lie(U ). Let T be a maximal torus of L. Let B be a Borel subgroup of G such that T 6 B 6 Q. Let Lie(H) = Lie(T ) ⊕α∈Φ gα be the Weyl decomposition of Lie(H) with respect to T . Let ∆ = {α1 , ..., αr } be the basis of the root system Φ that corresponds to B; here r ∈ N is the rank of Hk . Let ΦU be the subset of Φ such that Lie(U ) = ⊕α∈ΦU gα . For each i ∈ {1, ..., r}, there is α ∈ ΦU that is the sum of αi with an element of ΦU ∪ {0} which is a linear combination with coefficients in N ∪ {0} of elements of ∆ \ {αi }. As H is adjoint, this implies that the inner conjugation action of T on Lie(U ) is via characters of T that generate the group of characters of T . Thus the restriction of ρ to T is a closed embedding. So the identity component of Ker(ρK ) is a semisimple group over K that has rank 0. Thus Ker(ρK ) is a finite, ´etale subgroup of Z(HK ). As Z(HK ) 6 TK and as the intersection TK ∩ Ker(ρK ) is trivial, we get that Ker(ρK ) is trivial. So ρK is a closed embedding. From 1.1 (d) we get that ρ is a closed embedding, except perhaps when p = 2 and Lk has a normal subgroup that is an SO2n+1 group for some n ∈ N. So for the rest of the proof we can assume that p = 2 and that Lk has a normal subgroup Sk that is an SO2n+1 group for some n ∈ N, n ≤ r. This implies that r ≥ 2 and that Hk has a subgroup normalized by Tk and which is a P GL2 = SO3 group. So Hk is an SO2r+1 group, cf. [14, 3.8]. If Lie(Ker(ρk )) = {0}, then as in the end of 3.3 we argue that ρ is a closed embedding. So to end the proof, we only need to show that the assumption that Lie(Ker(ρk )) 6= {0} leads to a contradiction. As in 3.3 we argue that 1.1 (a) implies that Lie(Ker(ρk )) is formed by nilpotent elements. Based on 2.2.1 and its proof, we can assume that Sk is such that Lie(Sk ) ∩ Lie(Ker(ρk )) contains the ideal n of Lie(Sk ) generated by eigenspaces of the adjoint action of T on Lie(H ad ) that correspond to short roots; we have dimk (n) = 2n (cf. proof of 2.2.1). Let Uk− be the unipotent subgroup of Hk that is the opposite of Uk with respect to Tk ; so Lie(Uk− ) = ⊕α∈ΦU g−α ⊗V k. Let w0 ∈ Hk (k) be such that it normalizes both Tk and Lk and we have w0 Uk w0−1 = Uk− , cf. [3, (XI), PLATE II]. As n ⊂ Lie(Ker(ρk )), n centralizes Lie(Uk ). As n is a characteristic ideal of Sk , n is normalized by w0 and so it also centralizes Lie(Uk− ). Moreover, n is normalized by Lie(Lk ). Thus n is normalized by Lie(Hk ) = Lie(Uk ) ⊕ Lie(Lk ) ⊕ Lie(Uk− ) and so n is an ideal of Lie(Hk ). But Lie(Hk ) has a unique minimal ideal that has dimension 2r, cf. [8, (Br ) of 0.13]. Thus 2n = dimk (n) ≥ 2r ≥ 2n. Thus n = r and so Sk = Hk . This implies that Qk = Hk and so we reached a contradiction to the relation Qk 6= Hk . ¤

References [1] S. Bosch, W. L¨ utkebohmert and M. Raynaud: N´eron models, Springer-Verlag, 1990. [2] A. Borel: “Linear algebraic groups”, Grad. Texts in Math., Vol. 126, Springer-Verlag, 1991. [3] N. Bourbaki: Lie groups and Lie algebras, Springer-Verlag, 2002, Chapters 4–6.

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[4] F. Bruhat and J. Tits: “Groupes r´eductifs sur un corps local: I. Donn´ees radicielles ´ valu´ees”, Inst. Hautes Etudes Sci. Publ. Math., Vol. 41, (1972), pp. 5–251. [5] M. Demazure, A. Grothendieck and ´et al.: Sch´emas en groupes. Vol. I-III, Lecture Notes in Math., Vol. 151–153, Springer-Verlag, 1970. ´ ements de g´eom´etrie alg´ebrique. IV. Etude ´ [6] A. Grothendieck: “El´ locale des sch´emas ´ et des morphismes de sch´ema (Quatri`eme Partie)”, Inst. Hautes Etudes Sci. Publ. Math., Vol. 32, (1967). [7] G. Hiss: “Die adjungierten Darstellungen der Chevalley-Gruppen”, Arch. Math., Vol. 42, (1982), pp. 408–416. [8] J.E. Humphreys: Conjugacy classes in semisimple algebraic groups, In: Math. Surv. and Monog., Vol. 43, Amer. Math. Soc., Providence, 1995. [9] J.C. Jantzen: Representations of algebraic groups. Second edition., In: Math. Surveys and Monog., Vol. 107, Amer. Math. Soc., Providence, 2000. [10] H. Matsumura: Commutative algebra. Second edition., The Benjamin/Cummings Publ. Co., Inc., Reading, Massachusetts, 1980. [11] R. Pink: “Compact subgroups of linear algebraic groups”, J. of Algebra, Vol. 206, (1998), pp. 438–504. [12] G. Prasad and J.-K. Yu: On quasi-reductive group schemes, math.NT/0405381, 34 pages revision, June 2004. [13] A. Vasiu: “Integral canonical models of Shimura varieties of preabelian type”, Asian J. Math., Vol. 3(2), (1999), pp. 401–518. [14] A. Vasiu: “Surjectivity criteria for p-adic representations, Part I”, Manuscripta Math., Vol. 112(3), (2003), pp. 325–355.

CEJM 3(1) 2005 26–38

Quaternionic Geometry of Matroids Tam´as Hausel∗ Department of Mathematics, University of Texas at Austin, Austin TX 78712, USA

Received 5 July 2004; accepted 1 September 2004 Abstract: Building on a recent paper [8], here we argue that the combinatorics of matroids are intimately related to the geometry and topology of toric hyperk¨ahler varieties. We show that just like toric varieties occupy a central role in Stanley’s proof for the necessity of McMullen’s conjecture (or g-inequalities) about the classification of face vectors of simplicial polytopes, the topology of toric hyperk¨ahler varieties leads to new restrictions on face vectors of matroid complexes. Namely in this paper we will give two proofs that the injectivity part of the Hard Lefschetz theorem survives for toric hyperk¨ahler varieties. We explain how this implies the g-inequalities for rationally representable matroids. We show how the geometrical intuition in the first proof, coupled with results of Chari [3], leads to a proof of the g-inequalities for general matroid complexes, which is a recent result of Swartz [20]. The geometrical idea in the second proof will show that a pure O-sequence should satisfy the g-inequalities, thus showing that our result is in fact a consequence of a long-standing conjecture of Stanley. c Central European Science Journals. All rights reserved. ° Keywords: Face vectors, matroids, toric varieties, hyperk¨ ahler manifolds, Hard Lefschetz Theorem MSC (2000): 52B40, 52B05, 53C26, 14M25

∗

Email: [email protected]

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27

Introduction

McMullen [14] conjectured† in 1971 that the face vector‡ (f0 , . . . , fk−1 ) of a k-dimensional simplicial polytope P ⊂ Rk should satisfy, the following g-inequalities: gi ≥ 0, for 1 ≤ i ≤ ⌊ k2 ⌋, and, if one writes ¡ ¢ ¡ ¢ ¡ i−1 ¢ + · · · + nrr , gi = nii + ni−1

(1)

with ni > ni−1 > · · · > nr ≥ r ≥ 1, then ¡ r +1¢ ¡ i +1¢ ¡ni−1 +1¢ + · · · + nr+1 + gi+1 ≤ ni+1 i for 1 ≤ i < ⌊ k2 ⌋, where gi = hi − hi−1 and hi =

i X j=0

i−j

(−1)

µ

¶ k−j fj−1 . i−j

(2)

Stanley [17] in 1980 proved this conjecture using toric varieties. In a nutshell the proof goes as follows. First one perturbs the vertices of P a little bit so that P becomes a rational polytope. Because P is simplicial this does not change the face vector of P . The next step is to take the corresponding k-dimensional toric orbifold X(∆P ), where ∆P is the fan of cones over the faces of P . It is a well-known fact (see e.g. [6]) that the ith h-number hi = b2i (X(∆P )) agrees with the 2ith Betti number of X(∆P ). Now X(∆P ) has an ample class ω ∈ H 2 (X(∆P ), C), which induces a map L : H ∗ (X(∆P ), C) → H ∗ (X(∆P ), C), by multiplication with ω. Using the injectivity part of the Hard Lefschetz theorem (see e.g. [4]), which implies that L is an injection below degree k, we get that the degree 2ith part of the graded algebra H ∗ (X(∆P ), C)/(im(L))) has dimension dim(H 2i (X(∆P ), C)/(im(L))) = hi − hi−1 = gi

(3)

for 2i < k. Since H ∗ (X(∆P ), C) is generated by H 2 (X(∆P ), C) we also get that the algebra H ∗ (X(∆P ), C)/(im(L)) is generated in degree 2. Now, using (3), a well-known theorem of Macaulay (see e.g. [19, Theorem II.2.3]) proves the g-inequalities (1). See [6] or [19] for more details. He, in fact, conjectured a complete characterization, the sufficiency part of which was proven by an ingenious construction of Billera and Lee in [2]. ‡ fi is the number of i-dimensional faces. †

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Our starting point is the observation [8, Corollary 1.2] that the h-vectors of a rationally representable matroid MB agree hi (MB ) = b2i (Y (A, θ)) with the Betti numbers of a toric hyperk¨ahler variety Y (A, θ), for a generic choice of θ, where the toric hyperk¨ahler variety can be considered as a quaternionic analogue of a toric variety. Therefore any restriction on the cohomology of a toric hyperk¨ahler variety will yield restrictions on the face vectors of rationally representable matroid complexes and vice versa any known restriction on the face vectors of (rationally representable) matroids yields cohomological restrictions on toric hyperk¨ahler varieties. This two-way relationship between these two seemingly unrelated subjects, hyperk¨ahler geometry on one hand and combinatorics of matroids on the other, is what we call the “Quaternionic geometry of matroids”. A relationship of this flavor is exploited in a recent paper by Swartz and the author [9]. There the combinatorics of affine hyperplane arrangements yields the existence of many L2 harmonic forms on the corresponding toric hyperk¨ahler manifold, in harmony with conjectures by physicists in string theory. For details see [9]. In the present paper our purpose is to use intuition arising from the study of the geometry of toric hyperk¨ahler varieties to prove results in the combinatorics of matroids. We will proceed as follows: In Section 2 and Section 3 we recall some basic notations and results from [19] and from [8]. Then we go on and in Section 4 give two different proofs for the injectivity part of the Hard Lefschetz theorem for toric hyperk¨ahler varieties. The second one is basically taken from [19, Theorem 7.4], while the first proof could be easily generalized for other similar hyperk¨ahler manifolds, such as Nakajima’s quiver varieties [16] or Hitchin’s moduli of Higgs bundles§ [11]. In Section 5 we explain how the geometric idea in the first proof can be generalized to any matroid complexes, a result recently proven by Swartz in [20]. We show that the geometrical structure needed for the first proof for general matroids is provided by Chari’s decomposition theorem [3]. In fact this proof is similar to Swartz’s original proof in [20]. We conclude our paper by showing that the geometric structure which yielded the second proof of the injective Hard Lefschetz theorem is present for pure O-sequences. This way we find that the ginequalities we proved in the previous section are in fact a consequence of a long standing conjecture of Stanley [18]. This last result is a strengthening of a result of Hibi in [10].

2

Simplicial and matroid complexes

We collect here some basic definitions and results on simplicial complexes and in particular matroid complexes from [19]. A simplicial complex Σ on a finite set V = {1, . . . , n} is a set of subsets of V , i.e. Σ ⊂ 2V , such that {x} ∈ Σ for any x ∈ V and F ∈ Σ and F ′ ⊂ F implies F ′ ∈ Σ. We call F ∈ Σ a face of Σ, the dimension of the face is one less than its size. The dimension of Σ is then the maximum dimension of its faces, while its rank is 1 more. A facet is a A recent paper of the author [7] conjectures a strong version of the Hard Lefschetz theorem for the moduli space of Higgs bundles, generalizing the one in this paper; and also relates it to the Alvis-Curtis duality in the representation theory of finite groups of Lie-type. §

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face of maximal dimension. A simplicial complex is called pure if its maximal faces are all facets. The f -vector of a rank-k simplicial complex is (f0 , f1 , . . . , fk−1 ), where fi is the number of i-dimensional faces in Σ. The h-vector of the simplicial complex is (h0 , . . . , hk ) given by (2). Define the Stanley-Reisner ring of a rank-k simplicial complex Σ as the graded ring given by: Y C[Σ] = C[x1 , . . . , xn ]/hxF = xi |F ∈ / Σi. i∈F

All our simplicial complexes in this paper will be Cohen-Macaulay, which will imply that we will always have a linear system of parameters or l.s.o.p for short, which is a sequence (θ) = (θ1 , . . . , θk ) of linear combinations of the xi , such that the graded ring C[Σ]/(θ) := C[Σ]/(θ1 C[Σ] + · · · + θk C[Σ]) is finite dimensional as a vector space over C and that the h-numbers hi (Σ) = (C[Σ]/(θ))i agree with the dimension of the corresponding graded piece of C[Σ]/(θ). We will use the following operation on simplicial complexes in Section 5. Given two simplicial complexes Σ with vertex set V and Θ with vertex set U we define their posettheoretic product (or join) Σ × Θ as a simplicial complex with vertex set U ∪ V and all faces of the form F ∪ F ′ where F ∈ Σ and F ′ ∈ Θ. The poset-theoretic product has the advantage that it behaves nicely after taking the corresponding Stanley-Reisner rings: C[Σ × Θ] ∼ = C[Σ] ⊗ C[Θ]. For examples of (Cohen-Macaulay) simplicial complexes we mention the boundary complex of a simplicial convex polytope, which was mentioned in the introduction. Another class for interest for us are matroid complexes or simply just matroids. A matroid complex M is a simplicial complex on a vertex set V such that for every W ⊂ V the induced subcomplex MW = {F ∈ M : F ⊂ W } is pure. The rank of the matroid is 1 more than its dimension. A vertex i ∈ V is a coloop of M if MV \i has rank smaller than the rank of M. The motivating example of a matroid complex MB on vertex set V = {1, . . . , n} is obtained from a vector configuration B = (b1 , . . . , bn ) ∈ Kk in a k-dimensional vector space over a field K, defined by F ∈ M iff {bi }i∈F is linearly independent. Such a matroid is called representable over K. For example, if K = Q then we call the matroid M rationally representable. For more details on these definitions consult [19], the poset-theoretic product was used in [3].

3

Toric hyperk¨ ahler varieties

Here we collect notation and terminology from [8] which we will need in the present paper. For more details see [8]. Let A = [a1 , . . . , an ] be a d × n-integer matrix whose d × d-minors are relatively prime. We choose an n × (n−d)-matrix B = [b1 , . . . , bn ]T which makes the following sequence

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exact: B

A

0 −→ Zn−d −→ Zn −→ Zd −→ 0. Taking θ ∈ NA, where A := {a1 , . . . , an } is a vector configuration in Zd , [8] constructs a quasi-projective variety Y (A, θ) (which sometimes we abbreviate as Y ), called a toric hyperk¨ahler variety. (This construction is an algebraic geometric version of the original construction of Bielawski and Dancer in [1].) By [19, Proposition 6.2] if θ ∈ NA is generic Y (A, θ) is an orbifold, while if, in addition, A is unimodular then Y (A, θ) is a smooth variety. The topology of Y (A, θ) is governed by an affine hyperplane arrangement denoted by H(B, ψ) of n planes in Rn−d . For example a key result in [19, Corollary 6.6] claims that the h-numbers of the matroid of the vector configuration B = {b1 , . . . , bn } agree with the Betti numbers of Y : hi (MB ) = b2i (Y (A, θ)). In the next section we will make use of a projective subvariety C(A, θ) of Y (A, θ), which is called the core of Y (A, θ). It is a reducible variety whose components are projective toric varieties, corresponding to top dimensional bounded regions in H(B, ψ). If the matroid of B is coloop-free than the core is a middle and pure dimensional projective subvariety of Y (A, θ). Finally we need to mention a result from [5]. They construct and study a certain residual U (1)-action on Y (A, θ), which comes from an algebraic C× -action. It follows from their results that, when B is coloop-free, one can always choose such a circle action, which makes Y (A, θ), a hyper-compact hyperk¨ahler manifold. It means that the U (1)action is Hamiltonian with proper moment map with a minimum, and also that the holomorphic symplectic form ωC is of homogeneity 1, meaning that for λ ∈ C× λ∗ ω = λω.

(4)

For further results about the topology and geometry of toric hyperk¨ahler varieties consult the papers [1], [5], [8], [9] and [12].

4

Injective Hard Lefschetz for hyperk¨ ahler manifolds

We are now ready to give two proofs of the following Theorem 4.1. For a smooth toric hyperk¨ahler variety Y (A, θ) of real dimension 4n−4d = 4k, such that B is coloop-free, we have that Lk−2i : H 2i (Y, C) → H 2k−2i (Y, C)

(5)

Lk−2i (α) = α ∧ ω k−2i is injective if 2i < k, where ω = [ωI ] is the cohomology class of the K¨ahler form corresponding to the complex structure I.

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Just like in Stanley’s proof of the McMullen conjecture, we also have the following numerical consequences: Corollary 4.2. The h-vector (h1 (M), . . . , hk (M)) of a coloop-free and rank k matroid M, which is (unimodularly and) rationally representable, satisfies hi (M) ≤ hj (M),

(6)

for i ≤ j ≤ k − i and the g-inequalities (1). Proof of Corollary. Let the (unimodular) vector configuration B = {b1 , . . . , bn } ∈ Zk ⊂ Qk represent the matroid M. Choosing a Gale dual configuration A = (a1 , . . . , an ) ∈ Zd and a generic θ ∈ NA, we can construct a smooth toric hyperk¨ahler variety Y (A, θ), whose Betti numbers agree with the h-numbers of M. Now Theorem 4.1 immediately implies (6). From Theorem 4.1 we can also deduce (1) exactly as in Stanley’s argument for simplicial convex polytopes. See the introduction or for more details [19, Theorem III.1.1]. Proof 1 of Theorem 4.1. As explained above we have a C× -action on Y := Y (A, θ), for which the corresponding U (1) ⊂ C× -action is hyper-compact. Recall that this means that it is Hamiltonian with a proper moment µR : Y → R map with respect to ω, and for which the holomorphic symplectic form ωC is of homogeneity 1 meaning (4). Suppose that the fixed point set of the circle action has f components, which are denoted by F1 , . . . , Ff . The numbering is such that µR (Fm ) > µR (Fl ) implies m > l. Now we define the Bialynicki-Birula stratification of Y with respect to our C× -action. Namely define Um = {p ∈ Y | limλ→0 λp ∈ Fm }, which is an affine bundle over Fm . Moreover we let U≤m = ∪j≤m Uj and U<m = ∪j<m Uj , which are open subvarieties of Y . Because the moment map µR is proper it follows that U≤f = Y , i.e. that we get this way a stratification of Y . Finally we denote by Nm the negative normal bundle of Fm . Because the holomorphic symplectic form is of homogeneity 1 with respect to our C× -action, it follows (cf. [15, Proposition 7.1]) that rankC (Nm ) + dimC (Fm ) =

1 dimC Y = k. 2

(7)

By induction on m we prove that the map Lk−2i in (5), when restricted to U≤m , is injective for 2i < k. For m = 1 the statement is clear because by (4) U1 = T ∗ F1 thus dimC (F1 ) = k and the statement follows from the traditional Hard Lefschetz theorem for the compact K¨ahler manifold F1 . Now suppose we have the required injectivity of the map Lk−2i on U<m . Then consider the decomposition U≤m = U<m ∪ Um . ¿From this decomposition, using the Thom isomorphism H 2i (U≤m , U<m ; C) ∼ = H 2i−2nm (Um , C), we get the cohomology exact sequence: τ

r

0 → H 2i−2nm (Um , C) → H 2i (U≤m , C) → H 2i (U<m , C) → 0,

(8)

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where nm = rankC (Nm ), τ is the Gysin map and r is the natural restriction map on cohomology. Now suppose 2i < k and 0 6= α ∈ H 2i (U≤m , C). If r(α) 6= 0, then by induction we can deduce that Lk−2i (α) 6= 0. If r(α) = 0, then there is a β ∈ H 2i−2nm (Um , C) such that τ (β) = α. However, Um is homotopy equivalent with the smooth compact K¨ahler manifold Fm and ω|Fm is a K¨ahler class. If we denote fm = dimC Fm , then the Hard Lefschetz theorem for Fm yields that 0 6= β ∧ ω fm −2(i−nm ) = β ∧ ω k−2i+nm |Fm , because fm + nm = k by (7). Since, τ is injective we get that τ (β ∧ ω k−2i |Fm ) = α ∧ ω k−2i |U≤m 6= 0. The result follows. Because we only used the hyper-compactness of the toric hyperk¨ahler variety, the same proof also yields the following Corollary 4.3. For a hyper-compact hyperk¨ahler manifold M (such as toric hyperk¨ahler varieties, Nakajima’s quiver varieties [16] or moduli spaces of Higgs bundles [11]) we have that Lk−2i : H 2i (M, C) → H 2k−2i (M, C) Lk−2i (α) = α ∧ ω k−2i is injective if 2i < k, where ω = [ωI ] is the class of the K¨ahler form corresponding to the complex structure I. Remark 4.4. In a recent work [7] the author explains a conjecture for a strong version of the Hard Lefschetz theorem for the moduli space of Higgs bundles, which is a theorem for rank 2 Higgs bundles. This completely unexpected conjecture is a generalization of the corollary above and has some intriguing relationship with the representation theory of finite groups of Lie type. We now recall our original proof of Theorem 4.1 from [8, Theorem 7.4] in the smooth case because we will use the idea in the final section. Proof 2 of Theorem 4.1. Let X1 , . . . Xr denote the irreducible components of the core of Y . Let φi : H ∗ (Y, C) → H ∗ (Xi , C) denote the natural restrictions. The heart of the proof of [8, Theorem 7.4] is that (φ1 ) ∩ (φ2 ) ∩ . . . ∩ ker(φr )

=

{0}.

(9)

In [8] we presented two proofs of this fact. One [8, Proposition 3.4] was a more general result for semi-projective toric orbifolds and the proof goes similarly to our first Proof 1 of Theorem 4.1 above, i.e. uses Morse theory type considerations with induction. It turns out that [8, Proposition 3.4] is equivalent with the fact that the bounded complex of the polytope (or in our case the bounded complex of the affine hyperplane arrangement H(B, ψ)) is always contractible. The second proof was given after equation (34) of [8], which showed that (9) is in fact equivalent with Stanley’s result [19, Proposition III.3.2] that the Stanley-Reisner ring of a matroid is level.

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33

Now we proceed as follows. For 2i < k take α ∈ H 2i (Y, C). Then, because of (9), we have a j so that φj (α) ∈ H 2i (Xj , C) is nonzero. But the traditional hard Lefschetz theorem for the smooth compact K¨ahler manifold Xj implies that φj (α ∧ ω k−2i ) 6= 0. The result follows. Remark 4.5. [8, Theorem 7.4] proves the same result, in the way sketched above, for a rationally representable matroid, i.e. for toric hyperk¨ahler orbifolds, not just for smooth toric hyperk¨ahler varieties. Here we restricted our attention to the smooth case, because the other Proof 1 only works in this case. The reason is that (7) could be false in the orbifold case. 2. Proof 1 works for any hyper-compact hyperk¨ahler manifold, however an extension of Proof 2 in the general case is not immediate. Indeed, the equivalent of (9) perhaps in intersection cohomology is not known for a general hyper-compact hyperk¨ahler manifold. 3. Another consequence of (9), explained in [8, Section 7], is that one can present the cohomology ring of Y in terms of cogenerator polynomials corresponding to the Xi , the components of the core. Indeed this algebraic presentation is rather similar to a presentation of a pure O-sequence, the only difference will be that we replace the cogenerator polynomials by monomials. This similarity will lead to the proof of Theorem 6.3 below.

5

Proof of the g-inequalities for matroid complexes

In this section we will use the geometrical idea from our first proof of Theorem 4.1 to prove the following generalization: Theorem 5.1. The h-vector (h1 (M), . . . , hk (M)) of a coloop-free and rank k matroid M satisfies (6) and the g-inequalities (1). Remark 5.2. This was first proven by Swartz [20], by using an algebraic version of Chari’s [3] decomposition theorem of matroids. Here we will show, that [3] gives us the geometrical structure for a general matroid so that we can repeat our Morse theory type first proof of Theorem 4.1. In fact this proof is similar to Swartz’s original proof. Proof 5.3. So let us first recall Chari’s result [3, Theorem 3]: Theorem 5.4 (Chari). A coloop-free matroid complex has a PS-ear decomposition. A PS-ear decomposition of a pure rank-k simplicial complex Σ on a vertex set {1, . . . , n} is a covering by pure rank-k simplicial subcomplexes Σ = ∪m i=1 Σi , such that • Σ1 is the poset-theoretic product of boundaries of simplices (a PS-k-sphere in the terminology of [3]), while for each i = 2, . . . , m, Σi is the poset-theoretic product of a simplex and a PS-sphere (called a PS-ball in [3]), and

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¡ ¢ • For i ≥ 2, Σi ∩ ∪i−1 j=1 Σj = ∂Σi , where ∂Σi denotes the pure rank-(k − 1) simplicial complex (which is just a PS-sphere in this case) whose facets are the rank-(k − 1) faces of Σi that are contained in only one facet of Σi . We will show that Theorem 5.1 holds for simplicial complexes having a PS-ear decomposition, a result which was also mentioned by Swartz in [20]. We will see that this PS-ear decomposition is in fact a very good combinatorial analogue of the Morse stratification of Y (or rather its Lagrangian core) used in Proof 1 of Theorem 4.1. We first make a Definition 5.5. Let R be a ring and M be a graded R-module. Then we say that M satisfies injective hard Lefschetz (IHL for short) around degree k/2 for ω ∈ R1 if the map Lk−2i : Mi → Mk−i Lk−2i (α) = αω k−2i is injective for 0 < i ≤ k/2. We will proceed by induction on m to show that there is an l.s.o.p (θ1 , . . . , θk ) so that the graded ring C[Σ]/(θ) P satisfies IHL around k/2 with ω = i xi .

(10)

When m = 1, then Σ is just a poset-theoretic product of boundaries of simplices. Therefore C[Σ] can be thought of as the torus equivariant cohomology ring of a product of projective spaces, while an l.s.o.p. (θ) can be chosen so that C[Σ]/(θ) is just the cohoP mology ring of the product of projective spaces. Then ω = xi is just a K¨ahler class, so the classical Hard Lefschetz theorem proves (10). Now suppose we know our statement for m − 1 and consider a pure rank-k simplicial complex with a PS-ear-decomposition. Let us denote Σ<m = ∪m−1 j=1 Σj . Consider the natural surjective map C[Σ] → C[Σ<m ]. We think of the kernel of this map as a graded C[x1 , . . . , xn ]-module and denote it by C[Σ, Σ<m ]. So we have the following exact sequence of graded C[x1 , . . . , xn ]-modules: 0 → C[Σ, Σ<m ] → C[Σ] → C[Σ<m ] → 0. We now claim that we can find an l.s.o.p (θ) = (θ1 , . . . , θk ) for C[Σ] such that in both graded C[x1 , . . . , xn ]-modules C[Σ<m ]/(θ) and C[Σ, Σ<m ]/(θ) the IHL for ω is satisfied around degree k/2 . By induction we know that the set of (θ) which is an l.s.o.p. for C[Σ<m ] and C[Σ<m ]/(θ) satisfies IHL for ω is non-empty and clearly Zariski open in Cnk . Because the set of (θ) which is l.s.o.p. for C[Σ] is also non-empty and Zariski open, the intersection of these two sets will also be non-empty and Zariski open. In summary we see that the set of (θ) which is an l.s.o.p for C[Σ] and C[Σ<m ]/(θ) satisfies IHL for ω is non-empty and Zariski open.

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35

It is also clear that the set of (θ) which is an l.s.o.p for C[Σ] and C[Σ, Σ<m ]/(θ) satisfies IHL around degree k/2 for ω is Zariski open. We now prove that it is in fact non-empty. Take the natural map C[Σm ] → C[∂Σm ] and denote by C[Σm , ∂Σm ] the kernel. We think of this kernel as an C[x1 , . . . , xn ]-module by letting the variables xj which correspond to vertices not in Σm acting trivially. Then it is easy to see that C[Σm , ∂Σm ] and C[Σ, Σ<m ] are isomorphic as graded C[x1 , . . . , xn ]-modules (this is the analogue of excision in cohomology). But Σm = ∆ × Φ is a poset-theoretic product of a k-simplex ∆ with a poset-theoretic product of boundary of simplices Φ. Now it is clear that C[Σm , ∂Σm ] ∼ = C[Φ] ⊗ C[∆, ∂∆] as graded C[x1 , . . . , xn ]-modules (this corresponds to the Thom isomorphism (8) in cohomology). If x1 , . . . , xl correspond to the vertices of ∆ then C[∆, ∂∆] is just a free C[x1 , . . . , xl ]-module generated by a degree l element x1 x2 . . . xl (which is the analogue of the Thom class). Recall that the set of (θ) = (θ1 , . . . , θk ) ∈ (C[Σ])k1 = Cnk for which C[Σ, Σ<m ]/(θ) := C[Σ, Σ<m ]/(θ1 C[Σ, Σ<m ] + · · · + θk C[Σ, Σ<m ]) P satisfies IHL around degree k/2 for ω = ni=1 xi is clearly Zariski open in Cnk . Now we show that it is non-empty. Take (θ) = (x1 , . . . , xl , θl+1 , . . . , θk ), so that (θl+1 , . . . , θk ) is P an l.s.o.p for C[Φ] and C[Φ]/(θl+1 , . . . , θk ) satisfies IHL around (k − l)/2 with ω = xi . For this choice we have C[Σ, Σ<m ]/(θ) = x1 x2 . . . xl C[Φ]/(θl+1 , . . . , θk ), and so IHL for C[Φ]/(θl+1 , . . . , θk ) around degree (k − l)/2 implies IHL for C[Σ, Σ<m ]/(θ) P around degree k/2 with ω = xi . As the intersection of non-empty Zariski subsets of Cnk is non-empty we can choose a (θ) = (θ1 , . . . , θk ), which is an l.s.o.p for C[Σ] and C[Σ<m ] and for which both C[Σ<m ]/(θ) P and C[Σ, Σ<m ]/(θ) satisfies IHL around k/2 with ω = xi . Now using the short exact sequence: 0 → C[Σ, Σ<m ]/(θ) → C[Σ]/(θ) → C[Σ<m ]/(θ) → 0, we can repeat the argument of Proof 1 of Theorem 4.1, to get that C[Σ]/(θ) satisfies IHL P around k/2 with ω = xi . Because Σ has a PS-ear decomposition it is shellable (see [3, Proposition 5]), and so Cohen-Macaulay, we have that hi (Σ) = dimC ((C[Σ]/(θ))i ) and so Theorem 5.1 follows. Remark 5.6. Because we have Hard Lefschetz theorem for boundary complexes of simplicial convex polytopes the above proof would have worked equally well for simplicial complexes with a decomposition just like PS-ear-decomposition above, but changing P Sspheres, in the definition, with boundary complexes of simplicial convex polytopes. For a unimodularly and rationally representable matroid such a presentation always arises

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naturally. Namely we can consider the Morse stratification (for details on this see [5]) of a hyper-compact U (1)-action on the bounded complex of a generic hyperplane arrangement which represents our given matroid. In this case the above combinatorial proof of Theorem 5.1 would essentially agree with Proof 1 of Theorem 4.1.

6

Proof of the g-inequalities for pure O-sequences

First a definition: Definition 6.1. A sequence of non-negative integers (h1 , h2 , . . . , hk ) is called a pure Osequence, if hk > 0 and there exists monomials m1 , . . . , mhk of degree k in the degree one variables x1 , . . . , xh1 , so that hl is the number of monomials m of degree l in variables x1 , . . . , xh1 , such that m|mi for some 0 < i ≤ hk . Now we can state a long standing conjecture of Stanley [18]: Conjecture 6.2 (Stanley). The h-vector (h1 (M), . . . , hk (M)) of a rank k matroid M is a pure O-sequence. This conjecture is still open for general matroids, although recently it has been proved for cographic matroids using [13], i.e. for the Betti numbers of toric quiver varieties [8, Section 8]. Another attack on Stanley’s conjecture has been to deduce numerical inequalities between the numbers in a pure O-sequence and then prove these inequalities for the h-vector of a matroid complex. As an example, Hibi [10] proved that for a pure O-sequence one has hi ≤ hj ,

(11)

where i ≤ j ≤ k − i and in particular that h1 ≤ h2 ≤ · · · ≤ h⌊ k ⌋ , 2

this was in turn proven for h-vectors of matroid complexes by Chari [3]. Here we strengthen this result by proving the following Theorem 6.3. A pure O-sequence (h1 , h2 , . . . , hk ) satisfies (11) and the g-inequalities. Corollary 6.4. Theorem 5.1 is a consequence of Stanley’s Conjecture 6.2. Proof of Theorem 6.3. We are going to follow the structure of Proof 2 of Theorem 4.1. Namely take a pure O-sequence (h1 , h2 , . . . , hk ) with generating monomials m1 , . . . , mhk in variables x1 , . . . , xh1 . First we construct a graded ring R=

C[∂1 ,...,∂h1 ] I

I = ann(m1 ) ∩ · · · ∩ ann(mhk )

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37

which will be the analogue of the cohomology ring H ∗ (Y, C) of a toric hyperk¨ahler manifold. Here ∂i is a variable of degree one, which we think of as a differential operator, satisfying ∂i (xj ) = δij . The ideal in the denominator is the ideal I of polynomials in the ∂i which annihilate all the monomials mj . Clearly dim Rj = hj . Then we construct graded rings Rj =

C[∂1 ,...,∂h1 ] Ij

Ij = ann(mj ) for each monomial mj , which will be the analogue of H ∗ (Xj , C) (in fact it is useful to think about Rj as the cohomology ring of the product of projective spaces of dimension given by the exponents in the monomial mj ). Because I ⊂ Ij , we have a natural map pj : R → Rj . The equation I = ∩j Ij now implies the analogue of (4), i.e. that the map p = p 1 × · · · × p h k : R → R 1 × · · · × R hk P is injective. Now take the degree 1 class ω = j ∂j . It is clear that the map Lk−2i : j j j k−2i k−2i ) is injective for 2i < k. Indeed, think of Ri → Rk−i given by Lj (α) = αpj (ω j R as the cohomology ring of the product of projective spaces. Then pj (ω) corresponds to the natural ample class, so the hard Lefschetz theorem implies injectivity of Lk−2i . j Of course in this case one can check this result by hand for the explicitly defined rings Rj . The injectivity of p and of Lk−2i implies the injectivity of Lk−2i : Ri → Rk−i , j Lk−2i (α) = αω k−2i for 2i < k. The result follows.

7

Acknowledgment

This paper grew out from a project started with Bernd Sturmfels in [8]. Conversations with Edward Swartz were also useful. Financial support was provided by a Miller Research Fellowship at the University of California at Berkeley, and by NSF grants DMS0072675 and DMS-0305505.

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[6] W. Fulton: Introduction to Toric Varieties, Princeton University Press, New Jersey, 1993. [7] T. Hausel: “Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve”, preprint, arXiv:math.AG/0406380. [8] T. Hausel, B. Sturmfels: “Toric hyperk¨ahler varieties”, Documenta Mathematica, Vol. 7, (2002), pp. 495–534. [arXiv: math.AG/0203096] [9] T. Hausel, E. Swartz: “Intersection forms of toric hyperk¨ahler varieties”, preprint, arXiv:math.AG/0306369. [10] T. Hibi: “What can be said about pure O-sequences?”, J. Combin. Theory Ser. A, Vol. 50, (1989), pp. 319–322. [11] N. Hitchin: “The self-duality equations on a Riemann surface”, Proc. London Math. Soc., Vol. 55, (1987), pp. 59–126. [12] H. Konno: “Cohomology rings of toric hyperk¨ahler manifolds”, Internat. J. Math., Vol. 11, (2000), pp. 1001–1026. [13] C.M. Lopez: “Chip firing and the Tutte polynomial”, Ann. Combinatorics, Vol. 1, (1997), pp. 253–259. [14] P. McMullen: “The numbers of faces of simplicial polytopes”, Israel J. Math., Vol. 9, (1971), pp. 559–570. [15] H. Nakajima: Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18. American Mathematical Society, Providence, RI, 1999. [16] H. Nakajima: “Quiver varieties and finite-dimensional representations of quantum affine algebras”, J. Amer. Math. Soc., Vol. 14, (2001), pp. 145–238. [17] R. Stanley: “The number of faces of a simplicial convex polytope”, Adv. in Math., Vol. 35, (1980), pp. 236–238. [18] R. Stanley: Cohen-Macaulay complexes, in Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976), Reidel, Dordrecht, 1977, pp. 51–62. NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci., 31. [19] R.P. Stanley: Combinatorics and Commutative Algebra, 2nd ed., Birkh¨auser Boston, 1996. [20] E. Swartz: “g-elements of matroid complexes”, Journal of Comb. Theory Ser. B, Vol. 88, (2003), pp. 369–375..

CEJM 3(1) 2005 39–57

An existence result for a quadrature surface free boundary problem Mohammed Barkatou1∗ , Diaraf Seck2† , Idrissa Ly2,3‡ 1

D´epartement de Math´ematiques et Informatique, Facult´e des Sciences, Universit´e Chouaib Doukkali, B.P. 20, El Jadida, Maroc 2 Facult´e des Sciences Economiques et de Gestion, Universit´e Cheikh Anta Diop, B.P. 5683, Dakar, S´en´egal 3 Laboratoire d’Analyse Num´erique et d’Informatique, Universit´e Gaston Berger, B.P. 234, Saint-Louis, S´en´egal

Received 21 May 2004; accepted 24 October 2004 Abstract: The aim of this paper is to present two different approachs in order to obtain an existence result to the so-called quadrature surface free boundary problem. The first one requires the shape derivative calculus while the second one depends strongly on the compatibility condition of the Neumann problem. A necessary and sufficient condition of existence is given in the radial case. c Central European Science Journals. All rights reserved. ° Keywords: Dirichlet problem, Neumann problem, quadrature surface, maximum principle, shape optimization, optimality conditions MSC (2000): 35J05, 35A35

1

Introduction and main theorems

Assuming throughout that: D ⊂ RN (N ≥ 2) is a bounded ball which will contain all the domains we will use. If ω is an open subset of D, let ν be the outward normal to ∂ω and |∂ω| (resp. |ω|) be the perimeter (resp. the volume) of ω. Let k > 0 and f be ¡ ¢ a positive function in L2 RN with a non-empty compact support Suppf . By uω we ∗

† ‡

E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

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denote the solution of the Dirichlet problem:    −∆uω = f in ω, P (ω, f )   uω = 0 on ∂ω.

Consider the following free boundary problem:    Find an open subset Ω of D which contains strictly Suppf        and a function uΩ ∈ H01 (Ω) such that:      (QS) −∆uΩ = f in Ω         uΩ = 0 on ∂Ω        ∂uΩ   −  = k on ∂Ω (overdetermined condition). ∂ν This problem is known as the quadrature surface free boundary problem and arises in many areas of physics (free streamlines, jets, Hele-show flows, electromagnetic shaping, gravitational problems etc.) It has been intensively studied from different points of view, by several authors. In [17], A. Henrot used the method of subsolutions and supersolutions introduced by A. Beurling [6]. In [16], the method used by B. Gustafsson and H. Shahgholian consists of minimizing some functional. This method goes back to K. Friedrichs [14], or even to T. Carleman [10], and was developed by H. W. Alt and L. A. Caffarelli [1] (for more details about the methods used for solving this problem see the introduction in [16]). In [27], by using the moving plane method [15], the author shows that if the problem (QS) admits a solution (Ω, uΩ ) such that Ω is of class C 2 and uΩ ∈ C 2 (Ω), then all the inward normal rays to Ω meet C (the convex hull of Suppf ). Since we relate the existence of a solution for Problem (QS) to the existence of a minimum of some shape optimization problem, it is natural to resolve this one in a class of domains with this geometric normal property (see below). In [3], we study bounded domains with the property that we denote C-GNP (Geometric Normal Property w.r.t C). Namely, for a given compact convex set C, the bounded domain ω saisfies C-GNP iff (1) ω ⊃ int(C), (2) ∂ωÂC is locally Lipschitz, (3) for every x ∈ ∂ωÂC the inward normal ray to ω (if exists) meets C, (4) for any c ∈ ∂C there is an outward normal ray ∆c such that ∆c ∩ ω is connected. Remark 1.1. In [3], we prove that the boundary of a domain ω which satisfies C-GNP has a uniform cone property outside C. Moreover, even though cusps can be formed at the points of ∂ω ∩∂C (One can consider, in two dimensions, the convex C = [−1, 1]×{0} and

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the domain ω = B(−1, 1)∪B(1, 1), observe that ω satisfies C-GNP but it isn’t Lipschitz). It is then shown that these cusps are not sharper than (i.e. contain) a canonical cusp (which is obtained by revolving the cusp between two touching circles of large radius around its axis). In particular, this implies that every point of ∂ω is regular for the Dirichlet problem, as one can easily verify the Wiener citerion. Set OC = {ω ⊂ D : ω satisfies C-GNP} , ½ Z ¾ 1 Of,k = ω ∈ OC : |∂ω| ≤ f , k ω ½ Z ¾ 1 f . Oconv = ω ⊂ D : ω is convex and |∂ω| ≥ k ω Consider the following shape optimization problems:    Find Ω ∈ OC such that J1 (Ω) = min J1 (ω), ω∈OC (OP )   J1 (ω) = − 1 R |∇uω (x)|2 dx + k2 |ω| . 2 ω 2 ½ (OP )∗ Find Ω∗ ∈ Of,k such that J1 (Ω∗ ) = min J1 (ω). ω∈Of,k

(OP )∗∗

   Find Ω∗∗ ∈ Oconv such that J2 (Ω) = min J2 (ω), ω∈Oconv R  ∂u  J2 (ω) = k |∂ω| + ω . ∂ω ∂ν

(uω is the solution of the Dirichlet problem P (ω, f ).) Our aim here is to prove the following theorems.

Theorem 1.2. Suppose that N = 2 or 3, then the following hold : (1) There exist Ω ∈ OC and uΩ ∈ H01 (Ω) such that:    J1 (Ω) = min J1 (ω), and ω∈OC

  −∆uΩ = f in Ω, uΩ = 0 on ∂Ω.

(2) Let uC be the solution of P (C, f ) which we assume in C 2 (C). Suppose that Ω is of Ω class C 2 , uΩ ∈ C 2 (C) and ∂u is continuous on ∂Ω. If C satisfies an interior ball ∂ν condition and ∂uC > k on ∂C, (1) − ∂ν ∂uΩ then C is strictly contained in Ω and − = k on ∂Ω. ∂ν Remark 1.3. The optimality conditions given in the proof of Theorem 1.2, thanks to the tool of domain derivative, are very classical (e.g. [24] or [28]). The main difficulty

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in this part would be to prove that the minimizer we got is regular enough in order to perform such derivative calculus. In general, for the convergences we use in this paper, the limit domain is only Lipschitz even if the sequences of domains approaching it are of class C 2 . In [5] (see also [24]), the authors got, for a Lipschitz domain Ω, the following derivative for J1 : Z ©¡ ¢ ª dJ1 (Ω, V ) = div k 2 − |∇uΩ |2 V dx Ω

where V is any admissible displacement.

The aim of the following theorem is to prove the existence of a minimum of J1 which is of class C 2 . This is done in order to use the shape derivative and so to resolve Problem (QS). Theorem 1.4. Let L be a compact subset of RN . Let fn be a sequence of functions defined on L. We assume that the fn are of class C 3 and ¯ ¯ ¯ ¯ ¯ 2 ¯ 3 ¯ ∂fn ¯ ¯ ¯ ¯ ¯ ¯ ¯ ≤ M, ¯ ∂ fn ¯ ≤ M, ¯ ∂ fn ¯ ≤ M, ¯ ∂xi ¯ ¯ ∂xi ∂xj ¯ ¯ ∂xi ∂xj ∂xk ¯

where M is a strictly positive constant and is independent of n. Define a sequence Ωn , by Ωn = {x ∈ L : fn (x) > 0} and suppose there exists α > 0 such that |fn (x)| + |∇fn (x)| ≥ α for all x in L. If the Ωn have the C-GNP, then there exists Ω of class C 2 and a subsequence (still denoted by Ωn ) such that Ωn converges in the compact sense, to Ω (see Definition 2.3 below) and J1 (Ω) = min J1 (ω). ω∈OC

Theorem 1.5. Suppose that N = 2 or 3, then the following hold : (1) There exist Ω∗ ∈ Of,k and uΩ∗ ∈ H01 (Ω∗ ) such that: J1 (Ω∗ ) = min J1 (ω). ω∈Of,k

(2) If f ∈ C 1 (Ω∗ ) and − ∂u∂νΩ∗ ≤ k on ∂Ω∗ , then −∆uΩ∗ = f in Ω∗ , uΩ∗ = 0 and −

∂uΩ∗ = k on ∂Ω∗ . ∂ν

(3) Let uC be the solution of P (C, f ). Suppose that uC and uΩ∗ are in C 2 (C). If C satisfies an interior ball condition, then Condition (1) implies that C is strictly contained in Ω∗ . Theorem 1.6. Suppose that N ≥ 2, then the following hold : (1) There exist Ω∗∗ ∈ Oconv and uΩ∗∗ ∈ H01 (Ω∗∗ ) such that: J2 (Ω∗∗ ) = min J2 (ω). ω∈Oconv

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Ω∗∗ ≤ k on ∂Ω∗∗ , then (2) If f ∈ C 1 (Ω∗∗ ) and − ∂u∂ν

−∆uΩ∗∗ = f in Ω∗∗ , uΩ∗∗ = 0 and −

∂uΩ∗∗ = k on ∂Ω∗∗ . ∂ν

(3) Let uC be the solution of P (C, f ). Suppose that uC and uΩ∗∗ are in C 2 (C). If C satisfies an interior ball condition, then Condition (1) implies that C is strictly contained in Ω∗∗ . (4) If Ω∗∗ is of class C 2 , then H∂Ω∗∗ =

f on ∂Ω∗∗ (N − 1)k

where H∂Ω∗∗ is the mean curvature of Ω∗∗ . Theorem 1.7. If f is radially symmetric such that Suppf is some closed ball of radius R, then the corresponding free boundary problem (QS) has a solution if and only if Z

R

sN −1 f (s) ds > kRN −1 . 0

Remark 1.8. We would like to say that the minimum obtained in Theorem 1.2, Theorem 1.4, or Theorem 1.5, or Theorem 1.6 is a solution of Problem (QS). It is not so simple. In general, without any assumptions on f and k, Problem (QS) does not have a solution: Let (Ω, uΩ ) be a solution of the free boundary problem (QS), then Z Z −∆uΩ = f. Ω

Ω

By the Green formula we obtain Z

∂uΩ − = ∂ν ∂Ω

Z

f. Ω

Since

∂uΩ = k, ∂ν we deduce the following necessary condition of existence: Z f (x) dx = k|∂Ω|, −

Ω

which shows that if f has a too small total mass or if k is too large, the perimeter of Ω will not be large enough so that Ω includes C. In such a case, the minimum that we find comes to intersect the convex C, i.e. ∂Ω and ∂C have a common part and so Ω − ∂u ≤ k on ∂Ω ∩ ∂C (see Step 2. in the proof of Theorem 1.2). The condition (1) ∂ν given in Theorem 1.2, implies that ∂Ω ∩ ∂C = ∅. We refer the interested reader to [16] where other sufficient conditions of existence are obtained by using different techniques (see also Remark 9.7 in the end of this paper).

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The resolution of the shape optimization problems (OP ) and (OP )∗ is due to Theorems 2.6 and 2.7 given in Section 2. To resolve (OP )∗∗ , we use the fact that a convex set is uniformly Lipschitz (see [25]) and conclude with Lemma 2.11 (see Section 2). To prove 2. of Theorem 1.2 we use the shape derivative and obtain the overdetermined condition but not in the entire boundary ∂Ω. Then, by applying the maximum principle to uΩ and uC , the condition (1) implies that C is strictly contained in Ω and −

∂uΩ = k on ∂Ω. ∂ν

The aim of Theorems 1.4 and 1.5 is to resolve Problem (QS), without using the shape derivative. For the demonstrations, we use the maximum principle and the compatibility condition for the Neumann Problem (see Theorem 2.8). Two examples where the condition (1) is explicit are given in Section 7. The second one concerns the case where f is radially symmetric for which we obtain the inequality stated in Theorem 1.7 as a sufficient condition of existence for Problem (QS). Then, using the isoperimetric inequality it is shown that this condition is also a necessary one.

2

Preliminary results

This section is devoted to some definitions and results which will be useful to prove Theorems 1.2, 1.4 and 1.5 stated above. Definition 2.1. Let K1 and K2 be two compact subsets of D. We call a Hausdorff distance of K1 and K2 (or briefly dH (K1 , K2 )), the following positive number: dH (K1 , K2 ) = max [ρ(K1 , K2 ), ρ(K2 , K1 )] , where ρ(Ki , Kj ) = maxd(x, Kj ) i, j = 1, 2 and d(x, Kj ) = min |x − y| . x∈Ki

y∈Kj

Definition 2.2. Let ωn be a sequence of open subsets of D and ω be an open subset of D. Let Kn and K be their complements in D. We say that the sequence ωn converges in H the Hausdorff sense, to ω (or briefly ωn −→ ω) if lim dH (Kn , K) = 0.

n→+∞

Definition 2.3. Let ωn be a sequence of open subsets of D and ω be an open subset of D. K We say that the sequence ωn converges in the compact sense, to ω (or briefly ωn −→ ω) if • every compact subset of ω is included in ωn , for n large enough, and • every compact subset of ω c is included in ω cn , for n large enough. Definition 2.4. Let ωn be a sequence of open subsets of D and ω be an open subset of D. We say that the sequence ωn converges in the sense of characteristic functions, to ω

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L

(or briefly ωn −→ ω) if χωn converges to χω in Lploc (RN ), p 6= ∞, (χω is the characteristic function of ω). Lemma 2.5. ([8], [25]) If ωn is a sequence of open subsets of D, there exists a subsequence (still denoted by ωn ) which converges, in the Hausdorff sense, to some open subset of D. Theorem 2.6. If ωn ∈ OC , then there exists an open subset ω ⊂ D and a subsequence H K (again labeled ωn ) such that (i ) ωn −→ ω, (ii) ωn −→ ω, (iii) χωn converges to χω in L1 (D) and (iv) ω ∈ OC . For the proof of this theorem, see Theorem 3.1 in [3]. H

Theorem 2.7. Let {ωn , ω} ⊂ OC such that ωn −→ ω. Let un and uω be respectively the solutions of P (ωn , f ) and P (ω, f ). Then un converges strongly in H01 (D) to uω (un and uω are extended by zero in D). The second theorem is proven for N = 2 or 3 (see Theorem 4.3 in [3]). Theorem 2.8. (Theorem 12. in [23]) If f ∈ C 1 (ω), the Neumann problem    −∆vω = f in ω N (ω) ∂v   − ω = k on ∂ω, ∂ν has a solution if and only if the following compatibility condition holds: Z 1 |∂ω| = f. k ω In this case, there are infinitely many solutions differing by a constant.

Theorem 2.9. ([11]) If ωn is a sequence of uniformly Lipschitz domains, then there exists a domain ω which is uniformly Lipschitz and a subsequence (again denoted by ωn ) such that H (1) ωn −→ ω, L (2) ωn −→ ω, K (3) ωn −→ ω, and (4) un −→ uω strongly in H01 (D) , (un and uω are extended by 0 to D). Proposition 2.10. If χωn converges to χω in L1 (RN ), then (1) |ω| = lim |ωn | , and n→+∞

(2) |∂ω| ≤ lim inf |∂ωn | . n→+∞

The proof of this proposition is obvious and can be found in any book of measure theory, so it is omited here.

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Lemma 2.11. Let ωn , ω ⊂ D be convex with ωn −→ ω. Then |∂ωn | −→ |∂ω| . For the proof of this lemma see 4.4 of [9].

3

Proof of Theorem 1.2

Proof. Step 1. Resolution of Problem (OP ). Let uD and uω respectively denote the solution of P (D, f ) and P (ω, f ) . The maximum principle implies that 0 ≤ u ω ≤ uD . By using the variational formulation of P (ω, f ): Z Z 2 |∇uω (x)| dx = f uω χω . ω

D

R

Then J1 (ω) ≥ − D f uD and inf J1 exists. Let Ωn be a minimizing sequence in OC . Now according to Theorem 2.6, there exists a subsequence (still denoted by Ωn ) and H Ω ⊂ D such that Ωn −→ Ω and Ω ∈ OC . Then, Theorem 2.7 together with (iii) of Theorem 2.6 implies that Z Z Z Z 2 |∇uΩn (x)| dx = f un χΩn −→ f uΩ χΩ = |∇uΩ (x)|2 dx. Ωn

D

D

Ω

Hence, J1 (Ω) ≤ lim inf J1 (Ωn ). n→+∞

Step 2. Optimality conditions. We assume that the minimum Ω of the functional J1 is of class C 2 and uΩ ∈ H 2 (Ω) . Let ¢ ¡ us consider a deformation field V ∈ C 2 RN ; RN . Then, the classical Hadamard formula yields for the derivative of J1 with respect to the displacement V (or in the direction V ): ¶2 Z Z µ Z k2 1 ∂uΩ dJ1 (Ω; V ) = V · ν dσ − V · ν dσ − ∇uΩ .∇u0 (x)dx (2) 2 ∂Ω 2 ∂Ω ∂ν Ω where ν is the outward normal vector to ∂Ω and u0 the derivative of uΩ wich can be defined as the solution of the following problem:    −∆u0 = 0 in Ω (3)  u0 = − ∂uΩ V · ν on ∂Ω.  ∂ν

Now, using the Green formula, we can evaluate dJ1 (Ω; V ) by  ¶2 Z Z µ  k2 1 ∂uΩ   dJ1 (Ω; V ) = V · ν dσ − V · ν dσ Z 2 ∂Ω Z 2 ∂Ω 0 ∂ν ∂u    V · ν dσ. + uΩ ∆u0 (x)dx − uΩ ∂ν Ω ∂Ω

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According to (2) and (3) , k2 dJ1 (Ω; V ) = 2

Z

1 V · ν dσ − 2 ∂Ω

Z

∂Ω

µ

∂uΩ ∂ν

¶2

V · ν dσ.

Now since Ω is the minimum for the functional J1 , dJ1 (Ω; V ) ≥ 0 for every admissible displacement V. Therefore µ ¶2 ! Z Ã ∂u Ω k2 − V · ν dσ ≥ 0 for every admissible displacement V. ∂ν ∂Ω (We mean by admissible displacement the one which allows us to keep C-GNP). Let us denote by Nx the inward normal at some point x of ∂Ω. One can see that ∂Ω is decomposed as follows: ∂Ω = Γ0 ∪ Γ1 ∪ Γ where    Γ0 = ∂Ω ∩ C    Γ1 = {x ∈ ∂ΩÂC such that Nx is tangent to ∂C}      Γ = ∂ΩÂ (Γ0 ∪ Γ1 ) .

On Γ, all displacements V are admissible, so by using V (resp. −V ) and the density of the functions V · ν in L2 (Γ), one derives −

∂uΩ = k on Γ. ∂ν

(4)

In the same way, the admissible displacements V on Γ0 must satisfy V · ν ≥ 0. Hence, we get ∂uΩ − ≤ k on Γ0 . (5) ∂ν Lastly, if γ is some connected component of Γ1 as admissible displacement, one can take V such that for all x ∈ γ, V (x)·(x) = φ(x) (where φ is increasing or decreasing, according to the position of γ with respect to C). The optimality condition becomes ¶2 ! µ Z Ã ∂u Ω k2 − φ dσ ≥ 0. (6) ∂ν γ Ω = k on ∂Ω. Step 3. Ω contains strictly C and − ∂u ∂ν

∂Ω 6= ∂C, otherwise Ω = int(C) and uΩ = uC . But (5) gives −

∂uC ∂uΩ =− ≤ k on ∂C, ∂ν ∂ν

which is absurd. Now, suppose that ∂Ω ∩ ∂C 6= ∅. As uΩ and uC are in C 2 (C) , ∆uΩ = −f = ∆uC in int(C) and uΩ ≥ 0 = uC on ∂C,

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the maximum principle implies that uΩ ≥ uC in int(C). But uΩ 6= uC in int(C), then

uΩ > uC in int(C).

Now, since C satisfies the interior ball condition and uΩ = uC on ∂Ω ∩ ∂C, the maximum principle gives ∂uC ∂uΩ − <− on ∂Ω ∩ ∂C, ∂ν ∂ν Ω ≤ k on ∂Ω ∩ ∂C, then But − ∂u ∂ν

−

∂uC < k on ∂Ω ∩ ∂C, ∂ν

which is absurd. It then follows that C is strictly contained in Ω and thus, by (4) , −

∂uΩ = k on ∂ΩÂΓ1 . ∂ν

Now, ∂Ω 6= Γ1 , otherwise ∂C is the evolute to ∂Ω and so ∂Ω ∩ ∂C = 6 ∅, which contradicts ∂uΩ the fact that C is strictly contained in Ω. As we assume that ∂ν is continuous on ∂Ω, −

∂uΩ = k on ∂Ω. ∂ν ¤

Remark 3.1. In [4], using Steiner continuous symmetrization, the shape derivative and the maximum principle, we prove that if a ≥ 3.26k, then there exists an open and bounded set Ω in R2 which contains strictly the segment C (C = [−1, 1] × {0}) and such that the following overdetermined problem has a solution. −∆uΩ = aδC in Ω, uΩ = 0 and −

∂uΩ = k on ∂Ω, ∂ν

where δC denotes a uniform density supported by C, and a and k are two strictly positive constants. Observe that this condition is better than the one obtained in [16] (a > 24πk) for the same problem.

4

Proof of Theorem 1.4

Before proving this theorem, let us state and prove the following lemma.

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Lemma 4.1. Let L be a compact subset of RN . Let fn be a sequence of functions defined as in Theorem 1.4. Suppose that Ω is an open subset of L such that Ω = {x ∈ L : h(x) > 0} and

∂Ω = {x ∈ L : h(x) = 0} ,

where h is a continuous function defined in L. If the fn converge uniformly to h in L, then the Ωn converge in the compact sense, to Ω. Proof. (1) Let K1 be a compact subset of Ω. If β1 = inf h, β1 > 0 and there exists K1

n1 ∈ N such that for all n ≥ n1 , |fn − h|L∞ (K1 ) < β1 . This implies that for all x ∈ K1 , fn (x) > h(x) − β1 ≥ 0 and then K1 is contained in Ωn , for n ≥ n1 . c (2) Let K2 be a compact subset of Ω . By hypothesis, Ω = Ω∪∂Ω = {x ∈ L : h(x) ≥ 0} . If β2 = maxh, β2 < 0 and there exists n2 ∈ N such that for all n ≥ n2 , |fn − h|L∞ (K2 ) K2

< −β2 . This implies that for all x ∈ K2 , fn (x) < h(x) − β1 ≤ 0 and then K1 is c c contained in Ωn , for n ≥ n2 because {x ∈ L : h(x) < 0} ⊂ Ωn . Proof of Theorem 1.4 : Let C 2 (L) be the set of all functions of class C 2 defined on L. Consider the norm on C 2 (L) , defined by X X ∂g ∂2g ∞ + |g| := |g|L∞ + | ∂x | | ∞. | L ∂xi ∂xj L i

We will apply the Ascoli theorem to the set B = {fn }(n∈N) . We can easily verify that {fn } is bounded and closed. By the Mean Value Theorem, we show that {fn } n o n 2 o ∂fn ∂ fn .) B is is equicontinuous (We can obtain the same results for ∂xi and ∂xi ∂xj

compact, so there exist a function h ∈ C 2 (L) and a subsequence of fn (still labled fn ) such that fn converges to h. Now, let Ω = {x ∈ L : h(x) > 0} , and show that ∂Ω = {x ∈ L : h(x) = 0} . ∂Ω ⊂ {x ∈ L : h(x) = 0} is always true, it remains to show the other inclusion. Let x ∈ L such that h(x) = 0. As |fn (x)| + |∇fn (x)| ≥ α for all x in L, |∇h (x)| ≥ α > 0. Consider the function φ : R −→ R, φ(t) = h(x + t∇h(x)) with φ(0) = 0 and 0 φ (0) = |∇h (x)|2 . For xn = x+ n1 ∇h(x), φ( n1 ) > 0 and there exists n0 ∈ N such that for all n ≥ n0 , xn ∈ Ωn and xn converges to x ∈ Ω. This implies that x ∈ ∂Ω, that is {x ∈ L : h(x) = 0} ⊂ ∂Ω. It then obvious that Ω is of class C 2 . Now, according to Lemma 4.1, the Ωn converge in the compact sense, to Ω. Since Ωn have the C-GNP, Theorem 2.6 implies that Ωn converge in the Hausdorff sense, to Ω and Ω has the C-GNP. The same reasoning as in Theorem 1.2 allows us to get J1 (Ω) = min J1 (ω). ω∈OC

¤

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Proof of Theorem 1.5

Proof. Step 1. Resolution of (OP )∗ . According to Theorem 1.2, inf J1 exists. Let Ω∗n be a minimizing sequence in Of,k . By Theorem 2.6, there exists a subsequence (again labeled Ω∗n ) and an open subset Ω∗ ∈ OC H such that Ω∗n −→ Ω∗ . To prove that Ω∗ ∈ Of,k , it remains to verify that Z 1 ∗ f. |∂Ω | ≤ k Ω∗ (iii) of Theorem 2.6 implies that Z

lim

n→+∞

f= Ω∗n

Z

f. Ω∗

From (iii) of Theorem 2.6 and Proposition 2.10, we have |∂Ω∗ | ≤ lim |∂Ω∗n | . n→+∞

But since

|∂Ω∗n |

≤

1 k

R

Ω∗n

f, then 1 |∂Ω | ≤ k ∗

Z

f. Ω∗

Let u∗n be the solution of the Dirichlet problem P (Ω∗n , f ). Theorem 2.7 implies that u∗n converges strongly in H01 (Ω∗ ) to uΩ∗ the solution of P (Ω∗ , f ). This together with (iii) of Theorem 2.6, gives J1 (Ω∗ ) = lim inf J1 (Ω∗n ). n→+∞

Step 2.

− ∂u∂νΩ∗

= k on ∂Ω∗ .

Since − ∂u∂νΩ∗ ≤ k on ∂Ω∗ ,

∗

k |∂Ω | ≥

Z

∂Ω∗

³

´ − ∂u∂νΩ∗ dσ.

But as uΩ∗ is the solution of the Dirichlet problem P (Ω∗ , f ), by the Green formula Z Z ³ ´ ∂uΩ∗ f. − ∂ν dσ = Ω∗

∂Ω∗

Therefore

1 |∂Ω | ≥ k ∗

Z

f, Ω∗

which asserts the equality. Now, according to Theorem 2.8, there exists vΩ∗ (∈ H 1 (Ω∗ )) a solution to the Neumann problem N (Ω∗ ). But this implies that ∆uΩ∗ = −f = ∆vΩ∗ in Ω∗ , and thus uΩ∗ = vΩ∗ − hΩ∗ (hΩ∗ is harmonic on Ω∗ ). This together with the fact that uΩ∗ is the solution of P (Ω∗ , f ) and the Green formula, gives −∆uΩ∗ = f in Ω∗ , uΩ∗ = 0 and −

∂hΩ∗ ∂uΩ∗ =k+ on ∂Ω∗ . ∂ν ∂ν

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51

As − ∂u∂νΩ∗ ≤ k on ∂Ω∗ , Since hΩ∗

∂hΩ∗ ≤ 0 on ∂Ω∗ . ∂ν is harmonic on Ω∗ , using the Green formula Z

∂Ω∗

∂hΩ∗ = 0. ∂ν

Hence

∂hΩ∗ = 0 on ∂Ω∗ . ∂ν Step 3. C is strictly contained in Ω∗ . This part of the demonstration is the same as in Step 3. of the proof of Theorem 1.2, when we replace Ω by Ω∗ and uΩ by uΩ∗ . ¤

6

Proof of Theorem 1.6

Proof. Step 1. Resolution of (OP )∗∗ . As uω is the solution of the Dirichlet problem P (ω, f ), the Green formula gives Z Z ¡ ∂uω ¢ − ∂ν dσ = f. ∂ω

ω

Therefore

J2 (ω) = k |∂ω| −

Z

f. ω

∀ ω ∈ Oconv , J2 (ω) ≥ 0, so inf J2 = 0. Let Ω∗∗ n be a minimizing sequence in Oconv . As ∗∗ Ωn ⊂ D, by Lemma 2.5 there exists a subsequence (again labeled Ω∗∗ n ) and an open subset ∗∗ ∗∗ H ∗∗ Ω ⊂ D such that Ωn −→ Ω . Then it is simple to see that the compact convergence ∗∗ (3. of Theorem 2.9) and the fact that the Ω∗∗ is also convex. n are convex imply that Ω ∗∗ To prove that Ω ∈ Oconv , it remains to verify that Z 1 ∗∗ f. |∂Ω | ≥ k Ω∗∗ 1. of Theorem 2.9 implies that lim

n→+∞

Z

f= Ω∗∗ n

Z

f. Ω∗∗

From 2. of Theorem 2.9 and Lemma 2.11, we have |∂Ω∗∗ | = lim |∂Ω∗∗ n |. n→+∞

Therefore, J2 (Ω∗∗ ) = lim J2 (Ω∗∗ n ) = 0. n→+∞

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Hence

1 |∂Ω | = k ∗∗

Ω∗∗ Step 2. − ∂u∂ν = k on ∂Ω∗∗ .

Z

f. Ω∗∗

According to 3. and 4. of Theorem 2.4, we have −∆uΩ∗∗ = f in Ω∗∗ and uΩ∗∗ = 0 on ∂Ω∗∗ .

Now, as in the proof of Theorem 1.5, the compatibility condition of N (Ω∗∗ ) and the fact Ω∗∗ that − ∂u∂ν ≤ k on ∂Ω∗∗ imply that

∂uΩ∗∗ = k on ∂Ω∗∗ . ∂ν Step 3. C is strictly contained in Ω∗∗ . −

As in the previous proof. Step 4. H∂Ω∗∗ =

f (N −1)k 2

on ∂Ω∗∗ .

As Ω∗∗ is of class C , we can compute the domain derivative of J2 and obtain Z [k(N − 1)H∂Ω∗∗ − f ] V · ν dσ ≥ 0 ∂Ω∗∗

for every admissible displacement V. Since C is strictly contained in Ω∗∗ , if we use V and −V and the density of the functions V · ν in L2 (∂Ω∗∗ ) then we obtain f H∂Ω∗∗ = on ∂Ω∗∗ . (N − 1)k ¤ Remark 6.1. If f = χC then H∂Ω∗∗ = strictly C.

1 (N −1)k

on ∂Ω∗∗ and Ω∗∗ is a ball which contains

Remark 6.2. Let f ≡ 1 in RN . There exists a ball B which minimizes J2 on Oconv and B ≤ k on ∂B, then a function uB ∈ C 2 (B) such that if − ∂u ∂ν    −∆uB = 1 in B    (SP ) uB = 0 on ∂B      − ∂uB = k on ∂B. ∂ν Conversely, if the problem (SP ) admits a solution uB ∈ C 2 (B) and if B is convex, then B is a minimum of J2 on Oconv and therefore B is a ball, proving the result of J. Serrin [26]. Remark 6.3. Since Ω, Ω∗ and Ω∗∗ are solutions to Problem (QS) and Ω∗∗ is convex, if Suppf ⊂ Ω∩ Ω∗∗ (resp. Suppf ⊂ Ω∗ ∩ Ω∗∗ ) then Ω ⊂ Ω∗∗ (resp. Ω∗ ⊂ Ω∗∗ ) (see [18], [16] or [27]).

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7

53

Examples

In this section we give two explicit examples, where the convex C is some closed ball B(0, R). Example 1. Let EN be the fundamental solution of −∆ in RN . It is given by   1  EN (x) = − 2π log |x| for N = 2         EN (x) =

1 1 kN |x|N −2

for N ≥ 3,

(2(2−N )πN/2 ) where kN = . Γ(N/2) We know that the solution of the Dirichlet problem P (B(0, R), f ) is given by (see for instance [12]), ¶¸ · µ Z |y| x Ry − f (y) dy uC (x) = EN (x − y) − EN R |y| B(0,R) and ∂uC − (x) = cN ∂ν

where cN

Z

R2 − |y|2

R |x − y|N is a positive constant depending on N. B(0,R)

f (y) dy for |x| = R,

The condition(1) of Theorem 1.2, becomes Z R2 − |y|2 cN f (y) dy > k for N B(0,R) R |x − y|

|x| = R.

(7)

Example 2. Assume that f is radially symmetric. Then, the Dirichlet problem P (B (0, R) , f ) becomes, N −1 0 uC = f (r) for r ∈ ]0, R[ , r uC (R) = 0. ¡ N −1 0 ¢0 By the first equation, r uC = −rN −1 f (r) . As uC (R) = 0, we get Z R N −1 0 N −1 0 r uC (r) = R uC (R) + sN −1 f (s) ds. 00

−uC −

r

0

As r → 0, we shall have r N −1 uC (r) → 0 (otherwise we get a distributional contribution to ∆uC at the origin). Thus Z R 1 0 sN −1 f (s) ds, −uC (R) = N −1 R 0 and the condition(1) of Theorem 1.2, is equivalent to Z R sN −1 f (s) ds > kRN −1 . 0

(8)

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Proof of Theorem 1.7

Proof. According to Example 2. the inequality given in Theorem 1.7 is a sufficient condition of existence of a solution for Problem (QS). It remains now to prove that it is a necessary one. If (Ω, uΩ ) is a solution to Problem (QS), then by the Green formula, Z f (x) dx = k |∂Ω| . B(0,R)

According to the isoperimetric inequality, we have Z N −1 f (x) dx ≥ kN (VN )1/N |Ω| N , B(0,R)

where VN denotes the volume of the unit ball. As B (0, R) ⊂ Ω, we get Z

B(0,R)

¯ ¯ N −1 f (x) dx > N k (VN )1/N ¯B (0, R)¯ N .

Now, using polar coordinates (N = 2), and spherical ones (N = 3), we deduce Z R sf (s) ds > kR, 0

and

9

Z

R 0

2 2 s2 f (s) ds > k √ R > kR2 . 3 3 ¤

Final remarks

Remark 9.1. The shape optimization result obtained in Theorem 1.2 can be established for a large class of domain functionals. Let F : D × R × RN −→ R be continuous in (r, p) ∈ R × RN and satisfying ¡ ¢ ∀ (x, r, p) ∈ D × R × RN , |F (x, r, p)| ≤ c a(x) + r 2 + |p|2 where c is a constant and a(x) is a function in L1 (D) . Consider the domain functional Z j (ω) = F (x, uω (x) , ∇uω (x)) dx ω

where uω is the solution of the Dirichlet problem P (ω, f ). We can show the following theorem Theorem 9.2. If inf j > −∞, then the following shape optimization problem has a solution: Find Ω ∈ OC such that j(Ω) = min j (ω) . ω∈OC

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55

Remark 9.3. The hypothesis in Theorem 1.2 about the local regularity is not too restrictive because of, for instance, results due to E. Dibenditto [13], J.L. Lewis [21] and G.M. Lieberman [22]. In the proof of Theorem 1.5, we didn’t use the shape derivative so we didn’t need to assume any regularity of ∂Ω∗ . ¡ ¢ Remark 9.4. If f ≥ 0 in L2 RN such that the interior of Suppf is empty, Theorems 1.2, 1.5 and 1.6 can be obtained by taking C as the smallest ball containing Suppf. But in this case, the condition (1) is worse than the one we can obtain when we work with Suppf as we can see in the case of uniform density supported by a segment [4]. Remark9.5. Consider the following free boundary problem.   −∆p uΩ = f in Ω    (pL) uΩ = 0 on ∂Ω      − ∂uΩ = k on ∂Ω, ∂ν where ∆p uΩ =div(|∇uΩ |p−2 ∇uΩ ) is the p-Laplace operator with p > 1 and p 6= 2. As far as the authors know, this problem is still open. Using the shape derivative, the γ-convergence [7] and the Hopf’s comparison principle [29], we can show that the problem C > k on C (see [2]) where C is the convex hull of Suppf.. (pL) has a solution if − ∂u ∂ν Remark 9.6. Theorem 12. in [23], is given for the operator div(a(x)∇v). For this kind of divergence operator, the continuity result is a simple consequence of Mosco convergence (see for instance [7]). So we can use the maximum principle and obtain the analogous of Theorems 1.2, 1.5 and 1.6 stated in Section 1. Remark 9.7. In [16], the authors show the existence of a minimizer u to the functional Z ¡ ¢ J(v) = |∇v|2 − 2f v + k 2 χ{v>0} dx RN

over all 0 ≤ v ∈ H 1 (RN ). They prove that (Ωu , u) (Ωu = {u > 0}) is a solution to Problem (QS) where the overdetermined condition is given in a weak sense: Z ¡ ¢ |∇u|2 − k 2 η · vdHN −1 = 0 lim ε&0

¡ ∞

∂{u>ε}

¢ N

for every η ∈ C0 RN , R . Then, they relate their minimization problem to quadrature domain Q(f, k) and show that Suppf ⊂ Ωu ⇔ Ωu ∈ Q(f, k).

They conclude their paper by giving 4.7) the following sufficient condition: If ´ ³ N (in Theorem R 6 N Suppf ⊂ BR and BR f dx > 3R |BR | k, then Ωu ∈ Q(f, k) with B3R ⊂ Ωu . (BR is some ball of radius R). Since in the case where f is radially symmetric the condition

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M. Barkatou et al. / Central European Journal of Mathematics 3(1) 2005 39–57

given in Theorem 1.7 is necessary and sufficient to have Suppf ⊂ Ω, it is better than the condition stated above for such f .

References [1] H.W. Alt and L.A. Caffarelli: “Existence and regularity for a minimum problem with free boundary“, J. Reine angew. Math., Vol. 325, (1981), pp. 105–144. [2] M. Barkatou, D. Seck and I. Ly: “An existence result for a free boundary problem for the p-Laplace operator“, submitted. [3] M. Barkatou: “Some geometric properties for a class of non Lipschitz-domains“, New York J. of Math., Vol. 8, (2002), pp. 189–213. [4] M. Barkatou: “Existence of quadrature surfaces for a uniform density supported by a segment“, submitted. [5] J.A. Bello, E. Fernandez-Cara, J. Lemoine and J. Simon: “On drag differentiability for Lipschitz domains“, Control of Part. Diff. Eq. and Appl., Lec. Notes In Pure and Applied Math. Series, Vol. 174, (1995), Dekker, New York. [6] A. Beurling: “On free-boundary problems for the Laplace equation“, Sem. Anal. Funct., Inst. Adv. Study Princeton, Vol. 1, (1957), pp. 248–263. [7] D. Bucur and P. Trebeschi: “Shape Optimization Problems Governed by Nonlinear State Equations“, Proc. Roy. Sc. Edinburgh, Vol. 128 A, (1998), pp. 945–963. [8] D. Bucur and J.P. Zolesio: “N-dimensional shape optimization under capacitary constraints“, J. Diff. Eq., Vol. 123(2), (1995), pp. 504–522. [9] G. Buttazzo, V. Ferone and B. Kawhol : “Minimum problems over sets of concave functions and related questions“, Math. Nachr., (1995), pp. 71–89. ¨ [10] T. Carleman: “Uber ein Minimumproblem der mathematischen Physik“, Math. Z., Vol. 1, (1918), pp. 208–212. [11] D. Chenais: “On the existence of a solution in a domain identification problem“, J. Math. Anal. Appl., Vol. 52, (1975), pp. 189–289. [12] R. Dautray and J.L. Lions : Analyse math´ematique et calcul num´erique pour les sciences et les techniques, Vol. 1, 2, Masson, Paris, 1984. [13] E. DiBendetto: “C 1+α local regularity of weak solutions of degenerate elliptic equations“, Nonlinear Analysis., Vol. 7, (1983), pp. 827–850. ¨ [14] K. Friedrichs: “Uber ein Minimumproblem f¨ ur Potentialstr¨omungen mit freiem Rand“, Math.Ann. , Vol. 109, (1934), pp. 208–212. [15] G. Gidas, Wei-Ming Ni and L. Nirenberg: “Symmetry and related properties via the maximum principle“, Comm. Math. Phys., Vol. 68, (1979), pp. 209–300. [16] B. Gustafsson and H. Shahgholian: “Existence and geometric properties of solutions of a free boundary problem in potential theory“, J. Reine angew. Math., Vol. 473, (1996), pp. 137–179. [17] A. Henrot: “Subsolutions and supersolutions in a free boundary problem“, Arkiv f¨or Math., Vol. 32(1), (1994), pp. 79–98.

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[18] H. Hosseinzadeh and H. Shahgholian: “Some qualitative aspects of a free boundary problem for the p-Laplacian“, Ann. Acad. Scient. Fenn. Math., Vol. 24, (1999), pp. 109–121. [19] D. Gilbarg and N.S. Trudinger: Elliptic partial equations of second order, SpringerVerlag, 1983. [20] M.V. Keldyˇs: “On the solvability and the stability of the Dirichlet problem“, Amer. Math. Soc. Trans., Vol. 51(2), (1966), pp. 1–73. [21] J.L. Lewis: “Regularity of the derivatives of solutions to certain degenerate elliptic equations“, Indiana Univ. Math. J., Vol. 32, (1983), pp. 849–858. [22] G.M. Lieberman: “Boundary regularity for solutions of degenerate elliptic equations“, Nonlinear Analysis., Vol. 12, (1988), pp. 1203–1219. ´ [23] V. Mikhailov: Equation aux d´eriv´ees partielles, Mir, Moscow, 1980. [24] F. Murat and J. Simon: “Quelques r´esultats sur le contrˆole par un domaine g´eom´etrique“, Publ. du labo. d’Anal. Num., Paris VI, (1974), pp. 1–46. [25] O. Pironneau: Optimal shape design for elliptic systems, Springer Series in Computational Physics, Springer, New York, 1984. [26] J. Serrin: “A symmetry problem in potential theory“, Arch. Rat. Mech. Anal., Vol. 43, (1971), pp. 304–318. [27] H. Shahgholian : “Quadrature surfaces as free boundaries“, Arkiv f¨or Math., Vol. 32(2), (1994), pp. 475–492. [28] J. Sokolowski and J.P. Zolesio: Introduction to shape optimization: shape sensitity analysis, Springer Series in Computational Mathematics, Vol. 10, Springer, Berlin, 1992. [29] P. Tolksdorf: “On the Dirichlet problem for quasilinear equations in domains with conical boundary points“, Comm. Partial Differential Equations, Vol. 8(7), (1983), pp. 773–817.

CEJM 3(1) 2005 58–75

On the homotopy type of (n − 1)-connected (3n + 1)-dimensional free chain Lie algebra Mahmoud Benkhalifa1∗ , Nabilah Abughazalah2 1

Department of Mathematics of College of Sciences, King Khalid University, P.O Box 9004, Abha, Saudi Arabia 2 Department of Mathematics, Girls College of Education, Abha, Saudi Arabia

Received 15 April 2004; accepted 19 October 2004 Abstract: Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ⊇ Q, then p = ∞. Denote by DGLnp n , n ≥ 1, the category of (n − 1)connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGLnp n . In this work we intend to answer the following two questions: Given an object (L(V ), ∂) in DGL3n+2 and denote by n S(L(V ), ∂) the class of objects homotopy equivalent to (L(V ), ∂). How we can characterize a free dgl to belong to S(L(V ), ∂)? Fix an object (L(V ), ∂) in DGL3n+2 . How many homotopy n ′ ) ∼ H (V, d) are there? equivalence classes of objects (L(W ), δ) in DGL3n+2 such that H (W, d = ∗ ∗ n np 3n+2 Note that DGLn is a subcategory of DGLn when p > 3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl. c Central European Science Journals. All rights reserved.

Keywords: Anick model, differential graded free Lie algebra, whitehead exact sequence, homotopy type MSC (2000): 55Q15, 55U40

1

Introduction

Let R be a subring ring of Q. We reserve the symbol p > 3 for the least prime which is not a unit in R; if R = Q, then p = ∞. A graded Lie algebra, L is a (positive) graded R-module {Li }i≥0 together with a linear map [, ] : L ⊗ L → L which satisfies ∗

E-mail: [email protected]

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59

graded anticommutativity and the Jacobi identity (cf [10], 5.2). In addition L shall be connected, i.e. L0 = 0. Recall that T (V ) denotes the tensor algebra on the graded R-module V = {Vi }i≥0 . This a graded Lie algebra with the commutator bracket. The sub Lie algebra generated by V is called the free graded Lie algebra generated by V . L(V ) is the direct sum of the subspaces L(k) (V ) of bracket length k ≥ 1. A differential graded Lie algebra, (L, ∂) (dgl for short), is a graded Lie algebra L equipped with a differential ∂ : Li → Li−1 satisfying ∂[x, y] = [∂x, y] + (−1)i [x, ∂y] , x ∈ Li , y ∈ Lj . A dgl of the form (L(V ), ∂) is called, by abuse of notation, a free-dgl. The linear part of the differential ∂ is the differential d : V → V defined by ∂(v) − d(v) ∈ ⊕k≥2 L(k) (V ). The category DGLnp n , n ≥ 1, is the full subcategory of the category of differential graded Lie algebras whose objects are a free-dgl, L(V ), satisfying Hi (V, d) = 0 if i < n and i ≥ np. This paper is motivated by the following results due to D. Anick, ([1, 2]): If X is an (n − 1)-connected CW complex of dimension np, there exists an object (L(V ), ∂) of DGLnp n and a natural homomorphism of differential graded Lie algebras ∼ = (L(V ), ∂) → A(X) which induces an isomorphism H(U (L(V ), ∂) → H(U A(X)) := H∗ (ΩX; R) where A denotes a “Quillen type”-functor. Moreover he showed that for every k < D = min(n + 2p − 3, np − 1) we have πk (X) ⊗ R∼ = Hk (s−1 V, d)), where s−1 is the desuspension graded = Hk−1 (L(V ), ∂) and Hk (X, R) ∼ homomorphism. Given an object (L(V ), ∂) in DGL3n+2 and denote by S(L(V ), ∂) the class of objects n homotopy equivalent to (L(V ), ∂). In this work we intend to answer the following two questions: 1- How we can characterize a free dgl to belong to S(L(V ), ∂)? 2- Fix an object (L(V ), ∂) in DGL3n+2 . How many homotopy equivalence classes of n 3n+2 objects (L(W ), δ) in DGLn are there such that H∗ (W, d′ ) ∼ = H∗ (V, d)? Recall that D. Anick has shown in [1] that there is a reasonable concept of “homotopy” among morphisms between two dgls, analogous in many respects to the topological notion of homotopy. Therefore we denote by DGLnp n ≃ the resulting homotopy category, i.e., the category whose objects are those of DGLnp n and whose morphisms are the homotopy classes of dgl morphisms [1, 2]. ≃ is a subcategory of DGLnp Note that the condition p > 3 implies that DGL3n+2 n ≃ . n Our tool to address this problem is the Whitehead exact sequence associated with (L(V ), ∂): bk+1

L(V )

· · · → Hk+1 (V, d) −→ Γk L(V )

b

k −→ Hk (L(V )) −→ Hk (V, d) −→ ···

where Γk = ker (Hk (L(V≤k )) −→ Vk ). This sequence was first introduced by J.H.C Whitehead for topological spaces in order to classify the homotopy types of 1-connected 4-dimensional CW-complexes [6, 12]. After these results, J.H. Baues proved in [4]that this sequence also exists for a free dgl (L(V ), ∂)

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and showed that it can establish a classification of the homotopy types of rational 4dimensional 1-connected dgls. The main results in this paper are the following theorems which give partial answers to questions 1 and 2 asked above: Theorem 3.15 Let (L(V ), ∂) and (L(W ), δ) be two objects in DGL3n+2 . If condin tion (19) is satisfied, then (L(V ), ∂) and (L(W ), δ) have the same homotopy type, i.e., (L(W ), δ) ∈ S(L(V ), ∂) if and only if their Whitehead exact sequences are isomorphic. Theorem 3.17 Fix an object (L(V ), ∂) in DGL3n+2 , then the number of homotopy equivn alence classes of objects (L(W ), δ) such that condition (19) is satisfied and H∗ (W, d′ ) ∼ = H∗ (V, d) is equal to the number of the equivalence classes (see definition 3.16) of tuples (H≤3n+1 (V, d), b3n+1 , π3n , ...., b2n+1 , π2n ) where bk+1 ∈ Hom(Hk+1 (V, d), Γk ) and where πk ∈ Ext(Hk (V, d), Coker bk+1 ), for every k ≤ 3n. This article is organized as follows. In section 2, the Whitehead exact sequence associated with a free dgl is defined and the essential properties are given. Section 3 is devoted to the main results in this paper. We conclude with some geometric applications and an example in the last section.

2

Whitehead exact sequence associated with a dgl

In this section we give the definition and the essential properties of the Whitehead exact sequence associated to a free dgl. Recall that Baues has constructed this sequence in [4] and he proved that it is an exact sequence. Let (L(V ), ∂) be a free dgl. For all n, let L(V≤n ) be the free sub-dgl of L(V ) generated by the graded module (Vi )i≤n . From the following long sequence: jn

βn

· · · → Hn (L(V≤n )) −→ Vn −→ Hn−1 (L(V≤n−1 )) → · · · where the connecting βn is defined by: βn (vn ) = ∂(vn ),

(1)

where ∂(vn ) ∈ Hn−1 (L(V≤n−1 )) is the homology class of the (n − 1)-cycle ∂(vn ) ∈ Ln−1 (V≤n−1 ), we define the graded module (Γn )n≥2 by setting: jn

ΓnL(V ) = ker(Hn (L(V≤n )) −→ Vn ).

(2)

Recall that the linear part d of the differential ∂ is given by: dn = jn−1 ◦ βn , ∀n ≥ 2,

(3)

The Whitehead exact sequence associated with (L(V ), ∂) is by definition the following exact long sequence: bn+1

b

n ··· · · · → Hn+1 (V, d) −→ ΓnL(V ) −→ Hn (L(V )) −→ Hn (V, d) −→

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61

where : bn+1 (z) = βn+1 (z) = ∂(z),

(4)

and where z is the class of the element z ∈ ker dn+1 . Remark 2.1. For the construction and the proof of the exactness of this sequence we refer to [3, 4, 5] Remark 2.2. Since Vn is free then for each n ≥ 2, from the short sequence: ΓnL(V ) ֌ Hn (L(V≤n )) ։ ker βn ⊂ Vn we deduce that: Hn (L(V≤n )) ∼ = ΓnL(V ) ⊕ ker βn

(5)

and in terms of the differential dn+1 : Vn+1 → Vn we deduce the following decomposition: Vn+1 ∼ = (Im dn+1 )′ ⊕ ker dn+1

(6)

where (Im dn+1 )′ ⊂ Vn+1 is a copy of Im dn+1 ⊂ Vn . Therefore the short exact sequence: dn+1

(Im dn+1 )′ ֌ ker dn ։ Hn (V, d) may be chosen as a free resolution of the module Hn (V, d). - Since dn+1 ((Im dn+1 )′ ) ⊂ ker βn , the short exact sequence: dn+1

(Im dn+1 )′ ֌ ker βn ։ ker bn

(7)

may be chosen as a free resolution of the sub-module ker bn ⊂ Hn (V, d).

- According to relations (5) and (6) , if (zn+1,σ )σ∈P and (ln+1,σ′ )σ′ ∈P′ denote, respectively, a basis of the free modules ker dn+1 and (Im d′n+1 )′ , the formula (1) can be written: βn+1 (zn+1,σ + ln+1,σ′ ) = bn+1 (zn+1,σ ) + ϕn (ln+1,σ′ ) + dn+1 (ln+1,σ′ ) L(V )

where ϕn : (Im dn+1 )′ → Γn

(8)

is the homomorphism given by the differential ∂.

Proposition 2.3. Let (L(V ), ∂) be a free dgl, then for every n ≥ 2, we have: Coker bn+1 ⊕ ker βn Hn (L(V )) ∼ = Imϕn ⊕ Im dn+1 ϕn

L(V )

where ϕn : (Im dn+1 )′ → Γn

։ Coker bn+1 .

Proof. From the following exact sequence: βn+1

Vn+1 −→ Hn (L(V≤n )) → Hn (L(V≤n+1 )) → 0

(9)

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and the relation (5) we get: L(V )

Γn Hn (L(V≤n+1 )) ∼ =

⊕ ker βn . Im βn+1

(10)

Substituting relation (8) into formula (10) we get: L(V )

Hn (L(V≤n+1 )) ∼ =

Γn ⊕ ker βn Imϕn + Im bn+1 ⊕ Im dn+1

(11)

but it’s well known that Hn (L(V≤n+1 )) = Hn (L(V )) and: L(V )

Coker bn+1 Γn ∼ = Imϕn + Im bn+1 Imϕn L(V )

via the isomorphism sending the element x + (Imϕn + Im bn+1 ), where x ∈ Γn (x + Im bn+1 ) + Imϕn . Thus (11) can be written as:

, to

Coker bn+1 ⊕ ker βn Hn (L(V )) ∼ = Imϕn ⊕ Im dn+1 as desired.



The Whitehead exact sequence is natural with respect to dgl morphisms. Namely a dgl morphism α : (L(V ), ∂) → (L(W ), δ) induces the following commutative diagram: bn+1

L(V )

. . . → Hn+1 (V, d) −→ Γn (A)

e

α γn

Hn+1 (α)

?

b′n+1

? L(W )

. . . → Hn+1 (W, d′ ) −→ Γn

b

n ... −→ Hn (L(V )) −→ Hn (V, d) −→

e

Hn (α)

Hn (α )

?

?

b′

n ... −→ Hn (L(W )) −→ Hn (W, d′ ) −→

where γ∗α and α e : (V, d) → (W, d′ ) are respectively the graded homomorphism and the chain map induced by the dgl morphism α. If (Hn (e α))∗ and (γnα )∗ are the following homomorphisms: (Hn (e α))∗ : Ext(ker b′n , Coker b′n+1 ) → Ext(ker bn , Coker b′n+1 )

(γnα )∗ : Ext(ker bn , Coker bn+1 ) → Ext(ker bn , Coker b′n+1 ),

then the commutativity of the above diagram induces the formula: (Hn (e α))∗ ([Hn (L(W ))]) = (γnα )∗ ([Hn (L(V ))])

(12)

where [Hn (L(V ))] (respectively [Hn (L(W ))]) is the extension represented by the short exact sequence Coker bn+1 ֌ Hn (L(V )) ։ ker bn extracted from the Whitehead exact sequence associated with (L(V ), ∂) (respectively the short exact sequence Coker b′n+1 ֌ Hn (L(W )) ։ ker b′n extracted from the Whitehead exact sequence associated with (L(W ), δ)).

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63

Remark 2.4. Formula (12) implies the following: d′n+1

dn+1

Consider (Im dn+1 )′ ֌ ker βn ։ ker bn and (Im d′n+1 )′ ֌ ker βn′ ։ ker b′n as two free resolutions of ker bn and ker b′n , respectively, which are defined in (7). Given extensions [Hn (L(V ))], [Hn (L(W ))] and homomorphisms γnα , Hn (e α), we have the corresponding diagrams: dn+1

(Im dn+1 )′ ֌ ker βn ։ ker bn

dn+1

(Im dn+1 )′ ֌ ker βn ։ ker bn ξn+1

ϕn ?

?

d′n+1

(Im d′n+1 )′ ֌ ker βn′ ։ ker b′n

Coker bn+1 γnα

ϕ′n ?

?

Coker b′n+1

Coker b′n+1

where [ϕn ] = [Hn (L(V ))] , [ϕ′n ] = [Hn (L(W ))] and where γnα (respectively ξn+1 ) is the homomorphism induced by γnα (respectively by Hn (e α)) on the quotient module Coker bn+1 ′ (respect. on the sub-module (Im dn+1 ) ). Recall that: [ϕn ] ∈

Hom((Im dn+1 )′ , Coker bn+1 )  ∼ = Ext(ker bn , Coker bn+1 ) ∗ (dn+1 ) Hom(ker dn+1 , Coker bn+1 )

Thus the homomorphisms (γnα )∗ and (Hn (e α))∗ are defined by (γnα )∗ ([Hn (L(V ))]) = [γnα ◦ ϕn ] and (Hn (e α))∗ ([Hn (L(W )]) = [ϕ′n ◦ ξn+1 ]. So the second condition is equivalent to the relation [γnα ◦ ϕn ] = [ϕ′n ◦ ξn+1 ] in Ext(ker bn , Coker b′n+1 ), or that there exists a homomorphism gn : ker βn −→ Coker b′n+1 satisfying the relation: γnα ◦ ϕn − ϕ′n ◦ ξn+1 = gn ◦ dn+1

3

(13)

The Γ − System3n+2 category n

Let R be a ring as in the introduction. A graded R-module H∗ is said to be (n − 1)connected with dimension (3n + 1), if Hi = 0 for i ≤ n − 1 and i ≥ 3n + 2. For every (n − 1)-connected module H∗ with dimension (3n + 1) and for each b2n+1 ∈ Hom(H2n+1 , [Hn , Hn ]), we associate the graded module (Γi )2n≤i≤3n defined as follows: Γk = ⊕ [Hi , Hj ] i+j=k j≥i≥n

Γ3n

2n ≤ k ≤ 3n − 1

  [Hn , Hn ] = ⊕ [Hi , Hj ] ⊕ Hn , (Im b2n+1 ) i+j=3n j≥i≥n

where [Hi , Hj ], [Hn , [Hn , Hn ]] ⊂ L(H∗ ) For each graded homomorphism f∗ : H∗ → H∗′ , which makes the following diagram commutative:

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b2n+1

H2n+1 f2n+1

- [Hn , Hn ]

[fn , fn ]

?

b′2n+1

′ H2n+1

? - [Hn′ , Hn′ ]

we associate the graded homomorphisms (γi )2n≤i≤3n defined as follows: γkα = ⊕ [fi , fj ] i+j=k j≥i≥n

α γ3n =

2n ≤ k ≤ 3n − 1

h i ⊕ [fi , fj ] ⊕ fn , [fn , fn ]

i+j=3n j≥i≥n

where: [fn , fn ] :

[Hn , Hn ] [Hn′ , Hn′ ] → (Im b2n+1 ) (Im b′2n+1 )

is induced by the homomorphism [fn , fn ] : [Hn , Hn ] → [Hn′ , Hn′ ] and the commutativity of the above diagram. Definition 3.1. (Γi )2n≤i≤3n is called the graded module associated with H∗ and the homomorphism b2n+1 . The homomorphisms (γi )2n≤i≤3n are called homomorphisms associated with f∗ . Definition 3.2. The category Γ − System3n+2 is defined as follows: n Object: A collection (Hk , bk+1 , πk )n≤k≤3n , called a Γ − system, is such that: - H∗ is a graded module (n − 1)-connected with dimension (3n + 1), - for each k such that n ≤ k ≤ 3n, bk+1 ∈ Hom (Hk+1 , Γk ) where Γ2n≤n≤3n is the module associated with H∗ and b2n+1 . - For each n ≤ k ≤ 3n, πk ∈ Ext(Hk , Coker bk+1 ). Morphism: A morphism between two Γ − systems (Hk , bk+1 , πk )n≤k≤3n and (Hk′ , b′k+1 , πk′ )n≤k≤3n is a graded homomorphism f∗ : H∗ → H∗′ of degree 0, called a Γ − morphism, satisfying the following two conditions: (fk )∗ (πk′ ) = (γk )∗ (πk ) and fk+1 ◦ b′k+1 = γk ◦ bk+1

(14)

for every k ≤ 3n, where γ≤3n are the homomorphisms associated with f∗ and where the homomorphisms (fk )∗ , (γk )∗ are: (γk )∗ : Ext(Hk , Coker bk+1 ) → Ext(Hk , Coker b′k+1 )

(fk )∗ : Ext(Hk′ , Coker b′k+1 ) → Ext(Hk , Coker b′k+1 ).

Note that for each k ≤ 3n, γk is the homomorphism induced by γk on the quotient module Coker bk+1 .

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65

Remark 3.3. As in Remark 2.4 the first condition in (14) implies the following: Let (V∗ , d) and (W∗ , d′ ) be two chain complexes satisfying H∗ (V∗ , d) = H∗ and H∗ (W∗ , d′ ) = H∗′ . By virtue of the homotopy extension theorem [9], there exists a chain map ξ∗ : (V, d) → (W, d′ ) such that H∗ (ξ∗ ) = f∗ . For each k ≤ 3n and according to the formula (6) we can consider: dk+1

(Imdk+1 )′ −→ ker dk ։ Hk

d′k+1

(Imd′k+1 )′ −→ ker d′k ։ Hk′

as two free resolutions of Hk and Hk′ , respectively. Given extensions πk , πk′ and the homomorphisms γk , fk , we have the corresponding diagrams: dk+1

dk+1

(Im dk+1 )′ ֌ ker dk ։ Hk

(Im d′k+1 ) ֌ ker dk ։ Hk ξk+1

ϕk

?

?

Coker bk+1

d′k+1

(Im d′k+1 )′ ֌ ker d′k ։ Hk′

γk

ϕ′k

?

?

Coker b′k+1

Coker b′k+1

where [ϕk ] = πk , [ϕ′k ] = πk′ and where (γk )∗ (πk ) = [γk ◦ ϕk ], (fk )∗ (πk′ ) = [ϕ′k ◦ ξk+1 ]. So the first condition in (14) is equivalent to the existence of a homomorphism hk : ker dk −→ Coker b′k+1 satisfying the relation: γk ◦ ϕk − ϕ′k ◦ ξk+1 = hk ◦ dk

(15)

3.1 Example of Γ − systems and Γ − morphisms In this paragraph we begin by the following proposition which we need in order to give an example of Γ − Systems. Recall that DGL3n+2 is the full subcategory of the category of n differential graded Lie algebras whose objects are a free-dgl, L(V ), satisfying Hi (V, d) = 0 if i < n and i ≥ 3n + 2. Proposition 3.4. Let (L(V ), ∂) be an object in DGL3n+2 and let: n bk+1

L(V )

· · · → Hk+1 (V, d) −→ Γk

→ Hk (L(V )) → Hk (V, d) → · · · L(V )

be its Whitehead exact sequence. Then Γ≤3n is the graded module associated with H∗ (V, d) and the homomorphism b2n+1 (see definition 3.1). L(V )

Proof. Recall that Γ∗

is defined, for each k ≤ 3n, by the formula:   jk L(V ) Γk = ker Hk (L(V≤k )) → Vk .

(16)

Since (L(V ), ∂) is an object in DGL3n+2 we can choose the chain complex (V∗ , d) such n that: Vk = 0 if k ≤ n − 1 and k ≥ 3n + 1. (17)

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In [2] D. Anick has shown that H 3, then 3n + 1 < np and: (18) H≤3n+1 ((L(V ), d)) ∼ = L(H≤3n+1 ((V∗ , d)), therefore by substituting Hi to Hi (V∗ , d) and for a degree’s reason (17) we deduce: Hk ((L(V ), d)) = Hk ⊕ ⊕ [Hi , Hj ] i+j=k j≥i≥n

H3n ((L(V ), d)) = H3n ⊕

2n ≤ k ≤ 3n − 1

⊕ [Hi , Hj ] ⊕ [Hn , [Hn , Hn ]] .

i+j=3n j≥i≥n

The first differential which is potentially non-trivial is ∂2n+1 : V2n+1 −→ [Vn+1 , Vn+1 ]. Passing to the quotient ∂2n+1 induces the homomorphism b2n+2 : H2n+1 −→ [Hn , Hn ] which implies, according to (16), that: L(V )

Γk

= ⊕ [Hi , Hj ] i+j=k j≥i≥n

2n ≤ k ≤ 3n − 1 L(V )

≥2 Finally for k = 3n, (16) implies that H3n (L(V≤3n )) = Γ3n and due to:   ∂3n+1 ≥2 ≥2 ≥2 H3n (L(V≤3n )) = Coker H3n+1 ((L(V≤3n ), d)) −→ H3n ((L(V≤3n ), d)) L(V )

we deduce that Γ3n

= Coker∂3n+1 . But by (18) we have:

≥2 H3n+1 (L(V≤3n )) = [Hn , H2n+1 ] ⊕ L3n+1 (H≤2n ) ≥2 (L(V≤3n )) = H3n

⊕ [Hi , Hj ] ⊕ [Hn , [Hn , Hn ]] .

i+j=3n j≥i≥n

The differential ∂3n+1 is trivial on L3n+1 ((H≤3n−1 h )) and oni [Hn , H2n+1 ] it’s equal to the homomorphism [1Hn , b2n+1 ] : [Hn , H2n+1 ] −→ Hn , [Hn , Hn ] . Hence we get: h i ⊕ [Hi , Hj ] ⊕ Hn , [Hn , Hn ] i+j=3n

Coker ∂3n+1 =

j≥i≥n

[Hn , Im b2n+1 ]

which induces the formula: L(V ) Γ3n

  [Hn , Hn ] . = ⊕ [Hi , Hj ] ⊕ Hn , (Im b2n+1 ) i+j=3n j≥i≥n

and the theorem is proved.



Furthermore, as a consequence of this result we derive the following well-known result (see [4], for example), which is the version of the Hurewicz theorem in the category DGL3n+2 . n

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67

Corollary 3.5. Let (L(V ), ∂) be an object in DGL3n+2 . Then the “Hurewicz” homon morphism hk : Hk (L(V )) −→ Hk (V, d) is an isomorphism for k ≤ 2n − 1 and surjective for k = 2n. Proof. One only need to apply theorem 3.4 to the Whitehead exact sequence associated with the free dgl (L(V ), ∂). Note that this corollary can also follow from the observation that Li (V ) = Vi , ∀i < 2n.  The geometrical version of corollary 3.5 is: Corollary 3.6. Let X be an n-connected (3n+2)-dimensional CW-complex, n ≥ 1. Then the Hurewicz homomorphism hk+1 : πk+1 (X) ⊗ R −→ Hk+1 (X, R) is an isomorphism for k < D = min(n + 2p − 3, np − 1, 2n − 1) and surjective for k = D. Proof. One only need to apply corollary 3.5 to the free dgl (L(V ), ∂) associated with X by using the Anick functor and theorem 3.7 in [2].  Now we are able to give an example of Γ − systems. Let (L(V ), ∂) be a dgl in DGL3n+2 and let: n bk+1

L(V )

· · · → Hk+1 (V, d) −→ Γk

b

k → Hk (L(V )) → Hk (V, d) −→ ···

be its Whitehead exact sequence. This sequence can be split to give homomorphisms L(V ) bk+1 : Hk+1 (V, d) → Γk and short exact sequences Coker bk+1 ֌ Hk (L(V )) ։ ker bk . But according to (9), we have for every k ≤ 3n : Hk (L(V )) ∼ =

Coker bk+1 ⊕ ker βk . Imϕk + Im dk+1

Since (3) implies ker βk ⊂ ker dk , then we set: πk =

h Coker b

⊕ ker dk i . Imϕk + Im dk+1 k+1

Therefore πk ∈ Ext(Hk (V, d), Coker bk+1 ) and by proposition 3.4 we conclude that (Hk (V, d), bk+1 , πk )n≤k≤3n is an object of the Γ − System3n+2 . n Definition 3.7. The object (Hk (V, d), bk+1 , πk )n≤k≤3n is called a Γ − system associated with (L(V ), ∂). Now we give an example of a Γ-morphism. Let (L(V ), ∂) and (L(W ), δ) be two objects in DGL3n+2 and let: n bk+1

L(V )

· · · → Hk+1 (V, d) −→ Γk b′k+1

L(W )

· · · → Hk+1 (W, d′ ) −→ Γk

b

k −→ Hk (L(V )) −→ Hk (V, d) −→ ···

b′

k −→ Hk (L(W )) −→ Hk (W, d′ ) −→ ···

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be their respective Whitehead exact sequences. Theorem 3.8. If for every n ≤ k ≤ 3n the following condition is satisfied:   Hk (V, d) ′ Ext , Coker bk+1 = 0, ker bk

(19)

then for every dgl morphism α : (L(V ), ∂) → (L(W ), δ) the induced morphism H∗ (e α) is a Γ-morphism. Recall that α e is the chain map induced by α on the indecomposables of the dgls (L(V ), δ) and (L(W ), ∂). Before giving the proof of this theorem we require the following lemma: Lemma 3.9. Let α : (L(V ), ∂) → (L(W ), δ) be a morphism in DGL3n+2 . The homon L(V ) L(W ) α morphisms γ≤3n : Γ≤3n → Γ≤3n , induced by α, are the homomorphisms associated with H∗ (e α) (see definition 3.1). Proof. The proof is very simple but also very long. It is just a simple computation and verification (see [5, 7])  Proof. ( of theorem 3.8) Indeed from the following short exact sequence and by the second isomorphic theorem: i

ker βk ֌ ker dk ։

ker dk ∼ Hk (V, d) = ker βk ker bk

we deduce that for every k ≤ 3n:   Hom(ker βk , Coker b′k+1 ) Hk (V, d) ′ , Coker bk+1 = ∗ = 0. Ext ker bk i (Hom(ker dk , Coker b′k+1 ))

(20)

From Remark 2.4 we know that the homomorphism γkα ◦ϕk −ϕ′k ◦ξk, satisfies the following relation γkα ◦ ϕk − ϕ′k ◦ ξk = gk ◦ dk , where gk : ker βk −→ Coker b′k+1 . So (20) implies that gk can also be extended to ker dk . So the relation (15) in Remark 3.3 is satisfied, and therefore H∗ (e α) is a morphism in the category Γ − System3n+2 . n α ′ Note that the homomorphisms γk , ϕk and ϕk are respectively given by the dgl morphism α and the differentials ∂ and δ. 

3.2 Main results This section is devoted to the proof of the main results (theorems 3.15 and 3.17 given in the introduction) in this paper. We begin by proving the following propositions. Proposition 3.10. For each object (Hk , bk+1 , πk )n≤k≤3n in Γ − System3n+2 , there exists n 3n+2 a free dgl (L(V ), ∂) in DGLn such that the given object is a Γ-system associated with (L(V ), ∂).

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69

Proof. Since the graded module H∗ is an (n − 1)-connected with dimension (3n + 1), then there exists a chain complex (V∗ , d) = (Vk , dk )k≤3n+2 such that H∗ (V∗ , d) = H∗ . Assume, by way of induction, that we have constructed a free dgl (L(V≤3n+1 ), ∂) such that (Hk , bk+1 , πk )k≤3n−1 is a Γ-system associated with (L(V≤3n+1 ), ∂) and such that the differential ∂ 3n+1 : V3n+1 → L3n (V≤3n+1 ) is linear. This means V3n+1 = (Im d3n+1 )′ ′ and ∂ 3n+1 = d3n+1 . We shall prove that there exists a free dgl (L(V≤3n+2 ), ∂) such that (Hk , bk+1 , πk )k≤3n is a Γ-system associated with it and such that the differential ′ ′ ∂ 3n+2 : V3n+2 → L3n+1 (V≤3n+2 ) is linear. d3n+1

Consider (Im d3n+1 )′ ֌ ker d3n+1 ։ ker H3n as a free resolution of H3n . Therefore for the given extension π3n ∈ Ext(H3n , Coker b3n+1 ) and the homomorphism b3n+1 there exist homomorphisms µ3n+1 , ϕ3n which make the following diagram commute: ker d3n+1

µ3n+1 - Z (L(V )) 3n ≤3n pr

?

H3n+1

b3n+1

? L(V )

- Γ 3n

ϕ3n ϕ3n    + 



(Im d3n+1 )′



։ Coker b3n+1

′ ), ∂). First put: where [ϕ3n ] = π3n . We shall now define the dgl (L(V≤3n+2 ′ V3n+2 = (Im d3n+2 )′ ′ = V3n+1 ⊕ ker d3n+1 = (Im d3n+1 )′ ⊕ ker d3n+1 V3n+1 ′ V≤3n = V≤3n

Then we define a differential ∂ on L(V≤3n+2 ) by the formulas: ∂3n+2 (l3n+2,σ′′ ) = d3n+2 (l3n+2,σ′′ ) ∂3n+1 (z3n+1,σ + l3n+1,σ′ ) = d3n+1 (l3n+1,σ′ ) + ϕ3n (l3n+1,σ′ ) + µ3n+1 (z3n+1,σ ) ∂≤3n = ∂ ≤3n

where (z3n+1,σ )σ∈P , (l3n+1,σ′ )σ′ ∈P′ and (l3n+2,σ′′ )σ′ ∈P′ denote, respectively, bases of the free modules ker d3n+1 , (Im d3n+1 )′ and (Im d3n+2 )′ . By the above diagram the element ϕ3n (l3n+1,σ′ ) + µ3n+1 (z3n+1,σ ) lives in the sub-module Z3n (L(V≤3n )) of 3n-cycles in L3n (V≤3n ), therefore ∂3n (ϕ3n (l3n+1,σ′ ) + µ3n+1 (z3n+1,σ )) = 0. Since d3n+2 (l3n+2,σ′′ ) ∈ (Im d3n+1 )′ and on the module (Im d3n+1 )′ we have ∂3n+1 = ∂ 3n+1 = d3n+1 and by definition ∂3n = ∂ 3n , then we deduce the following relations:
∂3n (d3n+1 (l3n+1,σ′ )) = ∂ 3n ∂ 3n+1 (l3n+1,σ′ ) = 0 ∂3n+1 (∂3n+2 (l3n+2,σ′′ )) = ∂3n+1 (d3n+2 (l3n+2,σ′′ )) = d3n+1 (d3n+2 (l3n+2,σ′′ )) = 0

′ Hence ∂ is a differential on L(V≤3n+1 ). By construction the differential ∂3n+2 : V3n+2 → ′ L3n+1 (V≤3n+1 ) is only linear and it is easy to see that the Whitehead exact sequence ′ associated with the dgl (L(V≤3n+2 ), ∂) can be written: b3n+1

L(V ′ )

H3n+1 (V ′ , d) −→ Γ3n

→ H3n (L(V ′ )) → H3n (V ′ , d) → · · ·

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Also it is easy to check that such (Hk , bk+1 , πk )n≤k≤3n is a Γ-system associated with ′ (L(V≤3n+2 ), ∂) and the proof is completed.  Definition 3.11. A dgl morphism α : (L(V ), ∂) → (L(W ), δ) between two free dgls is said to be k-diagonal if α maps the direct factor ker dk of Vk to the module Wk . Lemma 3.12. If α : (L(V ), ∂) → (L(W ), δ) is k-diagonal then according to the splitting Hk (L(V≤k )) ∼ = Γk ⊕ ker bk (respectively Hk (L(W≤k )) ∼ = Γ′k ⊕ ker b′k ), the homomorphism Hk (α) : Hk (L(V≤k )) → Hk (L(W≤k )) splits into: Hk (α) = γkα ⊕ ξk

(21)

where γkα is the homomorphism induced by α on the sub-module Γk and where ξk : ker βk → ker βk′ is the chain transformation induced by α on the indecomposables. Proof. (trivial)



′ Proposition 3.13. Let f∗ : (Hk+1 , bk+1 , πk )n≤k≤3n −→ (Hk+1 , b′k+1 , πk′ )n≤k≤3n be a morphism in Γ − System3n+2 . If (L(V ), ∂) and (L(W ), δ) are two dgls associated respecn tively to the given Γ − systems by proposition 3.10. Then there exists a dgl morphism θ : (L(V ), ∂) −→ (L(W ), δ) such that, for every n ≤ k ≤ 3n + 1, θ is k-diagonal and:

e = fk Hk (θ)

(22)

where θe : (V, d) → (W, d′ ) is the chain map induced by θ on the indecomposables of the dgls (L(V ), ∂) and (L(W ), δ). Proof. Assume, by way of induction, that we have constructed a dgl morphism α : (L(V≤3n ), ∂) −→ (L(W≤3n ), δ) which is k-diagonal for each k ≤ 3n and such that Hk (e α) = fk , ∀k ≤ 3n. We shall now construct a dgl morphism θ : (L(V ), ∂) −→ (L(W ), δ) such e = f3n+1 . that θ is (3n + 1)-diagonal and H3n+1 (θ) ′ Consider the homomorphism H3n (α) ◦ β3n+1 − β3n+1 ◦ ξ3n+1 . Since the dgl morphism α is 3n-diagonal, the relations (1) and (21) give us an explicit expression of H3n (α) ◦ ′ β3n+1 − β3n+1 ◦ ξ3n+1 and it’s easy to show that this homomorphism satisfies the following relation: ′ α H3n (α) ◦ β3n+1 − β3n+1 ◦ ξ3n+1 = γ3n ◦ ϕ3n − ϕ′3n ◦ ξ3n+1 . α α ◦ ϕ3n − ϕ′3n ◦ ξ3n+1 , then according to ◦ ϕ3n − ϕ′3n ◦ ξ3n+1 ) = γ3n Since pr ◦ (γ3n the conditions defining the morphism in the category Γ − System3n+2 and remark 3.3, n we deduce that there exists a homomorphism g3n which makes the following diagram commute:

M. Benkhalifa, N. Abughazalah / Central European Journal of Mathematics 3(1) 2005 58–75

d3n+1

0 -(Im d3n+1 )′ α γ3n ϕn − ϕ′3n ξ3n+1

   ? L(W )





   g3n

pr

Γ3n

71

- ker d3n 

- Coker b′3n+1

But V3n+1 = (Imd3n )′ ⊕ ker d3n then we can extend g3n to the module V3n and therefore we have: ′ H3n (α) ◦ β3n+1 − β3n+1 ◦ ξ3n+1 − g3n ◦ d3n+1 : (Imd3n+1 )′ → Im b′3n+1

or: ′ (H3n (α) − g3n ◦ j3n ) ◦ β3n+1 − β3n+1 ◦ ξ3n+1 : (Imd3n+1 )′ → Im b′3n+1

As a result there exists a homomorphism λ3n+1 which makes the upper triangle in the following diagram commute:  λ3n+1 

V3n+1 X

  

  

  XX X

XX

′

? - Im b′3n+1 ⊂ Γ′3n 6 6

Ψ3n+1

XX X

 ker d3n+1 :

(α − h3n )∂ − δξ3n+1 pr XX XX XXX XX X z Z3n (L(W≤3n )) X 3n

′ ◦ ξ3n+1 and where the homomorphism where Ψ3n+1 = (H3n (α) − g3n ◦ j3n ) ◦ β3n+1 − β3n+1 h3n : V3n → Z3n (L(W≤3n )) satisfies the relation pr ◦ h3n = g3n . Recall that Z3n (L(W≤3n )) is the sub-module of the 3n-cycles of the dgl L(W ). Choose (z3n+1,σ )σ∈P and (l3n+1,σ′ )σ′ ∈P′ , respectively, as bases of the free modules ker d3n+1 and (Im d3n+1 )′ . Recall that V3n+1 ∼ = (Im d3n+1 )′ ⊕ (ker d3n+1 ). By the commutativity of the above diagram and for each σ ∈ Σ and σ ′ ∈ Σ′ there exists an element t3n+1,σ,σ′ ∈ L3n+1 (W≤3n ) such that:

((α3n − h3n )∂ − δξ3n+1 )(z3n+1,σ + l3n+1,σ′ )

(23)

= δ ◦ λ3n+1 (l3n+1,σ′ ) + δ(t3n+1,σ,σ′ ).

If z3n+1,σ + l3n+1,σ′ ∈ V3n+1 , then we define θ : (L(V≤3n+1 ), ∂) −→ (L(W≤3n+1 ), δ) by setting: θ3n+1 (z3n+1,σ + l3n+1,σ′ ) = λ3n+1 (l3n+1,σ,σ′ ) + t3n+1,σ,σ′ + ξ3n+1 (z3n+1,σ + l3n+1,σ′ ) θ3n = α3n − h3n θi = αi

i 6= 3n, 3n + 1

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θ is a dgl morphism i.e. θ ◦ ∂ = δ ◦ θ. Indeed, since for each z3n+1,σ + l3n+1,σ′ ∈ V3n+1 we have ∂(z3n+1,σ + l3n+1,σ′ ) ∈ Z3n (L(W≤3n )) and since h3n is nil on ker d2n+1 , then according to the definition of θ we have: θ ◦ ∂(z3n+1,σ + l3n+1,σ′ ) = (α3n − h3n ) ◦ ∂(z3n+1,σ + l3n+1,σ′ ) and on the other hand, according to (23) and the definition of θ, we have: δ ◦ θ3n+1 (z3n+1,σ + l3n+1,σ′ ) = δ ◦ λ3n+1 (l3n+1,σ′ ) + δ(t3n+1,σ,σ′ ) + δ ◦ ξ3n+1 (z3n+1,σ + l3n+1,σ′ )

= ((α3n − h3n ) ◦ ∂ − δ ◦ ξ3n+2 )(z3n+1,σ + l3n+1,σ′ )

+ δ ◦ ξ3n+1 (z3n+1,σ + l3n+1,σ′ )

= (α3n − h3n ) ◦ ∂(z3n+1,σ + l3n+1,σ′ ).

Therefore θ is a dgl morphism. Finally, from the definitions it is clear that θ is (3n + 1)-diagonal and verifies the relation (22).  Proposition 3.14. Under the notation above, the morphism f∗ is an isomorphism in Γ−System3n+2 if and only if the dgl morphism U θ induces an isomorphism in homology. n e Proof. First it is clear that if U θ induces an isomorphism in homology then f∗ = H∗ (θ) is an isomorphism. To show the converse, according to (22), if f∗ is an isomorphism, then e : H∗ (V, d) → H∗ (W, d′ ) is an isomorphism. Therefore the graded homomorphism H∗ (θ) theorem 6.3 in [2] implies that the dgl morphism θ induces an isomorphism in homology.  We can summarize propositions 3.10, 3.13 and 3.14 in the following theorem which gives a partial answer to the two questions asked in the introduction: Theorem 3.15. Let (L(V ), ∂) and (L(W ), δ) be two objects in DGL3n+2 . If the condin tion (19) is satisfied, then (L(V ), ∂) and (L(W ), δ) have the same homotopy type if and only if their Whitehead exact sequences are isomorphic. Proof. First it’s well-known that the Whitehead exact sequence is a homotopy invariant. Conversely, given two objects (L(V ), ∂) and (L(W ), δ) in DGL3n+2 such that their n Whitehead exact sequences are isomorphic, we can deduce, under condition (19), that their Γ-systems are also isomorphic in Γ − System3n+2 . Therefore the free dgls (L(V ), ∂) n and (L(W ), δ) have the same homotopy type by propositions 3.13 and 3.14.  Definition 3.16. Let (L(V ), ∂) be an object in DGL3n+2 . We say that two tuples n ′ ′ ′ (H≤3n+1 (V, d), b3n+1 , π3n , ...., b2n+1 , π2n ) and (H≤3n+1 (V, d), b3n+1 , π3n , ...., b′2n+1 , π2n ), where ′ ′ bk+1 , bk+1 ∈ Hom(Hk+1 (V, d), Γk ) and where πk , πk ∈ Ext(Hk (V, d), Coker bk+1 ) for every k ≤ 3n, are equivalent if there exists a graded automorphism f≤3n+1 : H≤3n+1 (V, d) ∼ = H≤3n+1 (V, d) such that, for every n ≤ k ≤ 3n: (fk )∗ (πk′ ) = (γk )∗ (πk ), and γk ◦ bk+1 = b′k+1 ◦ fk+1

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where the homomorphisms (γk )∗ and (fk )∗ are as in (14) Theorem 3.17. Fix an object (L(V ), ∂) in DGL3n+2 , then the number of homotopy n equivalence classes of objects (L(W ), δ) such that condition (19) is satisfied and H∗ (W, d′ ) ∼ = H∗ (V, d) is equal to the number of the equivalence classes of tuples (H≤3n+1 (V, d), b3n+1 , π3n , ...., b2n+1 , π2n ) where bk+1 ∈ Hom(Hk+1 (V, d), Γk ) and where πk ∈ Ext(Hk (V, d), Coker bk+1 ), for every k ≤ 3n.

4

Example and Computation

As an example we give the following proposition which is a geometric application of the above theorem. Proposition 4.1. Let n ≥ 1 and X be a CW-complex such that Hk (X, R) = 0 for every positive k satisfying: 3n + 2 2 3n + 1 k≤ 2 k ≥ 3n + 3. k≤

if n is even

(24)

if n is odd

Put Y = ∨M (Hk (X, R), k), where M (Hk (X, R), k) denotes the Moore space. If we denote k

by A(X) and A(Y ) the free dgl associated to X and Y , respectively, by the Anick model, then A(X) and A(Y ) have the same homotopy type in DGL3n+2 . n

Proof. Following Anick in [1, 2] and by the hypothesis, we can associate to the spaces X and Y two (n)-connected (3n + 2)-dimensional free dgls A(X) = (L(V ), δ) and A(Y ) = (L(W ), ∂) which satisfy the following properties: Hk (V, d) = Hk+1 (X, R)

Hk (W, d′ ) = Hk (Y, R)

and for k < D = min(n + 2p − 3, np − 1): Hk (L(V )) = πk+1 (X) ⊗ R

Hk (L(W )) = πk+1 (Y ) ⊗ R. A(X)

A(Y )

From the hypothesis (24) and theorem 3.4 we deduce that the modules Γk and Γk are ′ trivial for all k ≤ 3n. Then the two Γ-systems (H∗ (W, d ), 0, 0), (H∗ (V, d), 0, 0), associated to A(X) and A(Y ), respectively, are isomorphic in the Γ − System3n+2 . Moreover the n relation (19) is trivially satisfied, so the free dgls A(X) and A(Y ) have the same homotopy type.  We end this work by giving an example showing how theorem 3.17 can be used to compute the homotopy types of certain free dgls which are associated with certain CWcomplexes by the Anick model .

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Example 4.2. Let R = Z[ 12 , 31 , 51 , 17 ] and X be a CW-complex having the following homology groups: H4 (X, R) = H8 (X, R) = Zp , H5 (X, R) = Zq , H7 (X, R) = H6 (X, R) = Zr , H10 (X, R) = Zt , H9 (X, R) = Zs , H11 (X, R) = Zw , Hi (X, R) = 0

otherwise.

where p, q, s, r, t, w > 7 are relatively prime numbers. Let (L(V ), δ) be the free dgl associated with the CW-complex X by Anick model. How many homotopy types (L(V ), δ) exist? Recall first that we have Hk (V, d) = Hk+1 (X, R). Since X is a 3-connected 11-dimensional CW-complex, then (L(V ), δ) is an object of the category DGL11 3 , and therefore we can apply theorem 3.17. Put Hk (X, R) = Hk . By using proposition 3.4 we get: Γ3 = Γ4 = Γ5 = 0, Γ6 = [H4 , H4 ] , Γ7 = [H4 , H5 ] ,   [H4 , H4 ] Γ8 = [H5 , H5 ] ⊕ [H4 , H6 ] , Γ9 = [H5 , H6 ] ⊕ H4 , Im b8 where b8 ∈ Hom(H8 , Γ6 ) = Zp . By hypothesis we deduce that Γ6 = Zp , Γ7 = 0, Γ8 = Zq . If b8 = 0 then Γ9 = Zp and if b8 6= 0 then b8 is onto and Γ9 = 0. Therefore an easy computation shows that: Hom(H8 , Γ6 ) = Zp , Hom(Hk+1 , Γk ) = 0, ∀k ≤ 9

Ext(Hk , Coker bk+1 ) = 0, ∀k ≤ 9   Hk , Coker bk+1 = 0, ∀k. Ext ker bk Hence we find p tuples: (0, 0; 0, 0; 0, 0; 0, 0; 0, 0; 0, 0; ), (0, 0; 0, 0; 1, 0; 0, 0; 0, 0; 0, 0; ), .. . (0, 0; 0, 0; p − 1, 0; 0, 0; 0, 0; 0, 0; ). Note that it is easy to check that the tuples:

(0, 0; 0, 0; 1, 0; 0, 0; 0, 0; 0, 0; ), · · · , (0, 0; 0, 0; p − 1, 0; 0, 0; 0, 0; 0, 0; ) are all isomorphic. Then according to theorem 3.17 we have two homotopy types of the free dgl A(X).

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References [1] D.J. Anick: “Hopf algebras up to homotopy”, J. Amer. Math. Soc., Vol. 2(3), (1989), pp. 417–452. [2] D.J. Anick: “An R-local Milnor-Moore Theorem”, Advances in Math, Vol. 77, (1989), pp. 116–136. [3] H.J. Baues: Homotopy Type and Homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1996. [4] H.J. Baues: “Algebraic homotopy”, Cambridge studies in advanced mathematics, Vol. 15, (1989). [5] M. Benkhalifa: Mod`eles algebriques et suites exactes de Whitehead, Thesis (PhD), Universit´e de Nice France, 1995. [6] M. Benkhalifa: “Sur le type d’homotopie d’un CW-complexe”, Homology, Homotopy and Applications, Vol. 5(1), (2003), pp. 101–120. [7] M. Benkhalifa: “On the homotopy type of a chain algebra”, Homology, Homotopy and Applications, Vol. 6(1), (2004), pp. 109–135. [8] Y. Felix, S. Halperin and J.C. Thomas: “Rational homotopy theory”, C.M.T, Vol. 205, (2000). [9] S. MacLane: Homology, Springer, 1967. [10] J. Milnor and J.C. Moore: “On the structure of Hopf algebras”, Ann Math., Vol. 81, (1965), pp. 211–264. [11] J.C. Moore: S´eminaire H. Cartan, Expos´e 3, 1954-1955. [12] J.H.C. Whitehead: “A certain exact sequence”, Ann. Math, Vol. 52, (1950), pp. 51– 110.

CEJM 3(1) 2005 76–82

Bolzano’s Intermediate-Value Theorem for Quasi-Holomorphic Maps Aboubakr Bayoumi∗ Department of Mathematics of College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Basic Research Institute, P.O. Box 1577, Palm Harbor, FL 34682, U.S.A.

Received 11 February 2004; accepted 4 September 2004 Abstract: We extend Bolzano’s intermediate-value theorem to quasi-holomorphic maps of the space of continuous linear functionals from lp into the scalar field, (0 < p < 1). This space is isomorphic to l∞ . c Central European Science Journals. All rights reserved.

Keywords: Bolzano’s Intermediate-value theorem, infinite-dimensional holomorphy, non locally convex spaces, inverse mapping theorem, F-spaces, quasi-holomorphic maps MSC (2000): 46A16, 46E50

1

Introduction

In this paper we extend the intermediate-value theorem (or Bolzano’s theorem) to quasiholomorphic maps on L(lp ;C), I the space of continuous linear functionals on lp (Section 4). This is achieved after we prove the inverse mapping theorem for quasi-differentiable maps between locally bounded spaces (Section 3). Here a vector space E with a p-norm is called a p-normed space [1,p.2]. Every locally bounded space E is a p-normed space (see Rolewicz [7]). If E is complete we call it a locally bounded F-space; and hence it is topologically equivalent to a p-Banach space (0 < p ≤ 1). We further discuss Bolzano’s theorem for Banach spaces of continuous maps (Section 2). Bolzano’s theorem says in its simplest form that : “A real-valued continuous map f of a closed interval [a, b] such that f (a) and f (b) ∗

E-mail: [email protected]

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have different signs, has a zero in (a, b)”. Shih [9] extends the intermediate-value theorem to the complex plane C. I He was the first to note that Bolzano’s theorem can be reformulated as follows : Assume a < 0 < b and f (a) < 0 < f (b), and consequently the condition f (a).f (b) < 0 becomes x.f (x) > 0 for x ∈ ∂U (1) where ∂U denotes the boundary of U = (a, b). Then f has at least one zero in U. He then proved that : ”If U is a bounded domain of C I containing the origin, and f : U− →C I

(2)

is analytic in U and continuous in U − and such that Re z − f (z) > 0,

for z ∈ ∂U

(3)

then f has one zero in U ”. In addition, Shih [10] obtained the following extension of Bolzano’s theorem to C In : “Let U be a bounded domain in C I n containing the origin. Let f : U− →C In be analytic in U and continuous in U − and assume that Re z − f (z) > 0,

for

z ∈ ∂U

Then f has exactly one zero in U ”. Wlodraczyk [12] obtained an extension of Bolzano’s intermediate-value theorem for holomorphic maps in complex Banach spaces of continuous linear maps.

2

Bolzano’s theorem in infinite dimensional spaces

Let E, F be complex Hilbert spaces, let L(E; F ) denote the Banach space of all continuous linear maps A : E → F with the mapping norm kAk = sup kA(x)k.

(4)

L1 (E, F ) ⊂ L(E; F )

(5)

kxk≤1

Let be a closed linear subspace of L(E; F ). In infinite dimensional spaces, Wlodraczyk [12] obtained the following extension of Bolzano’s intermediate-value theorem for holomorphic maps in L1 (E, F ).

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Theorem 2.1. [12](1991) Assume Ω ⊂ L1 (E, F ) is a bounded domain such that 0 ∈ Ω and let f : A ∈ Ω− → f (A) ∈ L1 (E, F )

(6)

be continuous in Ω− . (1) If Re {A∗ f (A)} > 0,

for

A ∈ ∂Ω

(7)

and (I − f )(Ω) is contained in a compact subset of L(E; F ), where A∗ is the complex conjugate of A, and Re {A∗ f (A)} is the vector of the real parts of the product A∗ f (A), then f has at least one zero in Ω, i.e. f (A) = 0 for at least one A ∈ Ω. (2) If, additionally, f is holomorphic in Ω and Df (A) has a bounded inverse for A ∈ Ω, then f has exactly one zero in Ω. Let E and F be a p-normed space and a q-normed space respectively (0 < p, q ≤ 1) and U open in E. A mapping f : U → F is said to be m quasi-differentiable (or m pq-differentiable) at a ∈ U, if there exists a continuous linear map Ta ∈ L(E; F ), such that lim k f (x) − f (a) − Ta (x − a) km/q / k x − a k1/p = 0, x→a

Ta is called the m quasi-differential of f at a and is denoted by Dm f (a). If m = 1, f is called quasi-differentiable (or pq-differentiable) at a ∈ U . If f has a Taylor series expansion at a ,and hence f is m quasi-differentiable for every m [1, p.149], then f is called quasi-holomorphic map at a ∈ U . For more on the properties of quasi-differentiable maps, see the author [1, 2, 3, 4] . To extend Bolzano’s theorem to quasi-holomorphic maps on the space L(lp ;C) I (0 < p ≤ 1) of continuous linear maps we need the following result on inverse maps.

3

Inverse maps

Let U, V be open subsets of E and F , a p-normed space and a q-normed space respectively, (0 < p, q ≤ 1). If the mapping f : U → V is a homeomorphism and is quasi-differentiable on U , it does not necessarily follow that f is a diffeomorphism, i.e. the inverse f −1 of f is quasi-differentiable on V . Consider for example, the function f : R I →R I defined by f (x) = x3 Then f is a homeomorphism of class C 1 but the inverse g = f −1 is not quasi-differentiable at the origin. The following theorem gives a sufficient condition to have inverse quasi-differentiable maps between locally bounded spaces (in particular locally bounded F -spaces).

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Theorem 3.1. (Inverse Mapping Theorem) Let U and V be open subsets of E and F , a p-normed and a q-normed space (0 < p, q < 1) respectively, and let f : U → V be a homeomorphism. Assume that f is pqdifferentiable at a point a ∈ U . Then the inverse g = f −1 is qp-differentiable at the point b = f (a) if and only if Df (a) ∈ L(E, F ) is a homeomorphism of E onto F ; i.e. Df (a) is a topological isomorphism. In that case, Dg(b) = (Df (a))−1

(8)

Proof. If g is qp-differentiable at b = f (a) then, by the chain rule, we have g ◦ f = 1U , f ◦ g = 1V and Dg(b) ◦ Df (a) = 1E , Df (a) ◦ Dg(b) = 1F

which proves that Df (a) is a topological isomorphism of E onto F . Conversely, assume that Df (a) is topological isomorphism from E into F. We want to show that g is qp-differentiable at b = f (a) . We will first prove that g has the Lipschitzian property at b. For simplicity, let A = Df (a) and ∆(x) = f (x) − f (a) − A(x − a)

(9)

for x ∈ U . Since A−1 ∈ L(F ; E), by applying A−1 to both sides of (9) we obtain x − a = A−1 (f (x) − f (a) − ∆(x))

(10)

kx − ak1/p ≤ σc(kf (x) − f (a)k1/q + k∆(x)k1/q )

(11)

with c = kA−1 k. Since f is pq-differentiable at a for ǫ = 1/ 2ac, there is, by definition, r > 0 such that B(a, r) ⊂ U and k∆xk1/q ≤ kx − ak1/p /2ac for x ∈ B(a, r). Hence from (11) we get kx − ak1/p ≤ 2σckf (x) − f (a)k1/q (12)

Since g is also continuous , for some s > 0 , g(B(b, s)) ⊂ B(a, r) . If we set x = g(y) in (12), we have kg(y) − g(b)k1/p ≤ 2σcky − bk1/q (13) for y ∈ B(b, s) , which shows that g is Lipschitzian at b. Now we show that g is qp-differentiable. From the relation (10), we have kx − a − A−1 (f (x) − f (a))k1/p ≤ ck∆xk1/q

(14)

Since f is pq-differentiable at a, for any ǫ > 0 there exists r > 0 such that B(a, r) ⊂ U , and if x ∈ B(a, r), k∆xk1/q ≤ ǫkx − ak1/p . Hence (14) becomes kx − a − A−1 (f (x) − f (a))k1/p ≤ cǫkx − ak1/q for x ∈ B(a, r). Now choose δ > 0 such that B(b, δ) ⊂ U and g(B(b, δ)) ⊂ B(a, r). Then if y ∈ B(b, δ) and x = g(y), using (13) we obtain kg(y) − g(b) − A−1 (y − b)k1/p ≤ cσǫkg(y) − g(b)k1/q ≤ 2σc2 ǫky − bk1/q This completes the proof.



80

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Bolzano’s theorem for quasi-holomorphic maps

Regarding the non-locally convex spaces we prove the following theorem for the space lp, ( 0 < p < 1). The proof will be established by making use of the inverse map theorem. Theorem 4.1. Let L1 (lp ;K) I ⊂ L(lp ;K) I ≃ l∞ be a closed linear subspace of l∞ . Assume Ω ⊂ L1 is a bounded domain such that 0 ∈ Ω and let f : Ω− → L1 (lp ;K) I

(15)

be continuous in Ω− . (1) If Re {A∗ f (A)} > 0

A ∈ ∂Ω

for

(16)

and (I − f )(Ω) is contained in a compact subset of L(lp ;K) I , then f has at least one zero in Ω, that is f (A) = 0 for at least one A ∈ Ω. (2) If, additionally, f is quasi-holomorphic in Ω and Df (A) has a bounded inverse for A ∈ Ω, then f has exactly one zero in Ω. Proof. (1) Let ht (A) = A − f (t, A), i.e.

where

ht (A) = (1 − t)A + tf (A),

f (t, A) = t [A − f (A)] A∈Ω

and

0≤t≤1

(17)

Then h0 (A) = A 6= 0, h1 (A) = f (A) 6= 0

for

A ∈ ∂Ω.

Moreover, ht (A) 6= 0

for

A ∈ ∂Ω

and

0 < t < 1,

since Re {A∗ ht (A)} = Re {(1 − t)A∗ A + tA∗ f (A)} > (1 − t) Re {A∗ A} > 0.

where A∗ A is the product of A∗ and A. Since 0 ∈ Ω, we apply the homotopy property to ht (A), ( see Schwartz [8, p.81]), to obtain deg(I − f (0, .), Ω, 0) = deg(I − f (1, .), Ω, 0) = 1

(18)

deg(f, Ω, 0) = deg(I, Ω, 0) = 1,

(19)

or equivalently Hence it follows that f −1 (0) is nonempty.

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(2) Since A → Df (A) is a quasi-holomorphic map of Ω into L(lp ;K) I it follows from the inverse map theorem for complex p-Banach spaces that the map f is locally quasi-biholomorphic in Ω, that is, f and f −1 are quasi-holomorphic . So for each A ∈ Ω, there exists a neighborhood UA of A in Ω such that f (UA ) = VA is open in L1 (lp ;K), I f −1 exists and is quasi-holomorphic in VA . We now prove that f −1 (0) contains one mapping : Let us first show that f −1 (0) is finite. Towards a contradiction , let Ak ∈ f −1 (0), for all k = 1, 2, ... We have h(Ak ) = Ak , k = 1, 2, ...,where h = I − f. Since h is compact and continuous in Ω− , there exists a subsequence of (Ak ) say (Aj ), and A ∈ Ω− such that kA − Ak k → 0

as

k→∞

and

h(A) = A

(20)

This yields f (A) = 0

and

Re {A∗ f (A)} = 0

(21)

Consequently,A ∈ Ω. But f is biholomorphic in UA and Ak ∈ UA for sufficiently large k. This yields a contradiction. Thus f −1 (0) is finite. If f −1 (0) = {A1 , ..., An } (22) then by Smart [11, properties 10.3.5&10.3.6, p.80)] , we have deg(f, Ω, 0) = deg(I, Ω/U, 0) =

n X

deg(f, Uk , 0)

(23)

k=1

where Uk are all small neighborhoods of Ak such that the sets Uk are pairwise disjoint , Uk− ⊂ Ω and U = Ω− /(∪nk=1 Uk ). Further we have that f = I − h, h has isolated fixed points in Ω, h is compact and L1 (E, F ) is complex. Thus the multiplicity of each eigenvalue of D(h)(Ak ) = D(I − f )(Ak ),

k = 1, 2, ..., n

(24)

is even by Kransnoseliski [6, Lemma 4.1 and Theorem 4.7]. This yields deg(f, Uk , 0) = deg(I − h, Uk , 0) = 1 k = 1, 2, ..., n. By (19), (24) and (26) we deduce that n = 1, which completes the proof.

(25) 

Remark 4.2. Every separable locally bounded F-space Ep is isomorphic to a quotient space of lp , (0 < p ≤ 1) i.e. Ep ≃ lp /M

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for some closed subspace M of lp , see Rolewicz [7]. That is, Ep is an image of lp by a continuous linear map T , where M is its kernel . Therefore the above theorem can be applied for Ep whenever its dual separates its points. We consider equivalently the F-space lp /M , and a bounded domain Ω of a closed subspace L1 (lp /M ;K) I ⊂ L(lp /M ;K). I

Acknowledgment The author would like to thank the referee for his fruitful comments.

References [1] A. Bayoumi: Foundations of complex analysis in non locally convex spaces. Functions theory without convexity conditions, Mathematics studies, Vol. 193, North Holland, 2003. [2] A. Bayoumi: “Mean-Value Theorem for complex locally bounded spaces”, Communication in Applied Nonlinear Analysis, Vol. 4(3), (1997). [3] A. Bayoumi: “Mean-Value Theorem for real locally bounded spaces”, Journal of Natural Geometry, London, Vol. 10, (1996), pp. 157–162. [4] A. Bayoumi: “Fundamental theorem of calculus for locally bounded spaces”, Journal of Natural Geometry, London, Vol. 15(1-2), (1999), pp. 101–106. [5] A. Bayoumi: “Mean-Value Theorem for definite integrals of vector-valued maps of p-Banach spaces“, (2005), to appear. [6] M. Kransnoseliski: Topological method in theory of nonlinear integral equations, Mcmillan, 1964. [7] S. Rolewicz: Metric linear spaces, Monografie Matematyczne, Instytut Matematyczny Polskiej Akademii Nauk, 1972. [8] J.T. Schwartz:Nonlinear functional analysis, Gordon and Breach, New York, 1969. [9] M.H. Shih: “An analogy of Bolzano’s theorem for functions of a complex variable”, Amer.Math.Monthly, Vol. 89, (1982), pp. 210–211. [10] M.H. Shih: “Bolzano’s theorem in several complex variables”, Proc.Amer.Math.Soc., Vol. 79, (1980), pp. 32–34 . [11] D. Smart: Fixed points theorems, Cambridge Tracts in Math., Vol. 66, 1974. [12] K. Wlodraczyk: “Intermediate value theorem for holomorphic maps in complex Banach spaces”, Math.Proc.Camb.Phil.Sco., (1991), pp. 539–540.

CEJM 3(1) 2005 83–97

Congruences, ideals and annihilators in standard QBCC-algebras∗ Radom´ır Halaˇs† , Luboˇs Plojhar Department of Algebra and Geometry, Palack´y University, Tomkova 40, 779 00 Olomouc, Czech Republic

Received 2 September 2004; accepted 16 November 2004 Abstract: We characterize congruence lattices of standard QBCC-algebras and their connection with the congruence lattices of congruence kernels. c Central European Science Journals. All rights reserved. ° Keywords: Congruence, ideal, annihilator, BCC-algebra, QBCC-algebra MSC (2000): 06F35, 06A11, 03G25

1

Introduction

When solving the problem whether the class of all BCK-algebras form a variety, Y. Komori [10] introduced the class of BCC-algebras. A. Wro´ nski [11, 12] characterized BCCalgebras as algebras isomorphic to a subalgebra of the left-residuation reduct of some integral monoid with left-residuation. Recall that an algebra (A, •, 1) of type (2,0) is a BCC-algebra if it satisfies the following identities: (x • y) • [(z • x) • (z • y)] = 1 (BCC1) x•x=1 (BCC2) x•1=1 (BCC3) 1•x=x (BCC4) (x • y = 1 & y • x = 1) ⇒ x = y. (BCC5) It was shown by W.A. Dudek [2] that BCC-algebras satisfying the quasi-commutation ∗ †

Work on the paper was supported by Council of Czech Government No J14/98:153100011. E-mail: [email protected]

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axiom x • (y • z) = y • (x • z)

(QC)

are just BCK-algebras. BCK-algebras satisfying the left-distributivity axiom x • (y • z) = (x • y) • (x • z)

(LD)

are known as Hilbert algebras, an algebraic counterpart of the logical connective implication in intuitionistic logic. Moreover, Hilbert algebras satisfying the axiom of contraction (x • y) • x = x

(C)

are implication algebras, describing properties of the logical connective implication in a classical two-valued propositional logic. Hilbert algebras were recently generalized in [5] as follows: A pre-logic is an algebra (A, •, 1) of type (2,0) satisfying the axioms x•x=1 (PL1) 1•x=x (PL2) x • (y • z) = (x • y) • (x • z) (PL3) x • (y • z) = y • (x • z) (PL4) In other words, pre-logics contrary to Hilbert algebras need not satisfy the axiom (BCC5). The axioms of a BCC-algebra (A, •, 1) allow to define a natural ordering on A as follows: x ≤ y iff x • y = 1. (1) From this point of view BCC-algebras are special ordered sets, see also [6]. On the other hand, the relation ≤ defined by (1) on a pre-logic is reflexive and transitive only, i.e. a quasiorder. We obtain the same result when we omit the axiom (BCC5) from the axiomatic system of BCC-algebras. This led us to a common generalization of BCCalgebras and pre-logics, namely to QBCC-algebras as algebras satisfying the axioms (BCC1)-(BCC4), see [7, 8]. In the paper [8] QBCC-algebras having the property that every subset containing the element 1 is a subalgebra were characterized, see also [3, 4]. Such QBCC-algebras are called standard (hence for a standard QBCC-algebra A = (A, •, 1) we have x • y ∈ {x, y, 1} whenever x, y ∈ A). We started from any quasiordered set (A, ≤, 1) with the greatest element 1 and constructed the operation • on A making the algebra (A, ≤, 1) a standard QBCC-algebra. To recall the main theorem in [8], we need some other notions. For any quasiordered set (A, ≤), a, b ∈ A, we adopt the following terminology: We write a ∼ b whenever a ≤ b and b ≤ a hold and call the pair a, b indistinguishable; the set C(a) = {x ∈ A | x ∼ a} is called the cell of a. We write a < b if a ≤ b and a 6∼ b,

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85

and a k b whenever a, b are incomparable. If A is finite then (A, ≤) can be visualized as a poset the elements of which are substituted by cells. For example, the diagram in Fig. 1 represents a qoset in which, excluding reflexivity, the relations a ≤ b, a ≤ c, b ≤ c, b ≤ d, d ≤ b, a ≤ d, d ≤ c hold.

Fig. 1

One can easily derive that the natural quasiordering ≤ on any QBCC-algebra (A, •, 1) has the properties 1≤x y ≤x•y

iff for all

x = 1, x, y ∈ A.

(2) (3)

The property (2) exactly means C(1) = {1}. For any qoset (A, ≤, 1) with the greatest element 1 and C(1) = {1}, a pair (x, y) ∈ A2 , x > y is called a bridge if for each z ∈ A the following dual conditions hold: z > y implies z ≥ x, (b1) z < x implies z ≤ y. (b2) It is clear that (x, y) being a bridge in A yields that x covers y, i.e. there is no z ∈ A with x > z > y. The notion ”bridge” is motivated by the diagram of A around the pair (x, y), which looks like a bridge between x and y. Now we are ready to recall the main theorem in [8] describing all standard QBCCalgebras: Proposition 1.1. Let (A, ≤, 1) be a qoset with the greatest element 1 and C(1) = Let us define the operation • on A as follows: x • y = 1 if x ≤ y, 1 • x = x, x • y = y if x k y, x • y = y if x > y and (x, y) is not a bridge, if (x, y) is a bridge in A and x 6= 1 one can set x • y = y or x • y = x; in the latter case for each z ≥ x we have either z ∼ x and z • y = z or z > x and z • x = x; for each z ≤ y we have either z ∼ y and x • z = x or z < y and y • z = z.

{1}. (q1) (q2) (q3) (q4) (q5)

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Then (A, •, 1) is a standard QBCC-algebra and each standard QBCC-algebra is of this form. A pair (x, y), x > y, for which x • y = y will be called normal. Proposition 1.1 allows us to construct a standard QBCC-algebra from a given qoset A. Example 1.2. Let us consider a qoset A with the diagram in Fig. 2.

Fig. 2

By setting d • c = d we get by (q5) and (q4) e • c = e, c • a = a, c • b = b, d • a = a. The rest of cases is given by (q1), (q2), (q3) and (q4). The operation • is completely determined in the following table:

2

•

1 a

b

c

d

e

f

1

1 a

b

c

d

e

f

a

1

1

1

1

1

1 f

b

1

1

1

1

1

1 f

c

1 a

b

1

1

1 f

d

1 a

b

d

1

1 f

e

1 a

b

e

1

1 f

f

1 a

b

c

d

e

1

Ideals, deductive systems and annihilators in standard QBCCalgebras

Now we are ready to describe congruence lattices of standard QBCC-algebras. For this we introduce the notion of an ideal:

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Definition 2.1. A subset ∅ 6= I ⊆ A of a standard QBCC-algebra A = (A, ≤, 1) satisfying the conditions x ∈ I, y ∈ A, x ≤ y imply y ∈ I, (x, y) being a bridge and x • y = x ∈ I imply y ∈ I, (I1) is called an ideal of A. The set of all ideals of A will be denoted by Id(A). As usual, for a congruence θ on A denote by [1]θ its congruence class containing the element 1, the so-called kernel of θ. The next lemma shows that congruence kernels of standard QBCC-algebras coincide with their ideals. Lemma 2.2. Let A = (A, ≤, 1) be a standard QBCC-algebra, θ ∈ Con(A), I ∈ Id(A). Then (a) Iθ = [1]θ is an ideal of A, (b) the relation θI on A defined by hx, yi ∈ θI iff x ∼ y or x, y ∈ I is the greatest congruence on A with [1]θI = I. Proof. (a) Suppose hx, 1i ∈ θ, y ≥ x. By compatibility we obtain hx•y, 1•yi = h1, yi ∈ θ. If (x, y) is a bridge and x•y = x ∈ Iθ , then again by compatibility hx • y, 1 • yi = hx, yi ∈ θ. Hence [x]θ = [y]θ = [1]θ and y ∈ Iθ . (b) We will show that θI is a congruence on A. Reflexivity and symmetry of θI are clear. To prove transitivity, let hx, yi, hy, zi ∈ θI . If x ∼ y ∼ z or x, y, z ∈ I occur, we have hx, zi ∈ θI . In the remaining case x ∼ y, y, z ∈ I we obtain by (I1) x ∈ I, hence also hx, zi ∈ θI . Now we prove compatibility of θI . Suppose hx, yi ∈ θI and let u ∈ A be an arbitrary element. It is enough to prove hu • x, u • yi, hx • u, y • ui ∈ θI . First let x, y ∈ I. Applying (I1) one gets u • x, u • y ∈ I, and so hu • x, u • yi ∈ θI . If x • u = 1, then x • u ∈ I and x ≤ u so by (I1) u ∈ I. This yields y • u ∈ I. Suppose now x • u = u. The case y • u = u gives us hx • u, y • ui = hu, ui ∈ θI . For y • u = y 6= u, 1 due to (I2) u ∈ I and hx • u, y • ui = hu, yi ∈ θI . Secondly let us suppose that x ∼ y. We will show that u • x ∼ u • y and x • u ∼ y • u hold. Let u k x. By (q3) u • x = x ∼ y = u • y, x • u = u ∼ u = y • u. The case u ∼ x leads to u • x = 1 ∼ 1 = u • y, x • u = 1 ∼ 1 = y • u. Suppose further u > x. The following two subcases can occur. If both pairs (u, x), (u, y) are normal we get u • x = x ∼ y = u • y, x • u = 1 ∼ 1 = y • u. If they are non-normal, then u • x = u ∼ u = u • y, x • u = 1 ∼ 1 = y • u. The last possibility is u < x. In this case we have also two subcases. The both pairs (x, u), (y, u) are either normal, then u • x = 1 ∼ 1 = u • y, x • u = u ∼ u = y • u, or non-normal and u • x = 1 ∼ 1 = u • y, x • u = x ∼ y = y • u.

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The equality [1]θI = I follows directly from the definition of θI . Let us show that θI is the greatest congruence on A with kernel I. Suppose θ ∈ Con(A), [1]θ = I, and let hx, yi ∈ θ. Using the substitution property of θ leads to hx • y, 1i, hy • x, 1i ∈ θ, i.e. x • y, y • x ∈ I. Assume further that x 6∼ y. If x, y are incomparable, we get x • y = y ∈ I and y • x = x ∈ I. In the case of comparability let e.g. x ≤ y. Then y 6≤ x (otherwise x ∼ y), hence y • x ∈ {x, y}. Having y • x = x ∈ I we obtain with respect to x ≤ y and (I1) also y ∈ I. For y • x = y ∈ I the pair (y, x) is a bridge, and so by (I2) also x ∈ I. In summary, we have proved that x 6∼ y implies x, y ∈ I. Finally, we have got θ ⊆ θI . ¤ Lemma 2.2 also shows that the congruences on standard QBCC-algebras are of a very special type: Lemma 2.3. Let A = (A, •, 1) be a standard QBCC-algebra, θ ∈ Con(A). If hx, yi ∈ θ and x, y 6∈ I = [1]θ , then x ∼ y. Proof. It results from Lemma 2.2.

¤

Example 2.4. In contrast to BCC-algebras, standard QBCC-algebras need not be congruence distributive: Consider the qoset (A, ≤) given in Fig. 3 and the corresponding standard QBCC-algebra A = (A, ≤, 1).

Fig. 3

Then Con(A) is visualized in Fig. 4,

Fig. 4

where θ1 ...{{0, b}, {a}, {1}},

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θ2 ...{{0, a}, {b}, {1}}, θ3 ...{{a, b}, {0}, {1}}, θ4 ...{{0, a, b}, {1}}. This example also shows that ideals of QBCC-algebras can be kernels of more than one congruence. We will show that congruence kernels in arbitrary QBCC-algebras (i.e. not necessarily standard ones) correspond to deductive systems: Definition 2.5. A subset D ⊆ A of a QBCC-algebra A = (A, •, 1) is called a deductive system if 1∈D (D1) x • (y • z) ∈ D and y ∈ D imply x • z ∈ D. (D2) Denote by Ck(A) or Ded(A) the set of all congruence kernels of A or the set of all deductive systems of A, respectively. Lemma 2.6. For an arbitrary QBCC-algebra A it holds Ck(A)=Ded(A). Proof. It is easy to prove that Ck(A) ⊆Ded(A). Indeed, (D1) is trivial, for (D2) we use the compatibility of θ: hy, 1i ∈ θ, hy • z, zi ∈ θ, hx • (y • z), x • zi ∈ θ. Conversely, let D ∈ Ded(A). Define the relation θD on A by hx, yi ∈ θD iff x • y, y • x ∈ D. We show that θD ∈ Con(A) with [1]θD = D. Reflexivity and symmetry of θD are clear. To prove transitivity of θD , assume that hx, yi, hy, zi ∈ θD , i.e. x • y, y • x, y • z, z • y ∈ D. Then by (D1) and (BCC1) we have 1 = (y • z) • [(x • y) • (x • z)] ∈ D, and since x • y ∈ D, (D2) yields (y • z) • (x • z) ∈ D. Applying (D2) once more to 1 • ((y • z) • (x • z)) ∈ D with y • z ∈ D, we obtain 1 • (x • z) = x • z ∈ D. The validity of z • x ∈ D can be proved analogously, and so θD is transitive. Further, let hx, yi ∈ θD and u ∈ A be an arbitrary element. Then x • y, y • x ∈ D. By (BCC1) 1 = (x • y) • [(u • x) • (u • y)] = 1 • ((x • y) • [(u • x) • (u • y)]) ∈ D. Since x • y ∈ D applying (D2) we obtain 1 • [(u • x) • (u • y)] = (u • x) • (u • y) ∈ D. Analogously we prove (u • y) • (u • x) ∈ D and hu • x, u • yi ∈ θD . Applying (D2) again for 1 = (x • u) • [(y • x) • (y • u)] ∈ D and y • x ∈ D one gets (x • u) • (y • u) ∈ D. Interchanging x and y we have also (y • u) • (x • u) ∈ D and hx • u, y • ui ∈ θD . Finally, using transitivity of θD this gives θD ∈ Con(A) with [1]θD = D. ¤

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Corollary 2.7. For an arbitrary standard QBCC-algebra A = (A, •, 1) it holds Id(A) = Ck(A) = Ded(A). Definition 2.8. Let A = (A, ≤, 1) be a standard QBCC-algebra and B, C be non-void subsets of A. The set hCi = {x ∈ A | x • c = c for each c ∈ C} is called the annihilator of C. The set hC, Bi = {x ∈ A | (x • c) • c ∈ B

for each

c ∈ C}

is called the relative annihilator of C with respect to B. Remark 2.9. Note that the term ”annihilator” is used since hCi is the set of all elements of A which are ”annihilized” from the left by the elements of C. If C = {c} is a singleton then hCi will be briefly denoted by hci. For a qoset (A, ≤) and ∅ 6= M ⊆ A put U (M ) = {x ∈ A | m ≤ x for each m ∈ M }. In case M = {a1 , . . . , an } we also write U (a1 , . . . , an ) instead of U (M ). Theorem 2.10. Let A = (A, ≤, 1) be a standard QBCC-algebra and I be an ideal of A. Then hIi is also an ideal and the pseudocomplement of I in the lattice Id(A). Moreover , hIi = {x ∈ A | x k i for each i ∈ I\{1}} ∪ {1}. Proof. First we prove that hIi = {x ∈ A | x k i for each i ∈ I\{1}} ∪ {1}. Suppose a ∈ hIi, i.e. a • i = i for each i ∈ I. Evidently, for each i ∈ I\{1} either a k i or a ≥ i. If there exists i ∈ I\{1} with a ≥ i then we have by (I1) a ∈ I and so 1 = a • a = a, proving that a ∈ {x ∈ A | x k i for each i ∈ I\{1}} ∪ {1}. The converse inclusion is clear. Further let us prove that hIi ∈ Id(A). Suppose x ∈ hIi and x ≤ y 6= 1. Then y ≤ i for some i ∈ I\{1} leads to x ≤ i, contadicting x ∈ hIi. The case y ≥ i for some i ∈ I\{1} means y ∈ I which is also impossible due to x ≤ y. This shows y ∈ hIi. We have to show that (I2) holds. For this let x • y = x ∈ hIi for some bridge (x, y). Let us note that x = 1 would imply y = 1 • y = 1, hence it holds x 6= 1. Since (x, y) forms a bridge, the property x k i for each i ∈ I\{1} yields also y k i for each i ∈ I\{1}, and so y ∈ hIi. It is evident that I ∩ hIi = {1}. Suppose that J is any ideal of A with the property I ∩ J = {1}. If j ∈ J\{1}, i ∈ I\{1} then i k j: otherwise either i ≤ j ∈ I ∩ J or j ≤ i ∈ I ∩ J, a contradiction. This means that J ⊆ hIi and hence hIi is the pseudocomplement of I in Id(A). ¤ Theorem 2.11. Let B, C be ideals of a standard QBCC-algebra A = (A, ≤, 1). Then hC, Bi is the relative pseudocomplement of C with respect to B in the lattice Id(A). Moreover , hC, Bi = {x ∈ A | x k c for each c ∈ C\B} ∪ B.

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Proof. At first we show that hC, Bi = {x ∈ A | x k c for each c ∈ C\B} ∪ B. It is easily seen that B ⊆ hC, Bi. Suppose x k c for each c ∈ C\B. Then (x • c) • c = c • c = 1 ∈ B for each c ∈ C\B. In the remaining case we have also d ≤ (x • d) • d ∈ B whenever d ∈ C ∩ B, and altogether x ∈ hC, Bi. Conversely, suppose y ∈ hC, Bi\B and assume y ∦ c for some c ∈ C\B. If y ≤ c then (y • c) • c = 1 • c = c ∈ B, a contradiction. In the case y ≥ c we conclude y ∈ C and, moreover, y = 1 • y = (y • y) • y ∈ B, which is also a contradiction. This proves y k c for each c ∈ C\B. Now we show that hC, Bi is an ideal of A. Let x ∈ hC, Bi and x ≤ y. We have y ∈ B ⊆ hC, Bi whenever x ∈ B. Suppose further x k c for each c ∈ C\B. This yields y 6≤ c for each c ∈ C\B (otherwise x ≤ c, a contradiction). Hence we have either y k c for each c ∈ C\B or y > c for some c ∈ C\B. The first case immediately gives y ∈ hC, Bi. In the second case we have by (I1) y ∈ C, which w.r.t. x ≤ y and x k c for each c ∈ C\B yield y ∈ B ⊆ hC, Bi. We prove that hC, Bi satisfies (I2). Let x • y = x ∈ hC, Bi for some bridge (x, y). Then y ∈ B whenever x ∈ B. Suppose further that x k c for each c ∈ C\B and assume y ∦ c for some c ∈ C\B. If y ≤ c we get x > c or x ≤ c (since (x, y) is a bridge), a contradiction. Similarly, c ≤ y < x contradicts c k x and thus, finally, hC, Bi is an ideal of A. It is clear that C ∩ hC, Bi ⊆ B. Let J be an ideal of A with the property C ∩ J ⊆ B and assume j ∈ J\B. Suppose further j ∦ c for some c ∈ C\B. If c ≤ j, then j ∈ C ∩ J ⊆ B, a contradiction. The case j ≤ c leads to the contradiction c ∈ J ∩ C ⊆ B. This means that J ⊆ hC, Bi and hC, Bi is the relative pseudocomplement of C with respect to B in Id(A). ¤ There is a natural question to find conditions under which the annihilator of every non-void subset M of A is equal to the annihilator of the ideal generated by M . We will show that the answer is closely connected with the following example: Example 2.12. Consider a qoset A with the greatest element 1 where A\{1} is composed by pairwise incomparable blocks Bi , i ∈ Ω, being either a cell or Bi = C(ai ) ∪ C(bi ) for ai < bi , with bi •ai = bi . Such a standard QBCC-algebra will be called quasi-implication algebra. Theorem 2.13. For a standard QBCC-algebra A = (A, ≤, 1) the following conditions are equivalent: (a) for each ∅ 6= M ⊆ A it holds hM i = hI(M )i, (b) A is a quasi-implication algebra. Proof. (a) ⇒ (b) : Take M = {c} for c ∈ A\{1}. We know that I(c), the principal ideal generated by {c}, is equal to U (c) if there is no non-normal pair (c, d) with c > d in A or I(c) = U (c) ∪ C(d) if such a pair exists. We will show that in both cases hI(c)i = {x ∈ A | U (x, c) = {1}}.

(∗)

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Suppose x ∈ hI(c)i and let y ∈ U (x, c). Then y ∈ I(c), hence 1 = x • y = y proving that U (x, c) = {1}. Suppose conversely that U (x, c) = {1} for some x ∈ A and let y ∈ U (c). If x ≤ y, then y ∈ U (x, c) = {1} and hence x • y = x • 1 = 1 = y. Otherwise we have either x k y and x • y = y or y < x and x ∈ U (x, c) = {1} and x • y = 1 • y = y. Altogether we proved that x ∈ hU (c)i. Finally, let (c, d) be a non-normal pair of A, so c • d = c. Let us prove that x ∈ hdi. We have either x = 1 and x • d = 1 • d = d or x k c. In the latter case since (c, d) is a bridge, also x k d and x • d = d, hence the equality (∗) is proved. Consider now b > c for some b ∈ A. If the pair (b, c) is normal, then b • c = c, hence b ∈ hci = hI(c)i which, by (∗), gives U (b, c) = U (b) = {1} and b = 1. By Theorem 2.11 this means that A contains at most three-element chains otherwise it would contain a normal pair (x, y) with x, y 6= 1. If 1 > b > c is a three-element chain of A, the pair (b, c) cannot be normal, hence b • c = b, i.e. b 6∈ hci verifying that A is a quasi-implication algebra. (b) ⇒ (a) : Suppose that c ∈ Bi for Bi being a cell. Then we have hci = A\Bi = hI(c)i = hU (c)i. Further let Bi = C(ai ) ∪ C(bi ) with ai < bi , and bi • ai = bi . In this case we have I(x) = Bi ∪ {1} = U (ai ) for each x ∈ Bi , hence hI(x)i = A\Bi = hxi. Evidently hM i = {1}∪{Bj | Bj ∩M = ∅}, and hI(M )i = {1}∪{Bj | Bj ∩I(M ) = ∅}. The condition Bj ∩M = ∅ is equivalent to Bj ∩I(M ) = ∅ since I(M ) = {1}∪{Bi | Bi ∩C 6= ∅}. ¤ Lemma 2.14. Let A = (A, ≤, 1) be a standard QBCC-algebra, I ∈ Id(A). Each congruence θE on A with [1]θE = I is of the form θE = I 2 ∪ (E∩ ∼), where E is an equivalence on A \ I. Proof. (a) We show, that θE = I 2 ∪ (E∩ ∼) is a congruence with [1]θE = I. Since I 2 is an equivalence on I and (E∩ ∼) is a equivalence on A \ I, I 2 ∪ (E∩ ∼) is an equivalence on A. Analogously as in Lemma 2.2 we prove the compatibility of θE . Let hx, yi ∈ θE and let u ∈ A be an arbitrary element. It is enough to prove hu • x, u • yi ∈ θE , hx • u, y • ui ∈ θE . At first let hx, yi ∈ I 2 . Applying (I1) we immediately get hu•x, u•yi ∈ I 2 . If x•u = 1 ∈ I, then by (I1) u ∈ I and y • u ∈ I, hence hx • u, y • ui ∈ I 2 . Suppose further x • u = u. The case y • u = u gives us hx • u, y • ui = hu, ui ∈ θE . For y • u = y ∈ I due to (I2) we have u ∈ I and hx • u, y • ui = hu, yi ∈ I 2 . Secondly let us suppose that hx, yi ∈ (E∩ ∼). For u k x we obtain hu•x, u•yi = hx, yi ∈ (E∩ ∼), hx•u, y•ui = hu, ui ∈ θE . The case u ∼ x leads to hu•x, u• yi = h1, 1i ∈ I 2 , hx • u, y • ui = h1, 1i ∈ I 2 . Let u > x. By (q5), both pairs (u, x), (u, y) are either normal, so hu • x, u • yi = hx, yi ∈ (E∩ ∼), hx • u, y • ui = h1, 1i ∈ I 2 , or non-normal and hu • x, u • yi = hu, ui ∈ θE , hx • u, y • ui = h1, 1i ∈ I 2 . In the case u < x both the pairs

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(x, u), (y, u) are either normal, hence hu • x, u • yi = h1, 1i ∈ I 2 , hx • u, y • ui = hu, ui ∈ θE , or both non-normal and hu • x, u • yi = h1, 1i ∈ I 2 , hx • u, y • ui = hx, yi ∈ (E∩ ∼). The equality [1]θE = I follows directly from the definition of θE . (b) We show that each conguence θ on A with the kernel I is of the form θE . Let hx, yi ∈ θ. If x, y ∈ I, then hx, yi ∈ I 2 . Suppose x, y 6∈ I. Using Lemma 2.3 we get x ∼ y. Further, E = θ \ I 2 is an equivalence on A \ I and hx, yi ∈ E. Finally we have hx, yi ∈ (E∩ ∼). ¤ Remark 2.15. (a) For E = (A \ I)2 (i.e. the relation ι on A \ I) we obtain θI , the greatest congruence on A with kernel I defined in Lemma 2.2. (b) We use Lemma 2.3 for describing the congruence lattices of standard QBCC-algebras. For each ideal (i.e. the congruence kernel) we construct the corresponding congruences with this kernel. See the following example. Example 2.16. Consider the qoset (A, ≤) given in Fig. 5 and the corresponding standard QBCC-algebra A = (A, ≤, 1) without non-normal pairs.

Fig. 5

The lattice of all ideals is visualized in Fig. 6, where I = {1}, J = {1, a, b, c}, K = {1, a, b, c, d, e}.

Fig. 6

For I there are 52 equivalences on A \ I but only 10 distinct congruences: θJ1 ...{J, {d, e}, {a, b, c}} θJ2 ...{J, {d, e}, {a}, {b, c}} θJ3 ...{J, {d, e}, {b}, {a, c}}

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θJ4 ...{J, {d, e}, {c}, {a, b}} θJ5 ...{J, {d, e}, {a}, {b}, {c}} θJ6 ...{J, {d}, {e}, {a, b, c}} θJ7 ...{J, {d}, {e}, {a}, {b, c}} θJ8 ...{J, {d}, {e}, {b}, {a, c}} θJ9 ...{J, {d}, {e}, {c}, {a, b}} θJ10 ...{J, {d}, {e}, {a}, {b}, {c}}. For K we have 2 congruences with kernel K: θK1 ...{K, {d, e}} θK2 ...{K, {d}, {e}}. For L there is only θL ...{K}. The congruence lattice Con(A) is visualized in Fig. 7.

Fig. 7

Lemma 2.17. Let A be a standard QBCC-algebra and θ1 , θ2 ∈ Con(A). Then [1]θ1 ∨θ2 = [1]θ1 ∪ [1]θ2 . Proof. The proof easily follows from the facts that the ideals are congruence kernels and that the union of ideals is again an ideal. ¤ For a given relation R let Rt denotes its transitive closure.

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Lemma 2.18. Let A = (A, ≤, 1) be a standard QBCC-algebra, θ1 , θ2 ∈ Con(A). Then θ1 ∨ θ2 = ([1]θ1 ∪ [1]θ2 )2 ∪ ((θ1 ∪ θ2 )∩ ∼)t . Proof. The proof is easy.

¤

Remark 2.19. If there are at most two-element cells in a standard QBCC-algebra A = (A, ≤, 1) then ((θ1 ∪ θ2 )∩ ∼)t = ((θ1 ∪ θ2 )∩ ∼) holds. Lemma 2.20. Let A be a standard QBCC-algebra and assume that Con(A) is not distributive, i.e. there are pairwise distinct congruences θ1 , θ2 , θ3 ∈ Con(A) with the property θ1 ∧ θ2 = θ1 ∧ θ 3 , θ1 ∨ θ2 = θ1 ∨ θ 3 . Then [1]θ2 = [1]θ3 . Moreover, if the triple {θ1 , θ2 , θ3 } generates the diamond, then [1]θ1 = [1]θ2 = [1]θ3 .

Fig. 8

Proof. From Lemma 2.17 and due to the relations between θ1 , θ2 , θ3 we derive [1]θ1 ∪ [1]θ2 = [1]θ1 ∨θ2 = [1]θ1 ∨θ3 = [1]θ1 ∪ [1]θ3 , [1]θ1 ∩ [1]θ2 = [1]θ1 ∧θ2 = [1]θ1 ∧θ3 = [1]θ1 ∩ [1]θ3 . Since the lattice Ck(A) of congruence kernels of A is distributive, we have [1]θ2 = [1]θ3 . Assume further that {θ1 , θ2 , θ3 } generates the diamond. Then aditionally θ2 ∧ θ3 = θ2 ∧ θ1 , θ2 ∨ θ3 = θ2 ∨ θ1 , which implies as before [1]θ3 = [1]θ1 . Altogether we have [1]θ1 = [1]θ2 = [1]θ3 .

¤

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Fig. 9

Lemma 2.21. Let A = (A, ≤, 1) be a standard QBCC-algebra with at most two-element cells. Then Con(A) does not contain a sublattice isomorphic to the pentagon. Proof. Assume on the contrary that Con(A) contains a sublattice isomorphic to the pentagon as given in Fig. 9. From Lemma 2.20 we have [1]θ3 = [1]θ2 and hence also ([1]θ1 ∪ [1]θ2 )2 = ([1]θ1 ∪ [1]θ3 )2 . We show that θ3 = θ2 . Since their kernels are equal, it is sufficient to prove θ2 ∩ ∼= θ3 ∩ ∼ (see Lemma 2.14). By Lemma 2.18 and the successive Remark we have

θ1 ∨ θ2 = ([1]θ1 ∪ [1]θ2 )2 ∪ ((θ1 ∪ θ2 )∩ ∼) = θ1 ∨ θ3 = ([1]θ1 ∪ [1]θ3 )2 ∪ ((θ1 ∪ θ3 )∩ ∼). Evidently (θ2 ∩ ∼) ⊆ (θ3 ∩ ∼). Let x ∼ y, hx, yi ∈ θ3 , hx, yi 6∈ θ2 . Then hx, yi ∈ θ1 , i.e. hx, yi ∈ θ1 ∧ θ3 = θ1 ∧ θ2 , a contradiction. So, we have θ2 ∩ ∼= θ3 ∩ ∼. ¤ Theorem 2.22. Let A = (A, ≤, 1) be a standard QBCC-algebra. The lattice Con(A) is distributive iff A contains at most two-element cells. Proof. (⇐) Suppose that A contains at most two-element cells. By Lemma 2.21 Con(A) does not contain pentagon as a sublattice. Assume further that Con(A) contains a sublattice isomorphic to diamond as given in Fig. 8. By Lemma 2.20, I = [1]θ1 = [1]θ2 = [1]θ3 , and since θi = I 2 ∪ (θi ∩ (A \ I)2 ) = I 2 ∪ ((θi ∩ (A \ I)2 )∩ ∼), there exists ha, bi ∈ (A \ I)2 , a ∼ b with the property ha, bi ∈ θi , ha, bi 6∈ θj for some i, j ∈ {1, 2, 3}. Further, θi ∨ θj = I 2 ∪ ((θi ∪ θj )∩ ∼) for all i, j ∈ {1, 2, 3}. Let {i, j, k} = {1, 2, 3}. Evidently, ha, bi ∈ θi ∨ θj , i.e. also ha, bi ∈ θj ∨ θk and since ha, bi 6∈ θj , we have ha, bi ∈ θk . This yields ha, bi ∈ θi ∧ θk = θi ∧ θj , i.e. ha, bi ∈ θj , a contradiction. (⇒) We show that for an at least three-element cell in A, the lattice Con(A) is not distributive. Suppose a ∼ b ∼ c, a 6= b 6= c 6= a and consider the following congruences with the same kernel I = {1} : θ1 ... {I, {a}, {b, c}, remaining classes are singletons} θ2 ... {I, {b}, {a, c}, remaining classes are singletons}

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θ1 ... {I, {c}, {a, b}, remaining classes are singletons}. These congruences are incomparable. One can easily verify, that {θ1 , θ2 , θ3 , θ1 ∧θ2 , θ1 ∨θ2 } is a diamond in Con(A), hence Con(A) is not distributive. ¤

References [1] W.J. Blok and D. Pigozzi: Algebraizable logics, Memoirs of the American Math. Soc., Vol. 396, Providence, Rhode Island, 1989. [2] W.A. Dudek: “The number of subalgebras of finite BCC-algebras”, Bull. of the Inst. of Math., Academia Sinica, Vol. 20(2), (1992), pp. 129–135. [3] W.A.Dudek: “On subalgebras in Hilbert algebras”, Novi Sad J. Math., Vol. 29(2) (1999), pp. 181–192. [4] W.A. Dudek: “Subalgebras in finite BCC-algebras, Bull. of the Inst. of Math., Academia Sinica, Vol. 28, (2000), pp. 201–206. [5] I. Chajda and R. Halaˇs: “Pre-logics BCC-algebras”, Math. Slovaca, Vol. 52(2), (2002), pp. 157–175. [6] R. Halaˇs: “BCC-algebras inherited from posets”, Multiple Valued Logic, Vol. 8, (2002), pp. 223–235. [7] R. Halaˇs and J. Ort: “Standard QBCC-algebras”, Demonstratio Math., Vol. 36(1), (2003), pp. 1–10. [8] R. Halaˇs and J. Ort: “QBCC-algebras inherited from qosets”, Math. Slovaca, Vol. 53(4), (2003), pp. 331–340. [9] Y. Imai and K. Is´eki: “On axiomatic system of propositional calculi”, XIV, Proc. Japan Acad., Vol. 42, (1966), pp. 19–22. [10] Y. Komori: “The class of BCC-algebras is not a variety”, Math. Japon., Vol. 29, (1984), pp. 391–394. [11] A. Wro´ nski: “An algebraic motivation for BCK-algebras”, Math. Japon., Vol. 30, (1985), pp. 183–193. [12] A. Wro´ nski: “BCK-algebras do not form a variety”, Math. Japon., Vol. 28, (1983), pp. 211–213.

CEJM 3(1) 2005 98–104

Oscillations of Linear Integro-Differential Equations ˇ Rudolf Olach1∗ , Helena Samajov´ a2 1

Department of Mathematical Analysis and Applied Mathematics, ˇ University of Zilina, ˇ 010 26 Zilina, Slovak Republic 2 Department of Mathematics, Slovak Technical University, 812 31 Bratislava, Slovak Republic

Received 8 July 2004; accepted 19 October 2004 Abstract: Sufficient conditions which guarantee that certain linear integro-differential equation cannot have a positive solution are established. c Central European Science Journals. All rights reserved. ° Keywords: Integro-differential equation, oscillation, sufficient condition MSC (2000): 34K15, 34C10

1

Introduction

We consider linear integro-differential equation of the form Z t x(t) ˙ + x(t − s) dr(t, s) = 0, t ≥ 0,

(1)

0

where the integral is in the sense of Riemann-Stieltjes, under the standing hypotheses: (H1 ) r(t, 0) = 0 for t ∈ [ 0, ∞); (H2 ) r(t, t) : [ 0, ∞) → R is continuous; (H3 ) r(t, s) is increasing with respect to s for s ∈ [ 0, t]. By a solution of Eq. (1) we mean a continuous function x, which is defined for t ≥ 0 and which satisfies Eq. (1) for t ≥ 0. Our aim is to obtain sufficient conditions which guarantee that Eq. (1) cannot have a solution x which is positive on [ 0, ∞). ∗

E-mail: [email protected]

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The problem of oscillation and nonoscillation for linear Volterra-type integro-differential equation Z t k(t − s)x(s) ds = 0, t ≥ 0, x(t) ˙ + 0

has received attention in the recent years; see e. g. Gy¨ori & Ladas [3], Ladas et al [4] and the references cited therein. In addition the cognate equations to (1) have found a variety of applications in several fields of natural sciences, i. e. in the theory of a circulating fuel nuclear reactor [2]. The present paper is an attempt to make a study of oscillatory properties of the integro-differential equation of the form (1).

2

Main results

The following lemma will be useful in the proof of the main theorems. Lemma 2.1. Suppose there exists a continuous function σ(t) such that 0 < σ(t) < t, t − σ(t) is nondecreasing and lim (t − σ(t)) = ∞,

t→∞

lim inf t→∞

Z

t

[r(s, s) − r(s, σ(s))] ds > 0

(2)

t−σ(t)

and let x(t) be a positive (or negative) solution of Eq. (1) on ( 0, ∞). Then there exists a T > 0 such that x(t − σ(t)) x(t) is bounded on [ T, ∞). Proof. Without loss of generality we may assume that x(t) is positive for t ∈ ( 0, ∞). Then in view of conditions (H1 )−(H3 ) and Eq. (1), x(t) must eventually be monotonically decreasing. So we obtain −x(t) ˙ ≥

Z

t

x(t − s) dr(t, s) ≥ [r(t, t) − r(t, σ(t))]x(t − σ(t)) σ(t)

for t > 0. According to condition (2) there exists a T > 0 and ε > 0 such that Z

t

[r(s, s) − r(s, σ(s))] ds ≥ 2ε t−σ(t)

for t ∈ [ T, ∞). Then for arbitrary t ≥ T we can find a t∗ > t such that Z

t

[r(s, s) − r(s, σ(s))] ds ≥ ε and t∗ −σ(t∗ )

Z

t∗

[r(s, s) − r(s, σ(s))] ds ≥ ε. t

(3)

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With regard to the inequality (3) and decreasing character of x(t) we get ∗

∗

x(t − σ(t )) − x(t) ≥ x(t − σ(t))

t

Z

[r(s, s) − r(s, σ(s))] ds

t∗ −σ(t∗ )

≥ εx(t − σ(t)), ∗

∗

∗

x(t) − x(t ) ≥ x(t − σ(t ))

Z

t∗

[r(s, s) − r(s, σ(s))] ds t

≥ εx(t∗ − σ(t∗ )),

t ≥ T.

Combining the above inequalities we obtain x(t) > ε2 x(t − σ(t)) for t ∈ [ T, ∞). The proof is complete. Theorem 2.2. Suppose there exists a continuous function σ(t) such that (2) holds and lim sup

Z

t

r(s, s) ds < ∞, ¡ Rt ¢ Rt exp λ r(ξ, ξ) dξ dr(t, s) 0 t−σ(s) >0 −λ + lim inf t→∞ r(t, t) t→∞

(4)

t−σ(t)

(5)

for all λ > 0. Then Eq. (1) cannot have a solution x which is positive (or negative) on [ 0, ∞). Proof. Assume for the sake of contradiction, that Eq. (1) has a solution x which is positive on [ 0, ∞). Set Λ = {λ > 0 : x(t) ˙ + λr(t, t)x(t) ≤ 0 for all large t}. By using the decreasing nature of x on [ 0, ∞), it follows from Eq. (1) that x(t) ˙ + x(t)

Z

t

dr(t, s) ≤ 0, 0

x(t) ˙ + r(t, t)x(t) ≤ 0,

t ≥ 0,

which means that 1 ∈ Λ and Λ is a subinterval of ( 0, ∞). Now we prove that sup Λ < ∞. By Lemma 2.1, x(t − σ(t))/x(t) is bounded on [ T, ∞), T > 0. Thus x(t − σ(t)) ≤c x(t)

for t ≥ T,

(6)

where constant c > 0 we can choose so large that exp(kc) > c, where due to the condition (2) we choose Z t r(s, s) ds for t ≥ t1 ≥ T. 0
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Now we claim that sup Λ ≤ c. Otherwise c ∈ Λ and we obtain ´ ´i £ ³ Z t ³ Z t ¤ dh r(s, s) ds ≤ 0, r(s, s) ds = x(t) ˙ + cr(t, t)x(t) exp c x(t) exp c dt t1 t1

since c ∈ Λ. This implies that the function

´ ³ Z t r(s, s) ds x(t) exp c t1

is decreasing on [t1 , ∞). Hence ³ Z x(t − σ(t)) exp c

t−σ(t) t1

³ Z t ´ r(s, s) ds ≥ x(t) exp c r(s, s) ds ,

³ Z x(t − σ(t)) ≥ x(t) exp c

´

t1

t

´

r(s, s) ds ≥ x(t) exp(kc) > cx(t), t−σ(t)

t ≥ t2 ≥ t1 , where t2 is sufficiently large. Then x(t − σ(t)) > cx(t) for t ≥ t2 , which is a contradiction to (6). Thus sup Λ ≤ c < ∞. Set λ∗ = sup Λ and let µ be an arbitrary number in the interval ( 0, λ∗ ). Then λ∗ − µ = β ∈ Λ, and there exists a T1 ≥ t2 such that x(t) ˙ + βr(t, t)x(t) ≤ 0

for t ≥ T1 .

Then for any t, s with t ≥ T1 and 0 ≤ s ≤ t we get ³ Z t x(ξ) ³ ´ x(t − s) x(t) ´ ˙ = exp − = exp − ln dξ x(t) x(t − s) t−s x(ξ) ´ ´ ³ Z t ³ Z t r(ξ, ξ) dξ , r(ξ, ξ) dξ ≥ exp β ≥ exp β t−σ(s)

t−s

that is

³ Z x(t − s) ≥ x(t) exp β

t t−σ(s)

´ r(ξ, ξ) dξ .

From Eq. (1) and above inequality it follows that for t ≥ T1 , hZ t ³ Z t ´ i 0 ≥ x(t) ˙ + exp β r(ξ, ξ) dξ dr(t, s) x(t) 0 t−σ(s) ¡ Rt ¢ Rt exp β t−σ(s) r(ξ, ξ) dξ dr(t, s) 0 r(t, t)x(t). = x(t) ˙ + r(t, t) According to (7) we claim that ¡ Rt ¢ Rt exp β r(ξ, ξ) dξ dr(t, s) 0 t−σ(s) ≤ λ∗ . lim inf t→∞ r(t, t)

(7)

(8)

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Otherwise there exist a λ∗1 > λ∗ and a T2 ≥ T1 such that ¡ Rt ¢ Rt exp β t−σ(s) r(ξ, ξ) dξ dr(t, s) 0 r(t, t)

≥ λ∗1

for all t ≥ T2 and therefore (7) yields that 0 ≥ x(t) ˙ + λ∗1 r(t, t)x(t)

for t ≥ T2 .

Hence λ∗1 ∈ Λ, which contradicts the hypothesis that λ∗1 > λ∗ . Thus (8) has been established. Finally (8) implies that £ ∗ ¤ Rt Rt exp (λ − µ) r(ξ, ξ) dξ dr(t, s) 0 t−σ(s) ≤ 0. −λ∗ + lim inf t→∞ r(t, t) As µ ∈ (0, λ∗ ) is arbitrary and with regard to (4) we obtain ¡ ∗Rt ¢ Rt exp λ r(ξ, ξ) dξ dr(t, s) 0 t−σ(s) −λ∗ + lim inf ≤ 0, t→∞ r(t, t) which contradicts (5) and completes the proof of the theorem. Corollary 2.3. Suppose there exists a continuous function σ(t) such that (2), (4) hold and RtRt r(ξ, ξ) dξ dr(t, s) 1 0 t−σ(s) lim inf > . t→∞ r(t, t) e Then Eq. (1) cannot have a solution x which is positive (or negative) on [0, ∞). Proof. Since we obtain

1 min eλa = ea, λ>0 λ ³ 1 inf exp λ λ>0 λ

Z

t

a > 0,

´

r(ξ, ξ) dξ = e t−σ(s)

Z

t

r(ξ, ξ) dξ, t−σ(s)

t ≥ T > 0, 0 ≤ s ≤ t. Then for λ > 0 we get ¡ Rt ¢ Rt exp λ t−σ(s) r(ξ, ξ) dξ dr(t, s) 0 −λ + lim inf t→∞ r(t, t) h

= λ − 1 + lim inf h

t→∞

Rt

≥ λ − 1 + e lim inf t→∞

1 0 λ

¡ Rt ¢ exp λ t−σ(s) r(ξ, ξ) dξ dr(t, s) i r(t, t) r(ξ, ξ) dξ dr(t, s) i t−σ(s)

RtRt 0

r(t, t)

> 0,

thus the condition (5) is satisfied and we can apply Theorem 2.2.

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Example. Consider the equation (1) where r(t, s) =

cs , (t + 1)2

c ∈ (0, ∞).

If σ(t) = 12 t, then lim inf t→∞

lim inf t→∞

h

Z

1 r(t, t)

t

c [r(s, s) − r(s, σ(s))] ds = ln 2, 2 t−σ(t) Z t lim sup r(s, s) ds = c ln 2, t→∞ t−σ(t) Z tZ t i r(ξ, ξ) dξ dr(t, s) = c(1 − ln 2). 0

t−σ(s)

By Corollary 2.3 for

1 e(1 − ln 2)

c>

equation (1) cannot have a solution x which is positive (or negative) on [0, ∞). In a special case when r(t, s) =

Z

s

k(ξ) dξ, 0

we can transform Eq. (1) into the form Z t k(s)x(t − s) ds = 0, x(t) ˙ + 0

which is equivalent to equation x(t) ˙ +

Z

t

k(t − s)x(s) ds = 0,

t ≥ 0.

(9)

0

Corollary 2.4. Suppose there exists a continuous function σ(t) such that 0 < σ(t) < t, t − σ(t) is nondecreasing, k : (0, ∞) → (0, ∞) is continuous and

lim inf t→∞

Z

t t−σ(t)

lim sup t→∞

lim inf t→∞

lim (t − σ(t)) = ∞, Zt→∞ s k(ξ) dξ ds > 0,

Z

σ(s)

t

Z

s

k(ξ) dξ ds < ∞,

t−σ(t) 0 Rξ Rt Rt k(s) t−σ(s) 0 k(u) du dξ 0 Rt k(s) ds 0

ds

>

1 . e

Then equation (9) cannot have a solution x which is positive (or negative) on [0, ∞).

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Corollary 2.5. Suppose that σ(t) = σ > 0, k : (0, ∞) → (0, ∞) is continuous and Z t Z s 1 lim < c < ∞. k(ξ) dξ ds = c, t→∞ t−σ 0 e Then Eq. (9) cannot have a solution x which is positive (or negative) on [0, ∞). For equation (9) we can use also the interesting Corollary 9.1.1 in [3]. Corollary 2.6. (9.1.1, [3]) Suppose that k : (0, ∞) → (0, ∞) is continuous. If Z ∞ 1 sk(s) ds > , e 0 then Eq. (9) cannot have a solution x which is positive (or negative) on [0, ∞).

Acknowledgements This research was supported by the grant 1/0026/03 of Scientific Grant Agency of Ministry of Education of Slovak Republic and Slovak Academy of Sciences.

References [1] L. Berezansky, E. Braverman: “On oscillation of equations with distributed delay”, Z. Anal. Anwendungen, Vol. 20, (2001), pp. 489–504. [2] W.K. Ergen: “Kinetics of the circulating fuel nuclear reactor”, Journal of Applied Physics, Vol. 25, (1954), pp. 702–711. [3] I. Gy¨ori, G. Ladas: Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. [4] G. Ladas, CH.G. Philos, Y.G. Sficas: “Oscillations of integro-differential equations”, Differential and Integral Equations, Vol. 4, (1991), pp. 1113–1120. [5] G.S. Ladde, V. Lakshmikantham, B.G. Zhang: Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York and Basel, 1987. [6] R. Olach: “Observation of a Feedback Mechanism in a Population Model”, Nonlinear Analysis, Vol. 41, (2000), pp. 539–544. [7] X.H. Tang: “Oscillation of first order delay differential equations with distributed delay”, J. Math. Anal. Appl., Vol. 289, (2004), pp. 367–378.

CEJM 3(1) 2005 105–124

Localization of LMn - algebras Florentina Chirte¸s∗ Department of Mathematics, University of Craiova, Al. I. Cuza Street, 13, 200585-Craiova, Romania

Received 29 July 2004; accepted 9 December 2004 Abstract: The aim of this paper is to define the localization LMn - algebra of an LMn − algebra L with respect to a topology F on L; In Section 5 we prove that the maximal LMn algebra of fractions (defined in [3]) and the LMn - algebra of fractions relative to an ∧− closed system (defined in Section 2) are LMn - algebras of localization. c Central European Science Journals. All rights reserved. ° Keywords: LMn - algebra,topology, F− multiplier, multiplier, LMn - algebra of fractions, maximal LMn - algebra of fractions, LMn - algebra of localization MSC (2000): 03D20, 06G30

Introduction The concept of multiplier for distributive lattices was defined by W. H. Cornish in [7]. J. Schmid used multipliers in order to give a non–standard construction of the maximal lattice of quotients for a distributive lattice (see [13]). A direct treatment of the lattices of quotients can be found in [14]. In [10], G. Georgescu exhibited the localization lattice LF of a distributive lattice L with respect to a topology F on L mimicking the familiar construction for rings (see [12]) or monoids (see [15]). For the case of Hilbert and Heyting algebras see [4] and [8]. The concepts of LMn - algebra of fractions relative to an ∧− closed system, LMn algebra of fractions and maximal LMn - algebra of fractions was defined by the author in Section 2 and [3]. ∗

E-mail: [email protected]

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Definitions and preliminaries

Let n be an integer, n ≥ 2. Definition 1.1. ([2])An n − valued Lukasiewicz − M oisil algebra (shortly, LMn - algebra) is an algebra L = (L, ∧, ∨, N, 0, 1, {ϕi }1≤i≤n−1 ) of type (2, 2, 1, 0, 0, {1}1≤i≤n−1 ) satisfying the following conditions: (1.1) (L, ∧, ∨, N, 0, 1) is a De M organ algebra, (1.2) ϕ1 , ..., ϕn−1 : L → L are bounded lattice morphisms such that for every x, y ∈ L: (1.2.1) ϕi (x) ∨ N ϕi (x) = 1 for every i = 1, ..., n − 1, (1.2.2) ϕi (x) ∧ N ϕi (x) = 0 for every i = 1, ..., n − 1, (1.2.3) ϕi ϕj (x) = ϕj (x) for every i, j = 1, ..., n − 1, (1.2.4) ϕi (N x) = N ϕj (x) for every i, j = 1, ..., n − 1 with i + j = n, (1.2.5) ϕ1 (x) ≤ ϕ2 (x) ≤ ... ≤ ϕn−1 (x), (1.2.6) If ϕi (x) = ϕi (y) for every i = 1, ..., n − 1, then x = y. The relation (1.2.6) is called the determination principle. As consequences of the determination principle we obtain: (1.2.7) If x, y ∈ L, then x ≤ y iff ϕi (x) ≤ ϕi (y) for all i = 1, ..., n − 1, (1.2.8.) ϕ1 (x) ≤ x ≤ ϕn−1 (x) for all x ∈ L. We denote an LMn -algebra L = (L, ∧, ∨, N, 0, 1, {ϕi }1≤i≤n−1 ) by its universe L. Remark 1.2. The endomorphisms {ϕi }1≤i≤n−1 are called chrysippian endomorphisms. Examples: 1 , ..., n−2 , 1}.We define x∨y = max{x, y}, x∧y = min{x, y}, N x = 1−x 1. Let Ln = {0, n−1 n−1 j n−1−j j (N ( n−1 ) = n−1 ) and ϕi : Ln → Ln , ϕi ( n−1 ) = 0 if i + j < n and 1 if i + j ≥ n, for i, j = 1, ..., n − 1. Then (Ln , ∧, ∨, N, 0, 1, {ϕi }1≤i≤n−1 ) is an LMn -algebra. 2. If (B, ∧, ∨,′ , 0, 1) is a Boolean algebra, then (B, ∧, ∨,′ , 0, 1, {ϕi }1≤i≤n−1 ) is an LMn algebra, where ϕi = 1B for every 1 ≤ i ≤ n − 1. ′ 3. Let (B, ∨, ∧, , 0, 1) a Boolean algebra and D(B) = {(x1 , ..., xn−1 ) ∈ B n−1 : x1 ≤ ... ≤ xn−1 }. We define pointwise the infimum and the supremum, N (x1 , ..., xn−1 ) = ′ ′ (xn−1 , ..., x1 ) and ϕi (x1 , ..., xn−1 ) = (xi , ..., xi ) for all i = 1, ..., n − 1. Then (D(B), ∧, ∨, N, 0, 1, {ϕi }1≤i≤n−1 ) is an LMn -algebra. In the rest of this paper, by L we denote an LMn -algebra. We denote by C(L) the set of all complemented elements of L and we call it the center of L; it is easy to see that (C(L), ∨, ∧, N, 0, 1) is a Boolean algebra. Lemma 1.3. ([2])Let L be an LMn -algebra.The following are equivalent: (i) e ∈ C(L),

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(ii) there are i ∈ {1, ..., n − 1} and x ∈ L such that e = ϕi (x), (iii) there is i ∈ {1, ..., n − 1} such that e = ϕi (e), (iv) e = ϕi (e) for every i = 1, ..., n − 1, (v) ϕi (e) = ϕj (e) for every i, j = 1, ..., n − 1. Remark 1.4. If x ∈ L, then ϕi (x) ∈ C(L) for every i = 1, ..., n − 1. Lemma 1.5. ([2])Let L be an LMn -algebra.The following are equivalent: (i) e ∈ C(L), (ii) N e ∈ C(L), (iii) e ∧ N e = 0, (iv) e ∨ N e = 1. Lemma 1.6. If L is an LMn -algebra, then for every x ∈ L, x ∧ ϕ1 (N x) = 0 which is equivalent to x ∧ N ϕn−1 (x) = 0. Proof. For every x ∈ L we have x ≤ ϕn−1 (x), so x ∧ ϕ1 (N x) = x ∧ N ϕn−1 (x) ≤ ϕn−1 (x) ∧ N ϕn−1 (x) = 0 (by(1.2.2)),

hence x ∧ ϕ1 (N x) = 0.

¤

Theorem 1.7. ([1]) For an LMn -algebra L (with 0 6= 1), the following are equivalent: (i) C(L) = {0, 1}, (ii) L is a chain, (iii) L is subdirectly irreducible. Corollary 1.8. ([2]) Every chain which is an LMn -algebra is finite. ′

′

Definition 1.9. ([2])Let L and L be LMn -algebras. A function f : L → L is a morphism of LMn -algebras iff it satisfies the following conditions, for every x, y ∈ L : (i) f (x ∨ y) = f (x) ∨ f (y), (ii) f (x ∧ y) = f (x) ∧ f (y), (iii) f (0) = 0, f (1) = 1, (iv) f (ϕi (x)) = ϕi (f (x)) for every i = 1, ..., n − 1. Remark 1.10. It follows (from 1.2.4 and 1.2.6) that f (N x) = N f (x) for every x ∈ L. We denote by LMn the category of LMn -algebras.

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Definition 1.11. ([2]) Let L an LMn -algebra. We say that a nonempty subset I ⊆ L in an n − ideal if I is an ideal of the lattice L and if x ∈ I, then ϕn−1 (x) ∈ I. Remark 1.12. From (1.2.5) we deduce that if I ⊆ L is an n-ideal and x ∈ I, then ϕi (x) ∈ I for every i ∈ {1, ..., n − 1}. We denote by Idn(L) the set of all n − ideals of the LMn - algebra L and by Id(C(L)) the set of all ideals of the Boolean algebra C(L). If X ⊆ L is a nonempty set, we denote by < X > the n-ideal generated by X. We have that: p

< X >= {y ∈ L : there exist p ≥ 1 and x1 , ..., xp ∈ X such that y ≤ ϕn−1 ( ∨ xi )}. i=1

In particular, for a ∈ L, < a >= {x ∈ L : x ≤ ϕn−1 (a)} and if a ∈ C(L), then < a >= {x ∈ L : x ≤ a} = (a]. Let I be an n-ideal and x ∈ L. We denote by (I : x) = {y ∈ L : x ∧ y ∈ I}. Lemma 1.13. The set (I : x) is an n-ideal. Proof. Let y1 , y2 ∈ (I : x). Then x∧y1 , x∧y2 ∈ I, hence x∧(y1 ∨y2 ) = (x∧y1 )∨(x∧y2 ) ∈ I, that is, y1 ∨ y2 ∈ (I : x). If y1 ∈ (I : x) and y2 ≤ y1 , then x ∧ y1 ∈ I and x ∧ y2 ≤ x ∧ y1 , hence x ∧ y2 ∈ I, that is, y2 ∈ (I : x). If y ∈ (I : x) then x ∧ y ∈ I, hence ϕn−1 (x) ∧ ϕn−1 (y) = ϕn−1 (x ∧ y) ∈ I. But x ∧ ϕn−1 (y) ≤ ϕn−1 (x) ∧ ϕn−1 (y), so x ∧ ϕn−1 (y) ∈ I, that is, ϕn−1 (y) ∈ (I : x). Remark 1.14. ([10]) If I is an n − ideal of L, then I b = I ∩ C(L) is an ideal of the Boolean algebra C(L); also, every ideal J of C(L) induce an n-ideal ϕ−1 n−1 (J) of L. The −1 b mappings I 7→ I , J 7→ ϕn−1 (J) establish a bijection between the n − ideals of L and the ideals of C(L). Definition 1.15. ([2]) A congruence on an LMn -algebra L is an equivalence relation on L compatible with the operations ∧, ∨, N, ϕi , for every i = 1, ..., n − 1. Proposition 1.16. ([2])For an equivalence relation ρ on an LMn -algebra L, the following conditions are equivalent: (1) ρ is a congruence on L, (2) ρ is compatible with ∧, ∨, ϕi , for every i = 1, ..., n − 1.

2

LMn −algebra of fractions relative to an ∧-closed system

Definition 2.1. A nonempty subset S ⊆ L is called ∧−closed system in L if (2.1) 1 ∈ S, (2.2) x, y ∈ S implies x ∧ y ∈ S,

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(2.3) x ∈ S implies ϕn−1 (x) ∈ S. We denote by S(L) the set of all ∧−closed systems of L (clearly {1}, L ∈ S(L)). For S ∈ S(L), on the LMn -algebra L we consider the relation θS defined by (x, y) ∈ θS iff there exists s ∈ S such that x ∧ ϕn−1 (s) = y ∧ ϕn−1 (s). Lemma 2.2. θS is a congruence on L. Proof. The reflexivity (since 1 ∈ S) and the symmetry of θS are immediately. To prove the transitivity of θS , let (x, y), (y, z) ∈ θS . Thus, there are s, s′ ∈ S such that x ∧ ϕn−1 (s) = y ∧ ϕn−1 (s) and y ∧ ϕn−1 (s′ ) = z ∧ ϕn−1 (s′ ). If denote s′′ = s ∧ s′ ∈ S, then x ∧ ϕn−1 (s′′ ) = x ∧ ϕn−1 (s ∧ s′ ) = (x ∧ ϕn−1 (s)) ∧ ϕn−1 (s′ ) = (y ∧ ϕn−1 (s)) ∧ ϕn−1 (s′ ) = y ∧ ϕn−1 (s′ ) ∧ ϕn−1 (s) = z ∧ ϕn−1 (s′ ) ∧ ϕn−1 (s) = z ∧ ϕn−1 (s ∧ s′ ) = z ∧ ϕn−1 (s′′ ), hence (x, z) ∈ θS . To prove the compatibility of θS with the operations ∧, ∨, and ϕi for every i = 1, ..., n − 1, let x, y, z, t ∈ L such that (x, y), (z, t) ∈ θS . Thus there are s, s′ ∈ S such that x ∧ ϕn−1 (s) = y ∧ ϕn−1 (s) and z ∧ ϕn−1 (s′ ) = t ∧ ϕn−1 (s′ ). If we denote s′′ = s ∧ s′ ∈ S, then (x ∧ z) ∧ ϕn−1 (s′′ ) = (y ∧ t) ∧ ϕn−1 (s′′ ), hence (x ∧ z, y ∧ t) ∈ θS . From x ∧ ϕn−1 (s′′ ) = y ∧ ϕn−1 (s′′ ), z ∧ ϕn−1 (s′′ ) = t ∧ ϕn−1 (s′′ ) and the distributivity of L we deduce (x ∨ z) ∧ ϕn−1 (s′′ ) = (y ∨ t)∧ ϕn−1 (s′′ ), that is, (x ∨ z, y ∨ t) ∈ θS . From x ∧ ϕn−1 (s) = y ∧ ϕn−1 (s) we deduce that ϕi (x ∧ ϕn−1 (s)) = ϕi (y ∧ ϕn−1 (s)), for every i = 1, ..., n−1, that is, ϕi (x)∧ϕn−1 (s) = ϕi (y)∧ϕn−1 (s), hence (ϕi (x), ϕi (y)) ∈ θS .¤ For x ∈ L we denote by x/S the equivalence class of x relative to θS and by L[S] = L/θS . By pS : L → L[S] we denote the canonical map defined by pS (x) = x/S, for every x ∈ L. Clearly, in L[S], 0 = 0/S, 1 = 1/S and for every x, y ∈ L, x/S ∧ y/S = (x ∧ y)/S x/S ∨ y/S = (x ∨ y)/S N (x/S) = (N x)/S ϕ¯i : L[S] → L[S], ϕ¯i (x/S) = (ϕi (x))/S for every i = 1, ..., n − 1, (for every i = 1, ..., n − 1, ϕ¯i is correct defined because for x, y ∈ L such that x/S = y/S, there exists s ∈ S such that x∧ϕn−1 (s) = y∧ϕn−1 (s), so, ϕi (x)∧ϕn−1 (s) = ϕi (y)∧ϕn−1 (s), that is, ϕi (x)/S = ϕi (y)/S). Remark 2.3. Since for every s ∈ S, ϕn−1 (s) ∧ ϕn−1 (s) = 1 ∧ ϕn−1 (s), we deduce that ϕn−1 (s)/S = 1/S = 1, hence pS (ϕn−1 (S)) = {1}.

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Proposition 2.4. If a ∈ L, then a/S ∈ C(L[S]) iff there exists s ∈ S such that a ∧ ϕn−1 (s) ∈ C(L). So, if a ∈ C(L), then a/S ∈ C(L[S]). Proof. For a ∈ L, we have a/S ∈ C(L[S]) iff ϕ¯i (a/S) = a/S for all i = 1, ..., n − 1, that is, ϕi (a)/S = a/S for all i = 1, ..., n − 1. So, (ϕi (a), a) ∈ θS , which it means that there exists s ∈ S such that ϕi (a)∧ϕn−1 (s) = a∧ϕn−1 (s), that is, ϕi (a∧ϕn−1 (s)) = a∧ϕn−1 (s), hence a ∧ ϕn−1 (s) ∈ C(L). If a ∈ C(L), since 1 ∈ S and a∧ϕn−1 (1) = a ∈ C(L), we deduce that a/S ∈ C(L[S]).¤ Theorem 2.5. If L is an LMn -algebra and f : L → L′ is a morphism of LMn -algebras such that f (ϕn−1 (S)) = {1}, then, there is a unique morphism of LMn -algebras f ′ : L[S] → L′ , such that the diagram L

p

S −→ L[S]

ց

ւ

f

f′

L

′

is commutative (i.e. f ′ ◦ pS = f ). Proof. If x, y ∈ L and pS (x) = pS (y), then (x, y) ∈ θS , hence, there is s ∈ S such that x ∧ ϕn−1 (s) = y ∧ ϕn−1 (s). Since f is morphism of LMn -algebras, we obtain that f (x ∧ ϕn−1 (s)) = f (y ∧ ϕn−1 (s)), that is, f (x) ∧ f (ϕn−1 (s)) = f (y) ∧ f (ϕn−1 (s)). But f (ϕn−1 (s)) = 1, so f (x) ∧ 1 = f (y) ∧ 1, that is, f (x) = f (y). We deduce that the map f ′ : L[S] → L′ defined for x ∈ L by f ′ (x/S) = f (x) is correct defined. Clearly, f ′ is a morphism of LMn -algebras. The unicity of f ′ follows from the fact that pS is an onto map. ¤ Remark 2.6. The previous theorem allows us to call L[S] the LMn −algebra of fractions relative to the ∧−closed system S. Examples: (1) If S = {1} then θS is the identity congruence of L, hence L[S] = L. (2) If S is an ∧ − closed system such that 0 ∈ S ( for example S = L or S = C(L)), then for every x, y ∈ L, (x, y) ∈ θS (since x ∧ ϕn−1 (0) = y ∧ ϕn−1 (0)), hence in this case L[S] = {0}.

3

Topologies on an LMn -algebra

Definition 3.1. ([10]) A non-empty set F of n-ideals of L will be called a topology on L if the following properties hold: (T1 ) If I ∈ F, x ∈ L then (I : x) ∈ F,

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(T2 ) If I1 , I2 ∈ Idn(L) and I2 ∈ F, if (I1 : x) ∈ F for all x ∈ I2 , then I1 ∈ F. Lemma 3.2. ([10]) If F is a topology on L, then: (i) If I1 ∈ F and I2 is an n-ideal with I1 ⊆ I2 , then I2 ∈ F, (ii) If I1 , I2 ∈ F, then I1 ∩ I2 ∈ F, (iii) (F ∪ {⊘}, L) is a topological space. Remark 3.3. ([10]) Any intersection of topologies on L is a topology, hence the set G(L) of the topologies of L is a complete lattice with respect to inclusion. Examples: 1. If I ∈ Idn(L), then the set F(I) = {I ′ ∈ Idn(L) : I ⊆ I ′ } is a topology on L. 2. A non-empty set I ⊆ L will be called regular if for every x, y ∈ L such that e∧x = e∧y for every e ∈ I, then x = y. If we denote R(L) = {I ⊆ L : I is a regular subset of L}, then Idn(L) ∩ R(L) is a topology on L . 3. For any ∧− closed subset S of L(see Definition 2.1) we set FS = {I ∈ Idn(L) : I ∩ S 6= ⊘}. Then FS is a topology on L .

4

F-multipliers and localization LMn -algebra

Let F be a topology on L. We consider the relation θF of L (x, y) ∈ θF iff there exists I ∈ F such that e ∧ x = e ∧ y for every e ∈ I. Lemma 4.1. θF is a congruence on L. Proof. The reflexivity and the symmetry of θF are immediate; in order to prove the transitivity of θF let (x, y), (y, z) ∈ θF . Then, there exists I1 , I2 ∈ F such that e∧x = e∧y for every e ∈ I1 and f ∧ y = f ∧ z for every f ∈ I2 . If we set I = I1 ∩ I2 ∈ F, then for every g ∈ I, g ∧ x = g ∧ z, hence (x, z) ∈ θF . For the compatibility of θF with the operations ∧, ∨ let x, y, z, t ∈ L such that (x, y), (z, t) ∈ θF , that is, there exists I, J ∈ F such that e ∧ x = e ∧ y for every e ∈ I, and f ∧ z = f ∧ t for every f ∈ J. If we denote K = I ∩ J, then K ∈ F and for every g ∈ K, g ∧ x = g ∧ y and g ∧ z = g ∧ t. Then g ∧ (x ∧ z) = g ∧ (y ∧ t) and g ∧ (x ∨ z) = g ∧ (y ∨ t), that is, (x ∧ z, y ∧ t) ∈ θF and (x ∨ z, y ∨ t) ∈ θF .

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For the compatibility of θF with ϕi for every i = 1, ..., n − 1, let (x, y) ∈ θF and i ∈ {1, ..., n−1} fixed. Then, there exists I ∈ F such that e∧x = e∧y for every e ∈ I. Since for every j ∈ {1, ..., n−1} and e ∈ I, ϕj (e) ∈ I we deduce that ϕj (e)∧x = ϕj (e)∧y. Then, ϕi (ϕj (e)∧x) = ϕi (ϕj (e)∧y) ⇔ ϕj (e)∧ϕi (x) = ϕj (e)∧ϕi (y) ⇔ ϕj (e∧ϕi (x)) = ϕj (e∧ϕi (y)) for every j ∈ {1, ..., n − 1}. By (1.2.6) we deduce that e ∧ ϕi (x) = e ∧ ϕi (y) for every e ∈ I. Therefore (ϕi (x), ϕi (y)) ∈ θF for every i = 1, ..., n − 1. ¤ We shall denote by x/θF the congruence class of an element x ∈ L, by L/θF the quotient LMn -algebra and by pF : L → L/θF the canonical morphism of LMn -algebras. We denote the chrysippian endomorphisms of L/θF by ϕi and we have ϕi (x/θF ) = ϕi (x)/θF . Proposition 4.2. For a ∈ L, a/θF ∈ C(L/θF ) iff there exists I ∈ F such that e∧ϕi (a) = e ∧ a for every e ∈ I and i ∈ {1, ..., n − 1}. So, if a ∈ C(L), then a/θF ∈ C(L/θF ). Proof. For a ∈ L, a/θF ∈ C(L/θF ) iff ϕi (a/θF ) = a/θF iff ϕi (a)/θF = a/θF for every i = 1, ..., n − 1. So, (ϕi (a), a) ∈ θF , that is, there exists I ∈ F such that e ∧ ϕi (a) = e ∧ a for every e ∈ I and i ∈ {1, ..., n − 1}. So, if a ∈ C(L), then for every I ∈ F and e ∈ I, e ∧ ϕi (a) = e ∧ a, hence a/θF ∈ C(L/θF ). ¤ Corollary 4.3. If F = Idn(L) ∩ R(L), then a ∈ C(L) iff a/θF ∈ C(L/θF ). We recall ([3]) some folklore about multipliers. Let (A, ≤) be a poset. A nonempty subset I ⊆ A is an order ideal ( also known as a down-set or decreasing set) in A whenever x ≤ y ∈ I implies x ∈ I; we denote by I(A) the set of all order ideals in A. If A is an inf-semilattice and I ∈ I(A), a map f : I → A is called a multiplier (alias a partial translation) if f (a ∧ x) = a ∧ f (x) for all a ∈ A and x ∈ I. Such maps have been studied extensively by Cornish in [7]. In this paper, we are concerned with multipliers on an LMn -algebra L; clearly Idn(L) ⊆ I(L). Definition 4.4. ([3])By a partial multiplier of L we mean a map f : I → L, where I ∈ Idn(L), which verifies the following condition: (2.1) f (e ∧ x) = e ∧ f (x), for every e ∈ L and x ∈ I. Sometime we will denote the domain of f by dom(f ); if I = L we say that f is total. To simplify the language, we will use multiplier instead of partial multiplier and total to indicate that the domain of a certain multiplier is L. Examples: 1. The map 0 : L → L defined by 0(x) = 0 for every x ∈ L, is a total multiplier of L.

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2. The map 1 : L → L defined by 1(x) = 1 for every x ∈ L, is also a total multiplier of L. 3. For a ∈ L and I ∈ Idn(L), the map fa : I → L defined by fa (x) = a ∧ x for every x ∈ I, is a multiplier of L (called principal ). If dom(fa ) = L, we denote fa by fa ; clearly, f0 = 0 and f1 = 1. Remark 4.5. ([3]) If I ∈ Idn(L), f : I → L is a multiplier of L, then for all x, y ∈ I, f (f (x)) = f (x), f (x) ≤ x, f (x ∧ y) = f (x) ∧ f (y) and x ∧ f (y) = y ∧ f (x). For I ∈ Idn(L), we denote M (I, L) = {f : I → L : f is a multiplier on L} and M (L) =

∪

I∈Idn(L)

M (I, L).

Remark 4.6. ([3])If we have f ∈ M (I, L) ∩ M (J, L), then I = J, that is, the relation f ∈ M (I, L) determines uniquely I. Definition 4.7. ([3])If I1 , I2 ∈ Idn(L) and fi ∈ M (Ii , L), i = 1, 2, we define f1 ∧ f2 , f1 ∨ f2 : I1 ∩ I2 → L by (f1 ∧ f2 )(x) = f1 (x) ∧ f2 (x), (f1 ∨ f2 )(x) = f1 (x) ∨ f2 (x), for every x ∈ I1 ∩ I2 . Lemma 4.8. ([3]) If I1 , I2 ∈ Idn(L) and fi ∈ M (Ii , L), i = 1, 2, then f1 ∧ f2 , f1 ∨ f2 ∈ M (I1 ∩ I2 , L). Definition 4.9. ([3])For I ∈ Idn(L) and f ∈ M (I, L) we define f ∗ : I → L by f ∗ (x) = x ∧ N f (ϕn−1 (x)), for every x ∈ I. Remark 4.10. For x ∈ L we have 0∗ (x) = x ∧ N 0 = x ∧ 1 = x, that is, 0∗ = 1, and 1∗ (x) = x ∧ N ϕn−1 (x) = 0 (by Lemma 1.6 ), that is, 1∗ = 0. Lemma 4.11. ([3]) If I ∈ Idn(L) and f ∈ M (I, L), then f ∗ ∈ M (I, L). Definition 4.12. ([3])For I ∈ Idn(L) and i = 1, ..., n − 1 we define ϕ˜i : M (I, L) → M (I, L) by ϕ˜i (f )(x) = x ∧ ϕi (f (ϕn−1 (x))), for every f ∈ M (I, L) and x ∈ I.

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Lemma 4.13. ([3])If I ∈ Idn(L) and f ∈ M (I, L), then ϕ˜i (f ) ∈ M (I, L) for all i = 1, ..., n − 1. Proposition 4.14. ([3])(M (L), ∧, ∨,∗ , 0, 1,ϕ˜1 , ..., ϕ˜n−1 ) is an LMn -algebra. Lemma 4.15. ([3])The map vL : L → M (L) defined by vL (a) = fa for every a ∈ L is a monomorphism in LMn . Definition 4.16. Let F be a topology on L. By an F -multiplier of L we mean a map f : I → L/θF , where I ∈ F, which verifies the following condition: (3.1) f (e ∧ x) = e/θF ∧ f (x), for every e ∈ L and x ∈ I. Remark 4.17. If f : I → L/θF is an F -multiplier of L then, for every x, y ∈ I, f (x) ≤ x/θF , f (x ∧ y) = f (x) ∧ f (y) and x/θF ∧ f (y) = y/θF ∧ f (x). If F = {L}, then θF is the identity congruence of L and an F− multiplier is a total multiplier of L in the sense of Definition 4.4. The maps 0, 1 : L → L/θF defined by 0(x) = 0/θF and 1(x) = x/θF for every x ∈ L are multipliers in the sense of Definition 4.16. Also, for a ∈ L and I ∈ F, fa : I → L/θF defined by fa (x) = a/θF ∧ x/θF for every x ∈ I, is an F− multiplier. If dom(fa ) = L, we denote fa by fa ; clearly, f0 = 0. We shall denote by M (I, L/θF ) the set of all the F− multipliers having the domain I ∈ F and by M (L/θF ) = ∪ M (I, L/θF ). I∈F

If I1 , I2 ∈ F , I1 ⊆ I2 , we have a canonical mapping ϕI1 ,I2 : M (I2 , L/θF ) → M (I1 , L/θF ) defined by ϕI1 ,I2 (f ) = f|I1 for f ∈ M (I2 , L/θF ). Let us consider the directed system of sets h{M (I, L/θF )}I∈F , {ϕI1 ,I2 }I1 ,I2 ∈F ,I1 ⊆I2 i and denote by LF the inductive limit (in the category of sets): M (I, L/θF ). LF =lim −−→ I∈F

[ For any F− multiplier f : I → L/θF we shall denote by (I, f ) the equivalence class of f in LF . Remark 4.18. We recall that if fi : Ii → L/θF , i = 1, 2, are F-multipliers, then \ \ (I 1 , f1 ) = (I2 , f2 ) (in LF ) iff there exists I ∈ F , I ⊆ I1 ∩ I2 such that f1|I = f2|I .

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Definition 4.19. If I1 , I2 ∈ Idn(L) and fi ∈ M (Ii , L/θF ), i = 1, 2 we define, f1 ∧ f2 , f1 ∨ f2 : I1 ∩ I2 → L/θF by (f1 ∧ f2 )(x) = f1 (x) ∧ f2 (x), (f1 ∨ f2 )(x) = f1 (x) ∨ f2 (x)

for every x ∈ I1 ∩ I2 . \ \ \ \ \ \ Let (I 1 , f1 ) ∧ (I2 , f2 ) = (I1 ∩ I2 , f1 ∧ f2 ) and (I1 , f1 ) ∨ (I2 , f2 ) = (I1 ∩ I2 , f1 ∨ f2 ). Definition 4.20. If I ∈ Idn(L) and f ∈ M (I, L/θF ) we define f ∗ : I → L/θF by f ∗ (x) = x/θF ∧ N (f (ϕn−1 (x))) for any x ∈ I. df )∗ = (I, [ Let (I, f ∗ ). Clearly, the definitions of the operations ∧, ∨ and

∗

on LF are correct.

Lemma 4.21. If I1 , I2 ∈ Idn(L) and fi ∈ M (Ii , L/θF ), i = 1, 2, then f1 ∧ f2 , f1 ∨ f2 ∈ M (I1 ∩ I2 , L/θF ). Proof. Routine.

¤

Remark 4.22. For x ∈ L we have 0∗ (x) = x/θF ∧ N (0/θF ) = x/θF ∧ 1/θF = x/θF , that is, 0∗ = 1, and similarly 1∗ = 0. Lemma 4.23. If I ∈ Idn(L) and f ∈ M (I, L/θF ), then f ∗ ∈ M (I, L/θF ). Proof. If x ∈ I and e ∈ L, then f ∗ (e ∧ x) = (e ∧ x)/θF ∧ N f (ϕn−1 (e ∧ x)) = e/θF ∧ x/θF ∧ N f (ϕn−1 (e) ∧ ϕn−1 (x)) = e/θF ∧ x/θF ∧ N (ϕn−1 (e)/θF ∧ f (ϕn−1 (x)))

= e/θF ∧ x/θF ∧ (N ϕn−1 (e)/θF ∨ N f (ϕn−1 (x)))

= (e/θF ∧ x/θF ∧ N ϕn−1 (e)/θF ) ∨ (e/θF ∧ x/θF ∧ N f (ϕn−1 (x))) = 0/θF ∨ (e/θF ∧ x/θF ∧ N f (ϕn−1 (x))) = e/θF ∧ f ∗ (x).

¤

Definition 4.24. For I ∈ Idn(L) and i = 1, ..., n − 1 we define ϕ˜i : M (I, L/θF ) → M (I, L/θF ) by ϕ˜i (f )(x) = x/θF ∧ ϕ¯i (f (ϕn−1 (x))) =x/θF ∧ ϕi (f (ϕn−1 (x)))/θF , for every f ∈ M (I, L/θF ) and x ∈ I. Lemma 4.25. If I ∈ Idn(L), f ∈ M (I, L/θF ), then ϕ˜i (f ) ∈ M (I, L/θF ) for all i = 1, ..., n − 1.

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Proof. If x ∈ I and e ∈ L, then for all i = 1, ..., n − 1 we have: ϕ˜i (f )(e ∧ x) = (e ∧ x)/θF ∧ ϕ¯i (f (ϕn−1 (e ∧ x)))

= e/θF ∧ x/θF ∧ ϕ¯i (f (ϕn−1 (e) ∧ ϕn−1 (x)))

= e/θF ∧ x/θF ∧ ϕ¯i (ϕn−1 (e)/θF ∧ f (ϕn−1 (x)))

= e/θF ∧ x/θF ∧ ϕ¯i (ϕn−1 (e)/θF ) ∧ ϕ¯i (f (ϕn−1 (x)))

= e/θF ∧ x/θF ∧ ϕi (ϕn−1 (e))/θF ∧ ϕ¯i (f (ϕn−1 (x)))

= e/θF ∧ x/θF ∧ ϕn−1 (e)/θF ∧ ϕ¯i (f (ϕn−1 (x))) = e/θF ∧ ϕ˜i (f )(x).

¤

F d \ Let ϕF ˜i (f )). i : LF → LF defined by ϕi (I, f ) = (I, ϕ F Proposition 4.26. (LF , ∧, ∨,∗ , 0, 1,ϕF 1 , ..., ϕn−1 ) is an LMn -algebra.

Proof. We verify the axioms of LMn − algebras. In the following we work with f ∈ M (I, L/θF ) and fi ∈ M (Ii , L/θF ) where I, Ii ∈ ′ Idn(L), i = 1, 2 and we denote I = I1 ∩ I2 ∈ Idn(L). (1.1). It is easy to verify that (LF , ∧, ∨, 0, 1) is a bounded distributive lattice. To prove that it is a De M organ algebra, we have for x ∈ I ′ : (f1 ∨ f2 )∗ (x) = x/θF ∧ N ((f1 ∨ f2 )(ϕn−1 (x)))

= x/θF ∧ N (f1 (ϕn−1 (x)) ∨ f2 (ϕn−1 (x)))

= x/θF ∧ N f1 (ϕn−1 (x)) ∧ N f2 (ϕn−1 (x))

= (x/θF ∧ N f1 (ϕn−1 (x))) ∧ (x/θF ∧ N f2 (ϕn−1 (x))) = f1∗ (x) ∧ f2∗ (x) = (f1∗ ∧ f2∗ )(x),

that is, (f1 ∨ f2 )∗ = f1∗ ∧ f2∗ . Also, for every x ∈ I : (f ∗ )∗ (x) = x/θF ∧ N f ∗ (ϕn−1 (x))

= x/θF ∧ N (ϕn−1 (x)/θF ∧ N f (ϕn−1 (ϕn−1 (x)))) = x/θF ∧ N (ϕn−1 (x)/θF ∧ N f (ϕn−1 (x))) = x/θF ∧ (N ϕn−1 (x)/θF ∨ f (ϕn−1 (x)))

= (x/θF ∧ N ϕn−1 (x)/θF ) ∨ (x/θF ∧ f (ϕn−1 (x)))

= 0/θF ∨ (x/θF ∧ f (ϕn−1 (x))) = ϕn−1 (x)/θF ∧ f (x)

= f (x), that is, (f ∗ )∗ = f. Then,

(f1 ∧ f2 )∗ = (f1∗∗ ∧ f2∗∗ )∗ = ((f1∗ ∨ f2∗ )∗ )∗ = (f1∗ ∨ f2∗ )∗∗ = f1∗ ∨ f2∗ . (1.2). ϕF i : LF → LF , for all i = 1, ..., n − 1, are bounded lattice morphisms that satisfy (1.2.1) − (1.2.6).

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For x ∈ I ′ we have: ϕ˜i (f1 ∨ f2 )(x) = x/θF ∧ ϕ¯i (f1 (ϕn−1 (x)) ∨ f2 (ϕn−1 (x)))

= x/θF ∧ (ϕ¯i (f1 (ϕn−1 (x))) ∨ ϕ¯i (f2 (ϕn−1 (x))))

= (x/θF ∧ ϕ¯i (f1 (ϕn−1 (x)))) ∨ (x/θF ∧ ϕ¯i (f2 (ϕn−1 (x))))

= ϕ˜i (f1 (x)) ∨ ϕ˜i (f2 (x))

= (ϕ˜i (f1 ) ∨ ϕ˜i (f2 ))(x),

\ \ hence ϕ˜i (f1 ∨ f2 ) = ϕ˜i (f1 ) ∨ ϕ˜i (f2 ) for all i = 1, ..., n − 1, that is, ϕF i ((I1 , f1 ) ∨ (I2 , f2 )) = F \ F \ F \ F \ \ \ ϕF i ((I1 , f1 ))∨ϕi ((I2 , f2 )) and similarly ϕi ((I1 , f1 )∧(I2 , f2 )) = ϕi ((I1 , f1 ))∧ϕi ((I2 , f2 )),for all i = 1, ..., n − 1. Also, for all x ∈ L and i = 1, ..., n − 1, ϕ˜i (0)(x) = x/θF ∧ ϕ¯i (0(ϕn−1 (x))) = x/θF ∧ d 0/θF = 0/θF = 0(x), that is, ϕ˜i (0) = 0 and similarly ϕ˜i (1) = 1. So, we have ϕF i (L, 0) = [ [ d (L, 0) and ϕF i (L, 1) = (L, 1). (1.2.1). For x ∈ I, then: (ϕ˜i (f ) ∨ (ϕ˜i (f ))∗ )(x) = ϕ˜i (f )(x) ∨ (ϕ˜i (f ))∗ (x)

= (x/θF ∧ ϕ¯i (f (ϕn−1 (x)))) ∨ (x/θF ∧ N ϕ˜i (f )(ϕn−1 (x)))

= x/θF ∧ (ϕ¯i (f (ϕn−1 (x))) ∨ N (ϕn−1 (x)/θF ∧ ϕ¯i (f (ϕn−1 (ϕn−1 (x))))))

= x/θF ∧ (ϕ¯i (f (ϕn−1 (x))) ∨ N ϕn−1 (x)/θF ∨ N ϕ¯i (f (ϕn−1 (x)))) = x/θF ∧ 1/θF = x/θF ,

F [ ∗ [ [ hence ϕ˜i (f )∧(ϕ˜i (f ))∗ = 1 for all i = 1, ..., n−1, that is, ϕF i ((I, f ))∧(ϕi ((I, f ))) = (L, 1). To prove (1.2.2) we use the De M organ relations and (1.2.1) :

ϕ˜i (f ) ∧ (ϕ˜i (f ))∗ = (ϕ˜i (f ))∗∗ ∧ (ϕ˜i (f ))∗ = [(ϕ˜i (f ))∗ ∨ ϕ˜i (f )]∗ = 1∗ = 0,

F [ ∗ [ [ that is, ϕF i ((I, f )) ∨ (ϕi ((I, f ))) = (L, 0).

(1.2.3). For x ∈ I and i, j ∈ {1, ..., n − 1} then: ϕ˜i ϕ˜j (f )(x) = x/θF ∧ ϕ¯i (ϕ˜j (f )(ϕn−1 (x)))

= x/θF ∧ ϕ¯i (ϕn−1 (x)/θF ∧ ϕ¯j (f (ϕn−1 (ϕn−1 (x)))))

= x/θF ∧ ϕ¯i (ϕn−1 (x)/θF ) ∧ ϕ¯i (ϕ¯j (f (ϕn−1 (x))))

= x/θF ∧ ϕn−1 (x)/θF ∧ ϕ¯j (f (ϕn−1 (x)))

= x/θF ∧ ϕ¯j (f (ϕn−1 (x))) = ϕ˜j (f )(x), F [ F [ that is, ϕ˜i ϕ˜j (f ) = ϕ˜j (f ), hence ϕF i (ϕj ((I, f ))) = ϕj ((I, f )).

(1.2.4). For i = 1, ..., n − 1 we have to prove that: ϕ˜i (f ∗ ) = (ϕ˜n−i (f ))∗ .

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Indeed, for x ∈ I : ϕ˜i (f ∗ )(x) = x/θF ∧ ϕ¯i (f ∗ (ϕn−1 (x)))

= x/θF ∧ ϕ¯i (ϕn−1 (x)/θF ∧ N f (ϕn−1 (ϕn−1 (x)))) = x/θF ∧ ϕ¯i (ϕn−1 (x)/θF ) ∧ ϕ¯i (N f (ϕn−1 (x)))

= x/θF ∧ ϕn−1 (x)/θF ∧ ϕ¯i (N f (ϕn−1 (x)))

= x/θF ∧ ϕ¯i (N f (ϕn−1 (x))), and (ϕ˜n−i (f ))∗ (x) = x/θF ∧ N ϕ˜n−i (f )(ϕn−1 (x))

= x/θF ∧ N (ϕn−1 (x)/θF ∧ ϕ¯n−i (f (ϕn−1 (ϕn−1 (x)))))

= x/θF ∧ N (ϕn−1 (x)/θF ∧ ϕ¯n−i (f (ϕn−1 (x))))

= x/θF ∧ (N ϕn−1 (x)/θF ∨ N ϕ¯n−i (f (ϕn−1 (x))))

= (x/θF ∧ N ϕn−1 (x)/θF ) ∨ (x/θF ∧ N ϕ¯n−i (f (ϕn−1 (x))))

= 0/θF ∨ (x/θF ∧ N ϕ¯n−i (f (ϕn−1 (x))))

= x/θF ∧ N ϕ¯n−i (f (ϕn−1 (x)))

= x/θF ∧ ϕ¯i (N f (ϕn−1 (x))),

F [ [ so, ϕ˜i (f ∗ ) = (ϕ˜n−i (f ))∗ , that is, ϕF i ((I, f ) ) = ϕn−i ((I, f )). ∗

(1.2.5). For x ∈ I we obtain successively: ϕ¯1 (f (ϕn−1 (x))) ≤ ... ≤ ϕ¯n−1 (f (ϕn−1 (x)))

x/θF ∧ ϕ¯1 (f (ϕn−1 (x))) ≤ ... ≤ x/θF ∧ ϕ¯n−1 (f (ϕn−1 (x))) ϕ˜1 (f )(x) ≤ ... ≤ ϕ˜n−1 (f )(x) ϕ˜1 (f ) ≤ ... ≤ ϕ˜n−1 (f ),

F [ [ that is, ϕF 1 ((I, f )) ≤ ... ≤ ϕn−1 ((I, f )). F \ \ \ (1.2.6). If ϕF ˜i (f1 )) = (I2\ , ϕ˜i (f2 )), for all i = i ((I1 , f1 ) = ϕi ((I2 , f2 ), that is, (I1 , ϕ 1, ..., n − 1, then we get in turn, for all i = 1, ..., n − 1, according to Remark 4.18 there exists Ji ⊆ I1 ∩ I2 such that ϕ˜i (f1|Ji ) = ϕ˜i (f2|Ji ). For x ∈ J1 ∩ ... ∩ Jn−1 we have:

ϕ˜i (f1 )(x) = ϕ˜i (f2 )(x), x/θF ∧ ϕ¯i (f1 (ϕn−1 (x))) = x/θF ∧ ϕ¯i (f2 (ϕn−1 (x))),

ϕn−1 (x)/θF ∧ ϕ¯i (f1 (ϕn−1 (ϕn−1 (x)))) = ϕn−1 (x)/θF ∧ ϕ¯i (f2 (ϕn−1 (ϕn−1 (x)))), ϕ¯i (ϕn−1 (x)/θF ∧ f1 (ϕn−1 (x))) = ϕ¯i (ϕn−1 (x)/θF ∧ f2 (ϕn−1 (x))), ϕ¯i (f1 (ϕn−1 (x))) = ϕ¯i (f2 (ϕn−1 (x))), f1 (ϕn−1 (x)) = f2 (ϕn−1 (x)).

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So, f1 (x) = f1 (x ∧ ϕn−1 (x)) = x/θF ∧ f1 (ϕn−1 (x)) = x/θF ∧ f2 (ϕn−1 (x))

= f2 (x ∧ ϕn−1 (x)) = f2 (x), hence f1 = f2 on J1 ∩ ... ∩ Jn−1 ⊆ I1 ∩ I2 ,

\ \ that is, (I 1 , f1 ) = (I2 , f2 ).

¤

Definition 4.27. The LMn -algebra LF will be called the localization LMn − algebra of L with respect to the topology F. \ Lemma 4.28. Let the map vF : L → LF defined by vF (a) = (L, fa ) for every a ∈ L. Then: (i) vF is a morphism of LMn -algebras, \ f ) ∈ C(L ), (ii) For every a ∈ C(L), (L, a

(iii) vF (L) ∈ R(LF ).

F

\ \ \ \ Proof. (i). Clearly, vF (0) = (L, f0 ) = (L, 0), vF (1) = (L, f1 ) = (L, 1). For a, b ∈ L we have: \ \ \ fa ) ∨ (L, fb ) = (L,\ fa ∨ fb ) = (L, f a∨b ) = vF (a ∨ b), vF (a) ∨ vF (b) = (L, and \ \ \ vF (a) ∧ vF (b) = (L, fa ) ∧ (L, fb ) = (L,\ fa ∧ fb ) = (L, f a∧b ) = vF (a ∧ b). Also, for every x ∈ L :

vF (ϕi (a)) = (L,\ f ϕi (a) )

and \ ϕF ei (fa )). i (vF (a)) = (L, ϕ

But ϕ ei (fa )(x) = x/θF ∧ ϕ¯i (fa (ϕn−1 (x))) = x/θF ∧ ϕ¯i (a/θF ∧ ϕn−1 (x)/θF ) = x/θF ∧ ϕi (a)/θF ∧ ϕn−1 (x)/θF = ϕi (a)/θF ∧ x/θF = f ϕi (a) (x), hence vF (ϕi (a)) = ϕF i (vL (a)) for all i = 1, ..., n − 1. Therefore vF is a morphism in LMn . (ii). Is a direct consequence of (i), since any LMn -algebra morphism preserves the boolean elements. [ (iii). To prove that vF (L) is a regular subset of LF , let (I i , fi ) ∈ LF , Ii ∈ F, i = 1, 2, \ \ \ \ fa ) ∧ (I such that (L, 1 , f1 ) = (L, fa ) ∧ (I2 , f2 ) for every a ∈ L. Then, for any a ∈ L there exists Ja ⊆ I1 ∩ I2 such that for any x ∈ Ja we have (f1 ∧ fa )(x) = (f2 ∧ fa )(x), hence f1 (x) ∧ a/θF ∧ x/θF = f2 (x) ∧ a/θF ∧ x/θF , that is, f1 (x) ∧ a/θF = f2 (x) ∧ a/θF . In particular, for a = 1 we obtain that f1 (x) = f2 (x) for every x ∈ Ja ⊆ I1 ∩ I2 , hence \ \ (I1 , f1 ) = (I ¤ 2 , f2 ), that is, vF (L) ∈ R(LF ).

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Applications

In the following we describe the localization LMn -algebra LF in some special instances. 1. If I ∈ Idn(L) and F is the topology F(I) = {I ′ ∈ Idn(L) : I ⊆ I ′ } (see Example 1 in Section 3), then LF is isomorphic with M (I, L/θF ) and v F : L → LF is defined by vF (a) = fa |I for every a ∈ L. 2. If F = Idn(L) ∩ R(L) is the topology of regular sets of Idn(L) (see Example 2 in Section 3), then θF is the identity congruence of L and LF =lim M (I, L), −−→ I∈F

where M (I, L) is the set of multipliers of L having the domain I (see [3]). We recall the construction of maximal LMn -algebra of fractions of L from [3](where is denoted by Qn (L)). We denote Mr (L) = {f ∈ M (L) : dom(f ) ∈ Idn(L) ∩ R(L)}. Lemma 5.1. ([3]) If I1 , I2 ∈ Idn(L) ∩ R(L), then I1 ∩ I2 ∈ Idn(L) ∩ R(L). Remark 5.2. ([3])By Lemma 5.1, we deduce that Mr (L) is an LMn -subalgebra of M (L). Definition 5.3. Define the relation ρL on the LMn − algebra Mr (L) by the prescription: (f1 , f2 ) ∈ ρL iff f1 and f2 agree on the intersection of their domains. Lemma 5.4. ([3]) ρL is a congruence on Mr (L). Definition 5.5. For f ∈ Mr (L) with I = dom(f ) ∈ Idn(L) ∩ R(L), we denote by [f, I] the congruence class of f modulo ρL and LM = Mr (L)/ρL . Remark 5.6. For every I ∈ Idn(L) ∩ R(L) and a ∈ L we have [fa , L] = [fa , I] (because for every x ∈ I ∩ L = I we have fa (x) = fa (x) = a ∧ x). Remark 5.7. ([3])As was proved by Cignoli [5](see also [2], Theorem 2.4), the class of LMn -algebras is equational, therefore LM is an LMn −algebra, where, for [f, I], [g, J] ∈ ee : LM , [f, I] ∧ [g, J] = [f ∧ g, I ∩ J], [f, I] ∨ [g, J] = [f ∨ g, I ∩ J], and for i = 1, ..., n − 1, ϕ i e LM → L M , ϕ ei ([f, I]) = [ϕ ei (f ), I].

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Lemma 5.8. ([3])Let the map vL : L → LM be defined by vL (a) = [fa , L] for every a ∈ L. Then: (i) vL is a monomorphism in LMn , (ii) For every a ∈ C(L), [fa , L] ∈ C(LM ), (iii) vL (L) ∈ R(LM ). Remark 5.9. ([3]) Since by Lemma 4.15 and Lemma 5.8, for every a, b ∈ L, [fa , L] = [fb , L] iff fa = fb iff a = b, the elements of L can be identified with the elements of the sets {[fa , L] : a ∈ L} and {fa : a ∈ L}. So, vL (L) ≈ vL (L) ≈ L (as LMn -algebras).

Definition 5.10. ([3]) An LMn -algebra L′ is called an LMn -algebra of fractions of L if: (3.1) L is an LMn -subalgebra of L′ , (3.2) For every a′ , b′ , c′ ∈ L′ , a′ 6= b′ , there is e ∈ L such that e ∧ a′ 6= e ∧ b′ and ϕn−1 (e) ∧ c′ ∈ L (where by ϕn−1 we denote the chrysippian endomorphism of L which is the restriction to L of the chrisippian endomorphism ϕ′n−1 of L′ ). Remark 5.11. As a notational convenience, we write L 4 L′ to indicate that L′ is an LMn −algebra of fractions for L. Remark 5.12. ([3]) If L 4 L′ , e ∈ L and a′ , b′ ∈ L′ are such that e ∧ a′ 6= e ∧ b′ , then ϕn−1 (e) ∧ a′ 6= ϕn−1 (e) ∧ b′ . Remark 5.13. ([3]) If L 4 L′ , e ∈ L, c′ ∈ L′ and ϕn−1 (e) ∧ c′ ∈ L, then e ∧ c′ = [e ∧ ϕn−1 (e)] ∧ c′ = e ∧ [ϕn−1 (e) ∧ c′ ] ∈ L. Theorem 5.14. ([3]) For every LMn -algebra L, the LMn -algebra LM in Definition 5.5 has the following properties: (i) vL (L) 4 LM , (ii) For every LMn -algebra L′ such that L 4 L′ , there exists a monomorphism of LMn algebras u : L′ → LM which induces the canonical monomorphism vL of L into LM . Theorem 5.14 provides the motivation for the following: Definition 5.15. ([3]) For any LMn -algebra L, LM is called a maximal LMn -algebra of fractions of L. To range whith the tradition ( [4], [10], [13], [14]) we denote LM by Qn (L). So, in the case of the topology F = Idn(L) ∩ R(L) we obtain: Proposition 5.16. For F = Idn(L) ∩ R(L), LF is exactly the maximal LMn -algebra Qn (L) of fractions of L introduced in [3](where it is denoted by LM ).

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3. If S ⊆ L is an ∧−closed system of L, we consider the following congruence on L : (x, y) ∈ θS iff there exists s ∈ S such that x ∧ ϕn−1 (s) = y ∧ ϕn−1 (s) (see Section 2). The quotient LMn -algebra L[S] = L/θS is called in Section 2 the LMn -algebra of fractions of L relative to the ∧−closed system Proposition 5.17. If FS is the topology associated with an ∧−closed system S ⊆ L (see Example 3 in Section 3), then the LMn -algebra LFS is isomorphic with L[S]. Proof. Let x, y ∈ L. If (x, y) ∈ θFS then there exists I ∈ FS (hence I ∩ S 6= ⊘) such that x∧e = y ∧e for any e ∈ I. Since I ∩S 6= ⊘ there exists e0 ∈ I ∩S such that x∧e0 = y ∧e0 . But e0 ∈ I implies that ϕn−1 (e0 ) ∈ I, so in particular x ∧ ϕn−1 (e0 ) = y ∧ ϕn−1 (e0 ), that is, (x, y) ∈ θS . So, θFS ⊆ θS . If (x, y) ∈ θS , then there exists e0 ∈ S such that x ∧ ϕn−1 (e0 ) = y ∧ ϕn−1 (e0 ). If we set I0 =< e0 >= {x ∈ L : x ≤ ϕn−1 (e0 )}, then I0 ∈ Idn(L). Since e0 ≤ ϕn−1 (e0 ), we have that e0 ∈ I0 , so e0 ∈ I0 ∩ S, hence I0 ∩ S 6= ⊘, that is, I0 ∈ FS . For every e ∈ I0 , e ≤ ϕn−1 (e0 ), then e = e ∧ ϕn−1 (e0 ), so x ∧ e = x ∧ (e ∧ ϕn−1 (e0 )) = (x ∧ ϕn−1 (e0 )) ∧ e = (y ∧ ϕn−1 (e0 )) ∧ e = y ∧ (e ∧ ϕn−1 (e0 )) = y ∧ e, hence (x, y) ∈ θFS , that is, θS ⊆ θFS . Therefore θFS = θS . Then L/θFS = L/θS = L[S], hence an FS −multiplier can be considered in this case (see Definition 4.16) as a mapping f : I → L[S] (I ∈ FS ) having the property f (e ∧ x) = e/S ∧ f (x) for every x ∈ I and e ∈ L (x/S denotes the congruence class of x relative to θS ). \ \ \ \ If (I M (I, L[S]) and (I 1 , f1 ), (I2 , f2 ) ∈ LFS =lim 1 , f1 ) = (I2 , f2 ) then there ex−−→ I∈F

ists I ∈ FS such that I ⊆ I1 ∩ I2 and f1|I = f2|I . Since I, I1 , I2 ∈ FS , there exists s ∈ I ∩ S, s1 ∈ I1 ∩ S and s2 ∈ I2 ∩ S. We shall prove that f1 (ϕn−1 (s1 )) = f2 (ϕn−1 (s2 )). If we denote s′ = ϕn−1 (s ∧ s1 ∧ s2 ) = ϕn−1 (s) ∧ ϕn−1 (s1 ) ∧ ϕn−1 (s2 ), then s′ ∈ I ∩ S and s′ ≤ ϕn−1 (s1 ), ϕn−1 (s2 ). Since ϕn−1 (s1 ) ∧ s′ = ϕn−1 (s2 ) ∧ s′ ∈ I then f1 (ϕn−1 (s1 ) ∧ s′ ) = f2 (ϕn−1 (s2 ) ∧ s′ ), hence f1 (ϕn−1 (s1 )) ∧ s′ /S = f2 (ϕn−1 (s2 )) ∧ s′ /S, so f1 (ϕn−1 (s1 )) ∧ 1 = f2 (ϕn−1 (s2 )) ∧ 1 (since s ∈ S implies ϕn−1 (s)/S = 1 by Remark 2.3), that is, f1 (ϕn−1 (s1 )) = f2 (ϕn−1 (s2 )). In a similar way, we can show that f1 (ϕn−1 (s1 )) = f2 (ϕn−1 (s2 )) for any s1 , s2 ∈ I ∩ S. In accordance with these considerations we can define the mapping: α : LFS =lim M (I, L[S]) → L[S] −−→ I∈F

by putting [ α((I, f )) = f (ϕn−1 (s0 )) ∈ L[S], where s0 ∈ I ∩ S. \ \ We have α(0) = α((L, 0)) = 0(ϕn−1 (s)) = 0/S = 0 and α(1) = α((L, 1)) = 1(ϕn−1 (s)) = ϕn−1 (s)/S = 1 by Remark 2.3 for every s ∈ S.

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[ Also, for every (I i , fi ) ∈ LFS , i = 1, 2 we have: \ \ \ α((I 1 , f1 ) ∧ (I2 , f2 )) = α((I1 ∩ I2 , f1 ∧ f2 )) = (f1 ∧ f2 )(ϕn−1 (s)) \ \ = f1 (ϕn−1 (s)) ∧ f2 (ϕn−1 (s)) = α((I 1 , f1 )) ∧ α((I2 , f2 )), and \ \ \ α((I 1 , f1 ) ∨ (I2 , f2 )) = α((I1 ∩ I2 , f1 ∨ f2 )) = (f1 ∨ f2 )(ϕn−1 (s)) \ \ = f1 (ϕn−1 (s)) ∨ f2 (ϕn−1 (s)) = α((I 1 , f1 )) ∨ α((I2 , f2 )) with s ∈ I1 ∩ I2 ∩ S. [ If (I, f ) ∈ LFS and s ∈ I ∩ S, for every i = 1, ..., n − 1 we have \ \ α(ϕF ei (f ))) = ϕ ei (f )(ϕn−1 (s)) = ϕn−1 (s)/S ∧ ϕi (f (ϕn−1 (ϕn−1 (s)))) i ((I, f ))) = α((I, ϕ [ = 1 ∧ ϕ¯i (f (ϕn−1 (s))) = ϕ¯i (f (ϕn−1 (s))) = ϕ¯i (α((I, f ))). Therefore, this mapping is a morphism of LMn -algebras. We shall prove that α is injective and surjective. To prove the injectivity of α, let \ \ \ \ (I1 , f1 ), (I 2 , f2 ) ∈ LFS such that α((I1 , f1 )) = α((I2 , f2 )). Then, for any s1 ∈ I1 ∩ S, s2 ∈ I2 ∩S we have f1 (ϕn−1 (s1 )) = f2 (ϕn−1 (s2 )). If f1 (ϕn−1 (s1 )) = x/S and f2 (ϕn−1 (s2 )) = y/S with x, y ∈ L, since x/S = y/S, there exists s ∈ S such that x ∧ ϕn−1 (s) = y ∧ ϕn−1 (s). If we consider s′ = s ∧ s1 ∧ s2 ∈ I1 ∩ I2 ∩ S, we have that ϕn−1 (s′ ) ≤ ϕn−1 (s1 ), ϕn−1 (s2 ). It follows that f1 (ϕn−1 (s′ )) = f1 (ϕn−1 (s′ ) ∧ ϕn−1 (s1 )) = ϕn−1 (s′ )/S ∧ f1 (ϕn−1 (s1 ))

= ϕn−1 (s′ )/S ∧ f2 ((ϕn−1 (s2 )) = f2 (ϕn−1 (s′ ) ∧ ϕn−1 (s2 )) = f2 (ϕn−1 (s′ )).

If we denote I =< s′ >= {x ∈ L : x ≤ ϕn−1 (s′ )}, then s′ ∈ I, so I ∈ FS , I ⊆ I1 ∩ I2 and f1|I = f2|I (since if x ∈ I, then x ≤ ϕn−1 (s′ ), hence x = x ∧ ϕn−1 (s′ ), so f1 (x) = f1 (x ∧ ϕn−1 (s′ )) = x/S ∧ f1 (ϕn−1 (s′ )) = x/S ∧ f2 (ϕn−1 (s′ )) = f2 (x ∧ ϕn−1 (s′ )) = f2 (x)), \ \ hence (I 1 , f1 ) = (I2 , f2 ), that is, α is injective. To prove the surjectivity of α, let a/S ∈ L[S] and f¯a : L → L[S] defined by f¯a (x) = a/S ∧ x/S = (a ∧ x)/S for every x ∈ L. \ It is easy to see that f¯a is an FS −multiplier and α((L, fa )) = f¯a (ϕn−1 (s)) = (a ∧ ϕn−1 (s))/S = a/S ∧ ϕn−1 (s)/S = a/S ∧ 1 = a/S, where s ∈ S. So α is surjective. Therefore, α is an isomorphism of LMn -algebras. ¤

Finally, it is a pleasure for author to express his gratitude for the helpful remarks of the referees of his paper.

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References [1] R. Balbes and Ph. Dwinger: Distributive Lattices, University of Missouri Press, 1974. [2] V. Boicescu, A. Filipoiu, G. Georgescu and S.Rudeanu: Lukasiewicz-Moisil Algebras, North Holland, 1991. [3] D. Bu¸sneag and F. Chirte¸s: LM n -algebra of fractions and maximal LM n -algebra of fractions, to appear in Discrete Mathematics. [4] D. Bu¸sneag: “F-multipliers and the localization of Hilbert algebras”, Zeitschr. f. math. Logik und Grundlagen d. Math. Bd., Vol. 36, (1990), pp. 331–338. [5] R. Cignoli: Algebras de Moisil, Notas de Logica Matematica, 27, Instituto de Matematica, Universidad del Sur, Bahia Blanca, 1970. [6] R. Cignoli: “An algebraic approch to elementary theory based on n-valued Lukasiewicz logics”, Z. Math Logic u. Grund. Math., Vol. 30, (1984), pp. 87–96, . [7] W.H. Cornish: “The multiplier extension of a distributive lattice”, Journal of Algebra, Vol. 32, (1974), pp. 339–355. [8] C. Dan: F-multipliers and the localization of Heyting algebras, Analele Universit˘a¸tii din Craiova, Seria Matematica-Informatica, Vol. XXIV, 1997, pp. 98–109. [9] G. Georgescu and C. Vraciu: “On the Characterisation of Centred Lukasiewicz Algebras”, Journal of Algebra, Vol. 16(4), (1970), pp. 486–495. [10] G. Georgescu: “F-multipliers and localizations of distributive lattices”, Algebra Universalis, Vol. 21, (1985), pp. 181–197. [11] J. Lambek: Lectures on Rings and Modules, Blaisdell Publishing Company, 1966. [12] N. Popescu: Abelian categories with applications to rings and modules, Academic Press, New York, 1973. [13] J. Schmid: “Multipliers on distributive lattices and rings of quotients”, Houston Journal of Mathematics, Vol. 6(3), (1980). [14] J. Schmid: “Distributive lattices and rings of quotients”, Coll. Math. Societatis Janos Bolyai (Szeged, Hungary), Vol. 33, (1980). [15] B. Strenstr¨om: “Platnes and localization over monoids”, Math. Nachrichten, Vol. 48, (1971), pp. 315–334.

CEJM 3(1) 2005 125–142

Left-sided quasi-invertible bimodules over Nakayama algebras Zygmunt PogorzaÃly∗ Faculty of Mathematics and Computer Science, Nicholaus Copernicus University, ul. Chopina 12/18, 87-100 Toru´ n, Poland

Received 3 November 2004; accepted 6 December 2004 Abstract: Bimodules over triangular Nakayama algebras that give stable equivalences of Morita type are studied here. As a consequence one obtains that every stable equivalence of Morita type between triangular Nakayama algebras is a Morita equivalence. c Central European Science Journals. All rights reserved. ° Keywords: Bimodules, Nakayama algebras, Morita equivalence MSC (2000): 16D20, 16G20

Introduction Let K be a fixed field. All considered algebras will be associative, finite-dimensional Kalgebras with an identity element. For a fixed K-algebra A we shall denote by Mod(A) (respectively, (A)Mod) the category of all right (resp, left) A-modules. Furthermore, we shall denote by mod(A) (resp, (A)mod) the full subcategory of Mod(A) (resp, (A)Mod) consisting of the A-modules which are finite-dimensional over K. Following Brou´e [2] we say that two algebras A, B are stably equivalent of Morita type if there is an A-B-bimodule A NB and a B-A-bimodule B MA such that the following conditions are satisfied: (i) M , N are projective as left and right modules, (ii) M ⊗A N ∼ = B ⊕ Π as B-B-bimodules for some projective B-B-bimodule Π, (iii) N ⊗B M ∼ = A ⊕ Π′ as A-A-bimodules for some projective A-A-bimodule Π′ . We shall say that an A-B-bimodule is left-right projective if it is a left projective ∗

E-mail: [email protected]

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A-module and a right projective B-module. The category of finite-dimensional, left-right projective A-B-bimodules was studied in [1, 8]. Since the category of all finite-dimensional, left-right projective bimodules is frequently of wild representation type, it is impossible to classify all left-right projective bimodules for given K-algebras. Nevertheless Rickard’s proof of Theorem 3.2 in [11] suggests that it is enough to study only finite-dimensional bimodules which yield stable equivalences of Morita type. Moreover, it suggests the following definition. A finitedimensional, indecomposable, left-right projective A-B-bimodule X is defined to be leftsided quasi-invertible if HomA (X, A) ⊗A X ∼ = B ⊕ Π as B-B-bimodules for a projective B-B-bimodule Π. In fact Rickard showed in his proof that if an A-B-bimodule A NB yields a stable equivalence of Morita type between A and B then it is left-sided quasi-invertible. The main aim of the paper is to study left-sided quasi-invertible A-B-bimodules in the case when both algebras A and B are Nakayama algebras which are factors of hereditary algebras. Recall that an algebra is a Nakayama algebra provided that the lattice of the submodules of every indecomposable (left or right) projective module is a chain. Our first main result is the following theorem. Theorem 1. Let K be an algebraically closed field. Let A, B be finite-dimensional Nakayama K-algebras which are factors of hereditary algebras. Then there are at most finitely many pairwise non-isomorphic, left-sided quasi-invertible A-B-bimodules. As a consequence of the above result one gets the following theorem. Theorem 2. Let K be an algebraically closed field. Let A, B be finite-dimensional Nakayama K-algebras which are factors of hereditary algebras. Then A and B are stably equivalent of Morita type if and only if A and B are Morita equivalent. The paper is organized as follows. In Section 1 we construct a nice projective resolution for a tensor product of bimodules. Section 2 is devoted to one point extensions. This seems to be a natural tool in our investigations. Some general properties of left-sided quasi-invertible bimodules are studied in Section 3. The proof of Theorem 1 is given in Section 4 as well as the proof of Theorem 2 is given in Section 5.

1

Projective resolutions

Let A, B be finite-dimensional K-algebras. Consider the tensor product algebra Ao ⊗K B, where Ao is the opposite algebra of A. It is well-known and easy to verify that every AB-bimodule X with K acting centrally is a right Ao ⊗K B-module. Conversely, every right Ao ⊗K B-module is in fact an A-B-bimodule. Therefore we shall frequently identify A-B-bimodules with right Ao ⊗K B-modules. Fix three finite-dimensional K-algebras A, B, C which are assumed to be basic. Let 1A ∈ A, 1B ∈ B, 1C ∈ C be the identity elements. Let {e1 , . . . , en } ⊂ A, {f1 , . . . , fm } ⊂ B, {g1 , . . . , gs } ⊂ C be fixed complete sets of primitive, pairwise orthogonal idempotents. Then every indecomposable projective A-B-bimodule is isomorphic to Aei ⊗K fj B for some

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i = 1, . . . , n; j = 1, . . . , m. Moreover, every indecomposable projective B-C-bimodule is isomorphic to Bfj ⊗K gl C for some j = 1, . . . , m; l = 1, . . . , s. We shall use the above notations to prove the following. Lemma 1.1. Let P ∈ mod(Ao ⊗K B), Q ∈ mod(B o ⊗K C) be projective. Then P ⊗B Q ∈ mod(Ao ⊗K C) is projective or zero. Proof. Since the tensor product −⊗B − is additive on both sides, it is enough to prove our lemma for indecomposable P and Q. Then we know from the above considerations that there are i0 ∈ {1, . . . , n}, j0 , j1 ∈ {1, . . . , m}, l0 ∈ {1, . . . , s} such that P ∼ = Aei0 ⊗K fj0 B, ∼ ∼ Q = Bfj1 ⊗K gl0 C. Therefore P ⊗B Q = Aei0 ⊗K fj0 B⊗B Bfj1 ⊗K gl0 C. But fj0 B⊗B Bfj1 ∼ = fj0 Bfj1 as K-linear spaces. If fj0 Bfj1 = 0 then P ⊗B Q = 0. If fj0 Bfj1 6= 0 then it is a finite-dimensional K-linear space. Thus Aei0 ⊗K fj0 B ⊗B Bfj1 ⊗K gl0 C is isomorphic to dimK (fj0 Bfj1 ) copies of Aei0 ⊗K gl0 C, and so P ⊗B Q is a right projective Ao ⊗K C-module. ¤ For the fixed above K-algebras A, B, C, consider X ∈ mod(Ao ⊗K B) and Y ∈ mod(B o ⊗K C). Assume that they are not projective. Fix a projective resolution p2

p1

p0

· · · → P2 → P1 → P0 → X → 0 in mod(Ao ⊗K B). Applying to this resolution the functor −⊗B Y , we obtain the following complex p2 ⊗id p1 ⊗id p0 ⊗id · · · −→ P2 ⊗B Y −→Y P1 ⊗B Y −→Y P0 ⊗B Y −→Y X ⊗B Y −→ 0,

because the functor − ⊗B Y is right exact. Now fix a projective resolution r

r

r

2 1 0 · · · → R2 → R1 → R0 → Y →0

in mod(B o ⊗K C). Then for every j = −1, 0, 1, 2, . . . we have the following complex idPj ⊗r0 idPj ⊗r1 idPj ⊗r2 · · · −→ Pj ⊗B R2 −→ Pj ⊗B R1 −→ Pj ⊗B R0 −→ Pj ⊗B Y −→ 0, because the functors Pj ⊗B − are right exact, where P−1 = X. Throughout this section we shall keep all the above assumptions and notations. We shall construct a projective resolution of X ⊗B Y in mod(Ao ⊗K C) with a help of the above complexes. Lemma 1.2. The morphism (idX ⊗ r0 )(p0 ⊗ idR0 ) = p0 ⊗ r0 is an epimorphism. Proof. Obvious by the fact that the tensor product is a right exact functor.

¤

Lemma 1.3. For every even i ≥ 2 we have ker(pi ⊗ ri ) = ker(pi ) ⊗B Ri + Pi ⊗B ker(ri ).

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Proof. Observe that applying the functor − ⊗B Ri to the short exact sequence 0 → u v ker(pi ) →i Pi →i im(pi ) → 0 we obtain the following short exact sequence vi ⊗idR ui ⊗idR 0 −→ ker(pi ) ⊗B Ri −→ i Pi ⊗B Ri −→ i im(pi ) ⊗B Ri −→ 0,

because Ri is also a left projective, finite-dimensional B-module hence it is a left flat B-module. If wi : im(pi ) → Pi−1 is the inclusion morphism then wi vi = pi and pi ⊗ idRi = (wi ⊗ idRi )(vi ⊗ idRi ). Furthermore, in a similar way we obtain the following short exact sequence 0 −→ Pi−1 ⊗B ker(ri )

idPi−1 ⊗ai idPi−1 ⊗bi −→ Pi−1 ⊗B Ri −→ Pi−1 ⊗B im(ri ) −→ 0,

since i − 1 ≥ 1, Pi−1 is a finite-dimensional right projective B-module, and so it is a right flat B-module. If ci : im(ri ) → Ri−1 is the inclusion morphism then ci bi = ri and idPi−1 ⊗ri = (idPi−1 ⊗ ci )(idPi−1 ⊗bi ). Thus pi ⊗ri = (idPi−1 ⊗ri )(pi ⊗idRi ) hence ker(pi ⊗ri ) = ker(pi )⊗B Ri +L, where L is a submodule of Pi ⊗B Ri such that (pi ⊗ idRi )(L) ⊂ ker(idPi−1 ⊗ ri ) = Pi−1 ⊗B ker(ri ). On the other hand (pi ⊗ idRi )(Pi ⊗B ker(ri )) ⊂ Pi−1 ⊗B ker(ri ). Consequently, L = Pi ⊗B ker(ri ) which finishes the proof of the lemma. ¤ Lemma 1.4. For every even i ≥ 2 it holds im(pi+1 ⊗ idRi , idPi ⊗ ri+1 ) = ker(pi ⊗ ri ). Proof. Since Pi is a right flat B-module, it holds im(idPi ⊗ri+1 ) = Pi ⊗B ker(ri ). Similarly im(pi+1 ⊗idRi ) = ker(pi )⊗B Ri , because Ri is a left flat B-module. Then for the morphism (pi+1 ⊗ idRi , idPi ⊗ ri+1 ) : Pi+1 ⊗B Ri ⊕ Pi ⊗B Ri+1 −→ Pi ⊗B Ri we obtain that im(pi+1 ⊗ idRi , idPi ⊗ ri+1 ) = ker(pi ⊗B Ri + Pi ⊗B ker(ri ) = ker(pi ⊗ ri ) by Lemma 1.3. ¤ Lemma 1.5. For every even i ≥ 0 it holds 



 idPi+1 ⊗ ri+1  ker(pi+1 ⊗ idRi , idPi ⊗ ri+1 ) = im   −pi+1 ⊗ idRi+1 . 



 idPi+1 ⊗ ri+1  Proof. First observe that im   ⊂ ker(pi+1 ⊗ idRi , idPi ⊗ ri+1 ). Indeed, −pi+1 ⊗ idRi+1    idPi+1 ⊗ ri+1  we have (pi+1 ⊗ idRi , idPi ⊗ ri+1 )   = pi+1 ⊗ ri+1 − pi+1 ⊗ ri+1 = 0. −pi+1 ⊗ idRi+1

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

129



P

 t x t ⊗ yt  P Furthermore, we have ker(pi+1 ⊗ idRi , idPi ⊗ ri+1 ) = { P  ; t pi+1 (xt ) ⊗ yt + x ⊗ y s s s   P P  t x t ⊗ yt  ; s xs ⊗ ri+1 (ys ) = 0} = { P s x s ⊗ ys P P t pi+1 (xt ) ⊗ yt = − s xs ⊗ ri+1 (ys )}. Then consider the pull-back diagram L

Pi+1 ⊗B Ri

−→

↓ pi+1 ⊗ idRi

↓

−idPi ⊗ri+1 −→ Pi ⊗B Ri . Pi ⊗B Ri+1 It is clear that L = ker(pi+1 ⊗idRi , idPi ⊗ri+1 ). But(pi+1 ⊗idRi )(id Pi ⊗ri+1 ) = pi+1 ⊗ri+1 = P  t x t ⊗ yt  (−idPi ⊗ ri+1 )(−pi+1 ⊗ idRi+1 ). Therefore, for any  P  ∈ ker(pi+1 ⊗ idRi , idPi ⊗ s x s ⊗ ys   P  idPi+1 ⊗ ri+1  P ′ ri+1 ) there is j x′j ⊗ yj′ ∈ Pi+1 ⊗B Ri+1 such that   ( j xj ⊗ yj′ ) = −pi+1 ⊗ idRi+1   P  t x t ⊗ yt  ¤ P  which shows our lemma. x ⊗ y s s s





 idPi+1 ⊗ ri+1  Lemma 1.6. For any even i ≥ 0 we have ker   = im(pi+2 ⊗ ri+2 ). −pi+1 ⊗ idRi+1 



 idPi+1 ⊗ ri+1  Proof. We start our proof with an easy observation that   (pi+2 ⊗ −pi+1 ⊗ idRi+1        idPi+1 ⊗ ri+1   pi+2 ⊗ ri+1 ri+2   0  ri+2 ) =  .  =  . Thus im(pi+2 ⊗ ri+2 ) ⊂ ker  −pi+1 ⊗ idRi+1 0 −pi+1 pi+2 ⊗ ri+2   P  idPi+1 ⊗ ri+1  Now we consider a nonzero element . Then j xj ⊗ yj ∈ ker  −pi+1 ⊗ idRi+1 P P P j xj ⊗ ri+1 (yj ) = 0 and − j pi+1 (xj ) ⊗ yj = 0. Hence j xj ⊗ yj ∈ ker(idPi+1 ⊗ ri+1 ) ∩ ker(pi+1 ⊗ idRi+1 ). But ker(idPi+1 ⊗ ri+1 ) = im(idPi+1 ⊗ ri+2 ) and ker(pi+1 ⊗ idRi+1 ) = P im(pi+2 ⊗ idRi+1 ). Thus x ⊗ yj ∈ im(idPi+1 ⊗ ri+2 ) ∩ im(pi+2 ⊗ idRi+1 ). TherePj j′ P ′ P ′ ′ fore there is an element s xs ⊗ ys such that s xs ⊗ ri+2 (ys ) = j xj ⊗ yj and

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P

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pi+1 (x′s ) ⊗ ri+2 (ys′ ) = 0. Then consider the following exact sequence ker(pi+1 ) ⊗B ker(ri+1 )

ui+1 ⊗idker(ri+1 )

−→

Pi+1 ⊗B ker(ri+1 )

vi+1 ⊗idker(ri+1 )

−→

im(pi+1 ) ⊗B ker(ri+1 ) → 0. P We deduce from this sequence that 0 = s pi+1 (x′s ) ⊗ ri+2 (ys′ ) ∈ im(pi+1 ) ⊗B ker(ri+1 ), P because ker(ri+1 ) = im(ri+2 ). This means that s x′s ⊗ ri+2 (ys′ ) ∈ ker(vi+1 ⊗ idker(ri+1 ) ). P P P Thus there is an element t x′′t ⊗yt′′ such that t pi+2 (x′′t )⊗ri+2 (yt′′ ) = s x′s ⊗ri+2 (ys′ ) = P j xj ⊗ yj , because ker(pi+1 ) ⊗B ker(ri+1 ) = im(pi+2 ) ⊗B im(ri+2 ). This finishes the proof of the lemma. ¤ Theorem 1.7. Let A, B, C be finite-dimensional, basic K-algebras. Let X ∈ mod(Ao ⊗K p0 p1 p2 B), Y ∈ mod(B o ⊗K C) be nonprojective. Let · · · → P2 → P1 → P0 → X → 0 be a r2 r1 r0 projective resolution in mod(Ao ⊗K B) and · · · → R2 → R1 → R0 → Y → 0 be a o projective resolution in mod(B ⊗K C). Then there is a projective resolution · · · → q0 q1 q2 Q2 → Q1 → Q0 → X ⊗B Y → 0 in mod(Ao ⊗K C) for X ⊗B Y which satisfies the following conditions: (1) Q0 = P0 ⊗B R0 . (2) For every i = 3k + 2, Qi = P2k+1 ⊗B R2k+1 . (3) For every i = 3k + 3, Qi = P2k+2 ⊗B R2k+2 . (4) For every i = 3k + 1, Qi = P2k+1 ⊗B R2k ⊕ P2k ⊗B R2k+1 . (5) q0 = p0 ⊗ r0 .    idP2k+1 r2k+1  (6) For every i = 3k + 2, qi =  . −p2k+1 idR2k+1

(7) For every i = 3k + 3, qi = p2k+2 ⊗ r2k+2 . (8) For every i = 3k + 1, qi = (p2k+1 ⊗ idR2k , idP2k ⊗ r2k+1 ).

Proof. We infer by Lemma 1.2 that p0 ⊗ r0 is an epimorphism. Repeating the arguments from the proof of Lemma 1.3 we obtain that there is an epimorphism from ker(p0 )⊗B R0 + P0 ⊗B ker(r0 ) onto ker(p0 ⊗ r0 ). Repeating the arguments from the proof of Lemma 1.4 we obtain that im(p1 ⊗ idR0 , idP0 ⊗ r1 ) = ker(p0 ⊗ r0 ). Applying successively Lemmas q2 q1 q0 1.3, 1.4, 1.5, 1.6 we obtain that the sequence · · · → Q2 → Q1 → Q0 → X ⊗B Y → 0 satisfying conditions (1) - (8) is exact. Then we infer by Lemma 1.1 that this is a projective resolution for X ⊗B Y in mod(Ao ⊗K C). ¤ Remark 1.8. (1) The above theorem is also true in the case when X is a right B-module p0 p1 p2 and · · · → P2 → P1 → P0 → X → 0 is a projective resolution in mod(B). (2) The above theorem is also true in the case when Y is a left B-module and · · · → r2 r1 r0 R2 → R1 → R0 → Y → 0 is a projective resolution in (B)mod.

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2

131

One point extensions

Now we shall recall one of the main working tools in representation theory. Following Ringel [12] (see also [13]) a finite-dimensional K-algebra A1 is said to be a one point extension of the algebra A by the bimodule  K MA ifA1 is of the form K A1 =  0

K MA

A

 .

′ Throughout we identify a module X in mod(A1 ) with a system X = (XK , XA′′ , ϕ : X ′ ⊗K MA → XA′′ ).

Lemma 2.1. Let A, Bbe finite-dimensional and K MB be a finite-dimensional   K-algebras 

K M   A M ⊗K A  K-B-bimodule. Then  =  ⊗K A ∼  as K-algebras. 0 B 0 B ⊗K A 



K M  Proof. In order to prove our lemma consider the additive map f :   ⊗K A → 0 B        A M ⊗K A  k m k · a m ⊗ a   given by the formula f (  ⊗ a) =   for any k ∈ K, 0 B ⊗K A 0 b 0 b⊗a m ∈ M , b ∈ B, a ∈ A. In an obvious way one checks that f is a K-linear morphism. Furthermore, we have that ³³³ ´ ´ ³³ ′ ′ ´ ´´ ³³ ´³ ′ ′´ ´ km k m km k m ′ ′ f ⊗ a ⊗ a = f ⊗ aa = ′ ′ 0 b 0 b 0 b 0 b ¶ µ ´ ³³ ′ ´ kk km′ + mb′ kk ′ aa′ (km′ + mb′ ) ⊗ aa′ ′ = = f ⊗ aa ′ 0 bb 0 bb′ ⊗ aa′ ´³ ′ ′ ′ ´ ³ ′ ′ ´ ³ k a m ⊗ a′ kk aa km′ ⊗ aa′ + mb′ ⊗ aa′ ka m ⊗ a = 0 b⊗a 0 b ′ ⊗ a′ = 0 bb′ ⊗ aa′ ´ ´ ´ ³³ ′ ′ ´ ³³ k m km ′ ⊗ a . ⊗ a f f 0 b 0 b′ ³³

´ ´ ´ ³ 10 10 0 1 ⊗ 1 = 0 1 . Thus the map f is a homomorphism of K-

Moreover, f algebras. Now we shall show f´ is an epimorphism. ´any ³ ´ Consider ³ ³ that the ´homomorphism ³ a1 m ⊗ a2 a1 m ⊗ a2 a1 0 A M ⊗K A = 0 0 + element of the form 0 b ⊗ a ∈ 0 B ⊗ A . Then 0 b ⊗ a 3 3 K ³ ´ ³ ´ 0 0 0 m ⊗ a2 0³³ 0 ´ + 0³b ⊗ a´3 = ´ ³ ´ 00 0m 10 ⊗ a + ⊗ a ⊗ a + f 2 3 . Thus we infer by the additivity of f that 1 0 0 00 0b ´ ³ K M it is an epimorphism. Since the K-dimensions of the algebras 0 B ⊗K A and ´ ³ A M ⊗K A 0 B ⊗ A coincide, the epimorphism f is an isomorphism and our lemma is proved. K

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¤ Let A, B be finite-dimensional K-algebras. If MB is a right B-module then M ⊗K A ′ is a right (B ⊗K A)-module with (m ⊗ a) · (b ⊗ a ) = mb ⊗ aa′ . Hence M ⊗K A is an o o  A M ⊗K A  A -B-bimodule. We identify a module X ∈ mod   with a system X = o 0 B ⊗K A o

′′ ′ o ′′ (XA′ o , XB⊗ o , ϕ : XAo ⊗Ao MB ⊗K A → XB⊗ Ao ). But there exists an isomorphism of KA K right B⊗K Ao -modules ψ : MB ⊗K Ao → Ao ⊗K MB given by the formula ψ(m⊗a) = a⊗m, because ψ((m ⊗ a)(b ⊗ a′ )) = ψ(mb ⊗ aa′ ) = a′ a ⊗ mb = (a ⊗ m)(b ⊗ a′ ) for any m ∈ M , a, a′ ∈ Ao , b ∈ B. Thus XA′ o ⊗Ao MB ⊗K Ao ∼ = XA′ o ⊗Ao Ao ⊗K MB ∼ = XA′ o ⊗K MB . Hence ϕ :A X ′ ⊗K MB →A XB′′ is an A-B-bimodule morphism.

³ ´ K M We would like to know when the A- 0 BB -bimodule X = (A X ′ ,A XB′′ , ϕ :A X ′ ⊗K MB →A XB′′ ) is left-right projective. Some necessary conditions are given in the lemma below. ′ ′′ ′ ′′ Lemma ³ o o 2.2. Let ´ X = (A X ,A XB , ϕ :A X ⊗K MB →A XB ) be a right finite-dimensional A A ⊗K MB 0 B ⊗K Ao -module which is left-right projective. Then the following conditions are satisfied: (a) A X ′ is a left projective A-module or zero. (b) A X ′′ is a left projective A-module or zero.

Proof. (a) Suppose that X ′ 6= 0. Consider a minimal projective cover p : P → X ′ in (A)mod. Then we have the following epimorphism p⊗idM : P ⊗K M → X ′ ⊗K M of left Amodules. This epimorphism gives rise to the homomorphism ϕ(p ⊗ idM ) : P ⊗K M → X ′′ of left A-modules. Thus we obtain the following commutative diagram P ⊗K M ↓ p ⊗ idM

ϕ(p⊗idM )

−→ ϕ

X ′′ ↓ idX ′′

X ′ ⊗K M −→ X ′′ of left A-modules. Therefore we obtain an epimorphism (p, idX ′′ ) : (P, X ′′ , ϕ(p ⊗ idM )) → (X ′ , X ′′ , ϕ) of left A-modules. Since the codomain of the epimorphism (p, idX ′′ ) is a left projective A-module, this epimorphism is splitable. Hence the morphism p is a split epimorphism. Thus X ′ is a left projective A-module. (b) Suppose that X ′′ 6= 0. Consider a minimal projective cover q : Q → X ′′ in (A)mod. Since X ′ is a left projective A-module or zero by (a), the left A-module X ′ ⊗K M is projective or zero. If X ′ = 0 then X ′′ is a left projective A-module obviously. If X ′ 6= 0 is projective then X ′ ⊗K M is a left projective A-module. Then there is a morphism ψ : X ′ ⊗K M → Q of left A-modules such that qψ = ϕ. Therefore we have the following commutative diagram of left A-modules and their morphisms

Z. PogorzaÃly / Central European Journal of Mathematics 3(1) 2005 125–142

X ′ ⊗K M

133

ψ

−→ Q

↓ idX ′ ⊗ idM

ϕ

↓q

X ′ ⊗K M −→ X ′′ which shows that the morphism (idX ′ , q) : (X ′ , Q, ψ) → (X ′ , X ′′ , ϕ) is an epimorphism of left A-modules. By our assumption the module (X ′ , X ′′ , ϕ) is a left projective A-module. Thus the epimorphism (idX ′ , q) splits. Therefore the epimorphism q splits, and so X ′′ is a left projective A-module. ¤ ´ Ao Ao ⊗K M o 0 B ⊗K A module which is left-right projective. If MB is a right projective B-module then X ′′ is a right projective B-module. Lemma 2.3. Let X = (A X ′ ,A XB′′ , ϕ :A X ′ ⊗K MB →A XB′′ ) be a right

³

Proof. Let p : P → X ′′ be a minimal projective cover in mod(B). Since MB is a right projective B-module, the right B-module X ′ ⊗K M is projective. Therefore there is a morphism ψ : X ′ ⊗K M → P of right B-modules such that pψ = ϕ. Then we obtain the following commutative diagram of right B-modules and their morphisms X ′ ⊗K M

ψ

−→ P

↓ idX ′ ⊗ idM

ϕ

↓p

X ′ ⊗K M −→ X ′′ which proves that the morphism (idX ′ , PB , ψ) → (A X ′ ,A XB′′ , ϕ) is an epimorphism of right B-modules. By our assumption the right B1 -module (A X ′ ,A XB′′ , ϕ) is projective. Hence the epimorphism (idX ′ , p) splits. Therefore the epimorphism p splits, and so X ′′ is a right projective B-module. ¤

3

Left-sided quasi-invertible bimodules

For two basic K-algebras A, B and an A-B-bimodule X we shall denote by ∗ X the BA-bimodule HomA (X, A) and by X ∗ the B-A-bimodule HomB (X, B). Denote by D the usual duality HomK (−, K). Lemma 3.1. Let A X ′ be a left A-module, MB be a right B-module and A XB′′ be an A-Bbimodule. Then any A-B-bimodule homomorphism ϕ : A X ′ ⊗K MB →A XB′′ induces a B-A-bimodule homomorphism ∗ ϕ : ∗ X ′′ → D(M ) ⊗K ∗ X ′ . Proof. If ϕ : X ′ ⊗K M → X ′′ is an A-B-bimodule homomorphism then HomA (ϕ, A) : ∗ X ′′ → HomA (X ′ ⊗K M, A) is a B-A-bimodule homomorphism. Applying known adjustment formulas (which can be found in [5, Chap. V]) we have isomorphisms HomA (X ′ ⊗K M, A) ∼ = HomK (M, HomA (X ′ , A)) = HomK (M, ∗ X ′ ) ∼ = D(M ) ⊗K

∗

X′

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of B-A-bimodules. Denote by ι the composition of the above isomorphisms. Then ∗ ϕ = ι ◦ HomA (ϕ, A) is the required morphism of B-A-bimodules. ¤ ´ Ao Ao ⊗K M Proposition 3.2. Every left 0 B ⊗ Ao -module is of the form Y = (YA′ ,B YA′′ , ψ), K where the morphism ψ :B YA′′ → D(M ) ⊗K YA′ is a B-A-bimodule homomorphism. ³

Proof. Every left B1 ⊗K Ao -module B1 YA is of the form (YA′ ,B YA′′ , κ), where κ : Ao ⊗K M ⊗B⊗K Ao Y ′′ → YA′ is a homomorphism of right A-modules. But Ao ⊗K M ⊗B⊗K Ao Y ′′ ∼ = M ⊗B Y ′′ as right A-modules. Hence κ : M ⊗B Y ′′ → Y ′ is a homomorphism of right A-modules. Then κ ∈ HomK−A (M ⊗B Y ′′ , Y ′ ). Then we infer by [5, Chap. V] that HomK−A (M ⊗B Y ′′ , Y ′ ) ∼ = HomB−A (Y ′′ , HomK (M, Y ′ )) ∼ = HomB−A (Y ′′ , D(M ) ⊗K Y ′ ). Thus the last isomorphisms determine a B-A-bimodule homomorphism ψ : Y ′′ → D(M ) ⊗K Y ′ and the proposition is proved. ¤ Lemma 3.3. Let X = (A X ′ ,A XB′′ , ϕ) be a right ³ o o ´ A A ⊗K MB ∗ ′ ∗ ′′ ∗ ( X , X , ϕ) as left 0 B ⊗ Ao -modules. K

³

´ Ao Ao ⊗K MB -module. Then ∗ X ∼ = o 0 B ⊗K A

Proof. Since X = (A X ′ ,A XB′′ , ϕ) ∼ =A X ′ ⊕A XB′′ as left A-modules, ∗ X ∼ = (∗ X ′ ,∗ X ′′ ,∗ ϕ) as B1 -A-bimodules by Lemma 3.1 and Proposition 3.2. ¤ From now on we shall assume that K is an algebraically closed field. All considered finite-dimensional K-algebras are basic and connected. Then we know from [3] that any such an algebra B is isomorphic to KQB /IB , where QB is a finite quiver and IB is an admissible two-sided ideal in the path algebra KQB of the quiver QB . An algebra B∼ = KQB /IB is said to be triangular ³ ´ if QB has no oriented cycle. It is well-known that K MB a one point extension B1 = 0 B of a triangular algebra B is also triangular. ´ K MB be a one 0 B a K-category B(B∞ ) as follows. ψ1 , ψ2 , ψ3 , ψ4 ), where K VK is a mensional left B-module, K Y¯ ′ B Let B1 =

³

point extension of a triangular K-algebra B. We define Its objects are the eight-tuples (K VK ,B Y ′ ,K Y¯ ′ B ,B Y ′′ , K

B

finite-dimensional K-linear space, B YK′ is a finite-diis a finite-dimensional right B-module and B YB′′ is a finite-dimensional B-B-bimodule. Moreover, ψ1 :B Y ′ → D(MB ) ⊗K V , ψ3 : V ⊗K MB → Y¯ ′ B , ψ2 :B Y ′ ⊗K MB →B YB′′ , ψ4 :B YB′′ → D(MB ) ⊗K Y¯ ′ are bimodule morphisms such that (idD(MB ) ⊗ ψ3 )(ψ1 ⊗ idMB ) = ψ4 ψ2 . For any two objects Y = ′ (K VK ,B YK′ ,K Y¯ ′ B ,B YB′′ , ψ1 , ψ2 , ψ3 , ψ4 ) and Z = (K WK ,B ZK ,K Z¯′ B ,B ZB′′ , κ1 , κ2 , κ3 , κ4 ) the K-linear space of the morphisms HomB(B∞ ) (Y, Z) consists of all possible four-tuples ′ (α1 , α2 , α3 , α4 ) such that α1 :K VK →K WK is a K-linear morphism, α2 :B YK′ →B ZK is a morphism of left B-modules, α3 :K Y¯ ′ B →K Z¯′ B is a morphism of right B-modules, α4 :B YB′′ →B ZB′′ is a B-B-bimodule morphism and the following conditions are satisfied: (i) (κ1 ⊗ idMB )(α2 ⊗ idMB ) = (idD(MB ) ⊗ α1 ⊗ idMB )ψ1 ,

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135

(ii) α4 ψ2 = κ2 (α2 ⊗ idMB ), (iii) (idD(MB ) ⊗ α3 )ψ4 = κ4 α4 , (iv) (idD(MB ) ⊗ κ3 )(idD(MB ) ⊗ α1 ⊗ idMB ) = (idD(MB ) ⊗ α3 )(idD(MB ) ⊗ ψ3 ).

A routine verification shows that B(B∞ ) is a well-defined K-category with the obvious composition of morphisms. We shall show that B(B∞ ) is equivalent to mod(B1 ⊗K B1o ). ´ K MB Proposition 3.4. Let B1 = 0 B be a one point extension of a triangular K-algebra B. Then there is an equivalence F : mod(B1 ⊗K B1o ) → B(B∞ )) of categories. ³

³ ´ ³ o o ´ K M B B ⊗ M Proof. Since B1 = 0 BB , we have that B1 ⊗K B1o ∼ = 01 B1 ⊗K B oB by Lemma 2.1 K 1 and remarks after Lemma 2.1. Then every right, finite-dimensional B1 ⊗K B1o -module Y is of the form (B1 Y˜ ′ ,B1 Y¯′′ B , ψ :B1 Y˜ ′ ⊗K MB →B1 Y¯′′ B ). On the other hand the left B1 ¯ : M ⊗B Y ′ → V ) ∼ module B1 Y˜ ′ is of the form (V,B Y ′ , λ = (V,B Y ′ , λ :B Y ′ → D(MB )⊗K V ). Similarly the B1 -B-bimodule B1 Y¯′′ B is of the form (Y¯ ′ B ,B YB′′ , ρ :B YB′′ → D(MB ) ⊗K Y¯ ′ B ). Then B1 Y˜ ′ ⊗K MB ∼ = (V ⊗K MB ,B Y ′ ⊗K MB , λ ⊗ idMB : B Y ′ ⊗K MB → D(MB ) ⊗K V ⊗K MB ). Thus the above morphism ψ is a pair ψ = (ψ3 , ψ2 ), where ψ3 : V ⊗K MB → Y¯ ′ B is a morphism of right B-modules and ψ2 :B Y ′ ⊗K MB →B YB′′ is a B-B-bimodule homomorphism. We define a functor F : mod(B1 ⊗K B1o ) → B(B∞ ) as follows. For every object Y ∈ mod(B1 ⊗K B1o ) we put F (Y ) = (V,B Y ′ , Y¯ ′ B ,B YB′′ , ψ1 , ψ2 , ψ3 , ψ4 ), where V , B Y ′ , Y¯ ′ B , B YB′′ , ψ2 , ψ3 are as above and ψ1 = λ, ψ4 = ρ. A routine verification shows that F (Y ) ∈ B(B∞ ). Assume now that α : Y → Z is a homomorphism of right, finite-dimensional B1 ⊗K B1o modules. Then α = (α¯1 , α¯2 ) : (B1 Y˜ ′ ,B1 Y¯′′ B , ψ) → (B1 Z˜′ ,B1 Z¯′′ B , κ) in the above notations, where α¯1 : B1 Y˜ ′ → B1 Z˜′ is a homomorphism of left B1 -modules and α¯2 :B1 Y¯′′ B → B1 Z¯′′ B is a B1 -B-bimodule homomorphism. Then B1 Y˜ ′ = (V,B Y ′ , λ), B1 Z˜′ = (W,B Z ′ , µ) and the homomorphism α¯1 is a pair α¯1 = (α1 , α2 ) of the homomorphisms α1 : V → W , α2 :B Y ′ → B Z ′ . Moreover, B1 Y¯′′ B = (Y¯ ′ B ,B YB′′ , ρ) and B1 Z¯′′ B = (Z¯′ B ,B ZB′′ , η) and the homomorphism α¯2 is a pair α¯2 = (α3 , α4 ) of the homomorphisms α3 : Y¯ ′ B → Z¯′ B , α4 : B YB′′ → B ZB′′ . Then we put F (α) = (α1 , α2 , α3 , α4 ). It is easy to see that F (α) is a morphism in B(B∞ ) and F : mod(B1 ⊗K B1o ) → B(B∞ ) is a well-defined functor. Keeping the above notations observe that if F (α) = 0 then α¯1 = 0 = α¯2 , and so α = 0. Thus the functor F is faithful. For any morphism (α1 , α2 , α3 , α4 ) : (V,B Y ′ , Y¯ ′ B ,B YB′′ , ψ1 , ψ2 , ψ3 , ψ4 ) → (W,B Z ′ , Z¯′ B ,B ZB′′ , κ1 , κ2 , κ3 , κ4 ) the morphism (α1 , α2 ) : B1 Y˜ ′ → B1 Z˜′ of left B1 -modules and the homomorphism (α3 , α4 ) : B1 Y¯′′ B → B1 Z¯′′ B give rise to a morphism α : Y → Z such that α = (α¯1 , α¯2 ) and α¯1 = (α1 , α2 ), α¯2 = (α3 , α4 ). Thus F (α) = (α1 , α2 , α3 , α4 ). Hence the functor F is full. Consider an object (V,B Y ′ , Y¯ ′ B ,B YB′′ , ψ1 , ψ2 , ψ3 , ψ4 ) ∈ B(B∞ ). Then B1 Y˜ ′ = (V,B Y ′ , ψ1 : B Y ′ → D(MB ) ⊗K V ) is a left B1 -module. In the same way B1 Y¯′′ B = (Y¯ ′ B ,B YB′′ , ψ4 : ′′ ′ B YB → D(MB ) ⊗K YB ) is a B1 -B-bimodule. Moreover, the morphisms ψ2 , ψ3 induce a morphism ψ : B1 Y˜ ′ ⊗K MB → B1 Y¯′′ B of B1 -B-bimodules. Consequently, Y ∼ = (B1 Y˜ ′ ,B1 Y¯′′ B , ψ) is a right B1 ⊗K B1o -module and clearly F (Y ) ∼ = (V,B Y ′ , Y¯ ′ B ,B YB′′ ,

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ψ1 , ψ2 , ψ3 , ψ4 ) in B(B∞ ). Therefore the functor F is dense, and so it is an equivalence of categories. ¤ ´ K MB be a one point extension of a triangular K-algebra B. 0 B Let X = (A X ′ ,A XB′′ , ϕ) be a right, finite-dimensional B1 ⊗K Ao -module. Then ∗ X ⊗A X ∼ = ∗ ∗ ′ ′ ∗ ′′ ′ ∗ ′ ′′ ∗ ′′ ′′ ∗ ( X ⊗A X , X ⊗A X , X ⊗A X , X ⊗A X , ϕ⊗A idX ′ , id∗ X ′′ ⊗A ϕ, id∗ X ′ ⊗A ϕ, ϕ ⊗A idX ′′ ) as B1 -B1 -bimodules.

Lemma 3.5. Let B1 =

³

Proof. A routine verification shows the lemma. We leave the details to the reader. ¤ ´ ³ K M Lemma 3.6. Let B1 = 0 BB be a one point extension of a triangular K-algebra B.

Let X = (A X ′ ,A XB′′ , ϕ) be an indecomposable, right B1 ⊗K Ao -module which is left-right projective. If X is left-sided quasi-invertible then the following conditions are satisfied: (1) The A-B-bimodule A XB′′ is left-sided quasi-invertible. (2) The left B-module ∗ X ′′ ⊗A X ′ is zero or is isomorphic to a direct sum of some copies of D(MB ) and a left projective B-module. (3) The right B-module ∗ X ′ ⊗A X ′′ is isomorphic to a direct sum of a right projective B-module and some copies of MB .

Proof. If X is left-sided quasi-invertible then ∗ X ⊗A X ′ ∼ = B1 ⊕ Π as B1 -B1 -bimodules for some projective B1 -B1 -bimodule Π. Then we infer by Lemma 3.5 that ∗ X ⊗A X ∼ = ∗ ′ ′ ∗ ′′ ′ ∗ ′ ′′ ∗ ′′ ′′ ∗ ∗ ( X ⊗A X , X ⊗A X , X ⊗A X , X ⊗A X , ϕ⊗A idX ′ , id∗ X ′′ ⊗A ϕ, id∗ X ′ ⊗A ϕ, ϕ ⊗A idX ′′ ). But it is easy to see that B1 ∼ = (K, 0, MB , ψ1 , ψ2 , ψ3 , ψ4 ) as B1 -B1 -bimodules. Moreover, if Π is a projective B1 -B1 -bimodule then Π ∼ = (K n , D(MB )n ⊕ Q, MNm ⊕ R, P, κ1 , κ2 , κ3 , κ4 ), where Q is a left projective B-module, R is a right projective B-module and P is a projective B-B-bimodule. Therefore ∗ X ⊗A X ∼ = B1 ⊕ Π implies that ∗ X ′′ ⊗A X ′′ ∼ = B ⊕P ′′ as B-B-bimodules. Hence X is a left-sided quasi-invertible A-B-bimodule, which proves (1). Since ∗ X ′′ ⊗A X ′ ∼ = D(MB ) ⊕ Q, condition (2) is clear. ∗ ′ ′′ ∼ Since X ⊗A X = MB ⊕ MBm ⊕ R, condition (3) is clear. ¤

4

Left-sided quasi-invertible bimodules for triangular Nakayama algebras

Let A, B be two finite-dimensional, triangular Nakayama K-algebras. Then A ∼ = KQ/I ′ ′ ∼ and B = KQ /I , where the quiver Q is of the form αs−1

α

α

βt−1

β2

β1

2 1 s −→ (s − 1) −→ · · · −→ 2 −→ 1,

the quiver Q′ is of the form t′ −→ (t − 1)′ −→ · · · −→ 2′ −→ 1′ ,

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and I, I ′ are generated by some paths. Moreover, Ao ∼ = KQo /I o , where the quiver Qo is of the form αos−1 αo1 αo2 1 −→ 2 −→ · · · −→ (s − 1) −→ s and the path αi . . . αj is a generator of I if and only if the path αjo . . . αio is a generator ¯ B⊗ Ao /I¯B⊗ Ao , where the of I o . With the above notations we have that B ⊗K Ao ∼ = KQ K K ¯ B⊗ Ao is of the form quiver Q K (t′ , 1)

(t′ ,αo 1)

−→

(βt−1 , 1) ↓ ((t − 1)′ , 1)

−→

↓ (2′ ,αo 1)

−→

(β1 , 1) ↓

((t − 1)′ , 2)

→ ··· →

−→

((t − 1)′ , s − 1)

↓

↓

↓

↓

(t′ , s)

−→

(βt−1 , s) ↓ ((t−1)′ ,αo s−1 )

−→

(2′ , 2)

→ ··· →

(2′ , s − 1)

. . .

↓ (2′ ,αo s−1 )

−→

(β1 , s − 1) ↓

(1′ , 2)

→ ··· →

(1′ , s − 1)

((t − 1)′ , s) ↓

. . .

(β1 , 2) ↓ (1′ ,αo 1)

(t′ ,αo s−1 )

(t′ , s − 1) (βt−1 , s − 1) ↓

. . .

. . .

(1′ , 1)

→ ··· →

(βt−1 , 2) ↓ ((t−1)′ ,αo 1)

↓

(2′ , 1)

(t′ , 2)

(2′ , s) (β1 , s) ↓

(1′ ,αo s−1 )

−→

(1′ , s)

¯ B⊗ Ao generated by (Q′ × I o ) ∪ (I ′ × (Qo )0 ) and and I¯B⊗K Ao is the two-sided ideal in K Q 0 K ′ o ′ o by the elements of the form (βi , l)(i , αl ) − ((i + 1) , αl )(βi , l + 1), where for any quiver T its set of the vertices is denoted by T0 . ˜ I) ˜ of the bound quiver (Q ¯ B⊗ Ao , I¯B⊗ Ao ) containing all vertices A full subquiver (Q, K K ′ ˜ ˜ ¯ o (i , j) with i ≤ j in the set of integers and I = K Q∩ IB⊗K A is said to be lower-triangular. ˆ I) ˆ of the bound quiver (Q ¯ B⊗ Ao , I¯B⊗ Ao ) is defined to be admissA full subquiver (Q, K K ible if the following conditions are satisfied: (1) (2) (3) (4)

ˆ 0. There is j ∈ {1, 2, . . . , s} such that (1′ , j) ∈ Q ˆ 0 and i > 1 then ((i − 1)′ , j − 1) ∈ Q ˆ 0. If (i′ , j) ∈ Q ˆ 0 it holds i ≤ j in the set of integers. For each (i′ , j) ∈ Q ˆ ∩ I¯B⊗ Ao . Iˆ = K Q K

¯ B⊗ Ao , I¯B⊗ Ao ) is admissIt is easy to see that the lower-triangular subquiver of (Q K K ¯ B⊗ Ao , I¯B⊗ Ao ) is a full subquiver of the lowerible and each admissible subquiver of (Q K K triangular one. Let C ∼ = KQC /IC be a finite-dimensional K-algebra. A finite-dimensional, right, indecomposable C-module X is said to be multiplicity-free if it has non-isomorphic simple composition factors (see [9]). For every finite-dimensional, right C-module Y its support is the full subquiver (supp(Y ), Isupp(Y ) ) of the bound quiver (QC , IC ) formed by all vertices z ∈ (QC )0 such that HomC (Pz , Y ) 6= 0, where Isupp(Y ) = Ksupp(Y ) ∩ IC and Pz is an indecomposable, right, projective C-module whose top is concentrated at the vertex z. Lemma 4.1. Let A ∼ = KQ′′ /I ′′ where = KQ′ /I ′ . Let B1 ∼ = KQ/I and B ∼ βt

βt−1

β1

Q′′ = (t + 1)′ → t′ → (t − 1)′ → · · · → 2′ → 1′

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and I ′′ is a two-sided ideal in KQ′′ generated by paths. Let X = (A X ′ ,A XB′′ , ϕ) be a right, finite-dimensional B1 ⊗K Ao -module which is left-sided quasi-invertible. If A XB′′ is a multiplicity-free right B ⊗K Ao -module whose support is an admissible subquiver of ¯ B⊗ Ao , I¯B⊗ Ao ) then X is a multiplicity-free B1 ⊗K Ao -module whose support is an (Q K K ¯ B1 ⊗ Ao , I¯B1 ⊗ Ao ). admissible subquiver of (Q K K Proof. We know from Lemma 3.6(1) that the A-B-bimodule X ′′ is left-sided quasiinvertible. Using the assumption that X ′′ is multiplicity-free and its support is an ad¯ B⊗ Ao , I¯B⊗ Ao ) we deduce that its support is of the following missible subquiver of (Q K K form (d′ , j0 + d − 1) → · · · → (d′ , ld − 1) → (d′ , ld ) .. .. . . (3′ , j0 + 2) → ··· → (3′ , l2 ) → (3′ , l3 ) ↓ ↓ (2′ , j0 + 1) → (2′ , j0 + 2) → ··· → (2′ , l1 ) → (2′ , l2 ) ↓ ↓ ↓ (1′ , j0 ) → (1′ , j0 + 1) → (1′ , j0 + 2) → ··· → (1′ , l1 ) Moreover, we know from Lemma 3.6(2) that ∗ X ′′ ⊗A X ′ ∼ = HomA (X ′′ , X ′ ) is zero or is a left B-module which is isomorphic to a direct sum of a left projective B-module and D(MB )n for some n ≤ 1. Furthermore, we have a nonzero A-B-bimodule morphism ϕ :A X ′ ⊗K MB → X ′′ . Therefore it is easily seen that X ′ is a direct sum of some of the left projective A-modules of the form Pj0 +d−1 , . . . , Pld . But X is indecomposable. Thus we obtain that X ′ is isomorphic to one of the modules Pj0 +d−1 , . . . , Pld . Since HomA (X ′′ , Pj0 +d−1 ) is a simple B-module, the right B-module MB is simple and HomA (X ′′ , Pj0 +d−1 ) ∼ = ∗ ′ D(MB ) as left B-modules. Furthermore, we deduce from Lemma 3.6(3) that X ⊗A X ′′ ∼ = HomA (Pj0 +d−1 , X ′′ ) is an indecomposable right B-module, and = HomA (X ′ , X ′′ ) ∼ so it is isomorphic to MB . Then we infer by Lemma 3.5 that (K, D(MB ), MB , B ⊕ Π, ρ1 , ρ2 , ρ3 , ρ4 ) ∼ 6= = (K, 0, MB , B, ψ1 , ψ2 , ψ3 , ψ4 ) ⊕ Π′ which is impossible. Thus X ′ ∼ ′ ′′ Pj0 +d−1 . Hence X is isomorphic to one of the modules Pj0 +d , . . . , Pld . Then HomA (X , X ′ ) = 0 and HomA (X ′ , X ′′ ) is a right indecomposable B-module. Therefore HomA (X ′ , X ′′ ) ∼ =∗ X ′ ⊗A X ′′ ∼ = MB by Lemma 3.6(3). Then we get that X is multiplicity-free. We have to check additionally whether the support of X is admissible. But this support is of the form ((d + 1)′ , z) → · · · → ((d + 1)′ , ld ) → · · · → ((d + 1)′ , ld+1 ) ↓ (d′ , j0 + d − 1) → · · · →

↓

(d′ , z)

→ ··· →

(d′ , ld )

.

.

.

↓ ··· →

.

.

If ld+1 > ld then X is not a right projective B-module since it has a direct summand isomorphic to the right simple B-module S(d+1)′ . Consequently, ld+1 ≤ ld and it is clear ¯ B1 ⊗ Ao , I¯B1 ⊗ Ao ). ¤ that the support of X is an admissible subquiver of (Q K K

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Proof (of Theorem 1). Using a Morita equivalence we may assume that A and B are triangular algebras. Let A ∼ = KQ/I, B ∼ = KQ′ /I ′ , where (Q′ , I ′ ), (Q, I) are as in the beginning of this section. We shall prove by induction on t′ that every left-sided quasi-invertible A-B-bimodule is a multiplicity-free one whose support is an admissible ¯ B⊗ Ao , I¯B⊗ Ao ). subquiver of (Q K K ′ ′ If t = 1 then it is obvious that B ⊗K Ao ∼ = Ao , because B ∼ = K. Then every left-sided quasi-invertible A-K-bimodule is a left projective A-module. Thus the required condition holds. Assume that for all triangular Nakayama K-algebras A ∼ = KQ′ /I ′ such = KQ/I, B ∼ ′ ′ ∼ that t′ ≤ t′0 the required condition holds. Let A ∼ = KQ/I, ³ B1 =´KQ /I be triangular K M Nakayama K-algebras such that t′ = (t0 +1)′ . Then B1 ∼ = KQ′′ /I ′′ = 0 BB , where B ∼ βt

−1

β1

0 with Q′′ = t′0 → (t0 − 1)′ → · · · → 2′ → 1′ and I ′′ = KQ′′ ∩ I ′ . We know from the inductive assumption that there are only finitely, many pairwise non-isomorphic, left-sided quasi-invertible A-B-bimodules. They are multiplicity-free and their supports ¯ B⊗ Ao , I¯B⊗ Ao ). Then we infer by are admissible subquivers of the bound quiver (Q K K Lemma 3.6 and Lemma 4.1 that the required condition holds for every left-sided quasiinvertible A-B1 -bimodule. Consequently, there are only finitely many left-sided quasiinvertible, pairwise non-isomorphic A-B1 -bimodules and Theorem 1 follows. ¤

Corollary 4.2. Let A, B be triangular Nakayama algebras. Then every left-sided quasiinvertible A-B-bimodule is a multiplicity-free bimodule whose support is an admissible ¯ B⊗ Ao , I¯B⊗ Ao ). subquiver of the bound quiver (Q K K Proof. The corollary is a direct consequence of the proof of Theorem 1.

5

¤

Stable equivalences of Morita type between Nakayama algebras

Now we are interested in the special case. Let A ∼ = KQ′ /I ′ be triangular = KQ/I, B ∼ Nakayama algebras. Moreover, assume that in the notations used in the beginning of Section 4 we have s=t. This means that the numbers of pairwise non-isomorphic, right simple A-modules and B-modules coincide. We are interested in the problem when there are left-sided quasi-invertible A-B-bimodules. Moreover, we want to know the precise number of left-sided quasi-invertible A-B-bimodules in this special case. Throughout the sequel we shall keep the above assumptions. Lemma 5.1. For any left-sided quasi-invertible A-B-bimodule X its support is an ad¯ B⊗ Ao , I¯B⊗ Ao ) such that all vertices (i′ , j) satisfying i = j belong missible subquiver of (Q K K to supp(X). Proof. We infer by Corollary 4.2 that each left-sided quasi-invertible A-B-bimodule X is multiplicity-free whose support (supp(X), Ksupp(X)∩ I¯B⊗K Ao ) is an admissible subquiver

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¯ B⊗ Ao , I¯B⊗ Ao ). Suppose that (t′ , t) 6∈ supp(X). Then for a of the bound quiver (Q K K minimal projective resolution · · · → R2 → R1 → R0 → X → 0 of X it holds that Aei ⊗K et B does not appear as a direct summand in R0 for every i = 1, . . . , t. Furthermore, in an obvious way none of the simple A-B-bimodules S(t′ ,1) , . . . , S(t′ ,t) is a direct summand in the top of ∗ X. Let · · · → P2 → P1 → P0 →∗ X → 0 be a minimal projective resolution of ∗ X. Then none of the projective B-A-bimodules Bet ⊗K e1 A, . . . , Bet ⊗K et A appears as a direct summand in P0 . Thus we deduce from Theorem 1.9 that for the projective resolution · · · → Q2 → Q1 → Q0 →∗ X ⊗A X → 0 it holds Q0 ∼ = P0 ⊗A R0 . Observe that Bet ⊗K et B is not a direct summand in Q0 . Indeed, if Bet ⊗K et B appears as a direct summand in Q0 then it is isomorphic to a direct summand of Bet ⊗K ej A ⊗A Aei ⊗K et B, where j, i < t. But none of the bimodules Aei ⊗K et B appears as a direct summand of R0 . Therefore Bet ⊗K ey B is not a direct summand of Q0 . On the other hand Happel showed in [4] that for the B-B-bimodule B there is a minimal projective resolution in mod(B ⊗K B o ) of the form · · · → T2 → T1 → T0 → B → Lt 0 such that T0 ∼ = i=1 Bei ⊗K ei B. This gives a contradiction to the above claim. ′ Consequently, (t , t) ∈ supp(X). Then using the definition of the admissible subquiver of ¯ B⊗ Ao , I¯B⊗ Ao ) one gets that ((t − 1)′ , t − 1) ∈ supp(X), ...,(1′ , 1) ∈ supp(X) and our (Q K K lemma follows. ¤ Lemma 5.2. If there is an A-B-bimodule which is left-sided quasi-invertible then A ∼ = B. Proof. Assume that there is an A-B-bimodule X which is left-sided quasi-invertible. Then we know from Lemma 5.1. that (i′ , i) ∈ supp(X) for each i = 1, . . . , t. Furthermore, we know from Corollary 4.2 that X is multiplicity-free and its support is an admissible ¯ B⊗ Ao , I¯B⊗ Ao ). subquiver of the bound quiver (Q K K In order to show that A ∼ = B it is enough to prove that I = I ′ . We know that X is a left-right projective A-B-bimodule. Then suppose that a path αio · · · αjo is a generator of I o . We are going to show that βj · · · βi ∈ I ′ . If αio · · · αjo is a generator of I o then (i′ , αio ) · · · (i′ , αjo ) is a generator of I¯B⊗K Ao for every i = 1, . . . , t. Consider the path ¯ B⊗ Ao . If ω 6∈ I¯B⊗ Ao then X as a representation of the bound ω = (βj , j) · · · (βi , j) in Q K K ¯ B⊗ Ao , I¯B⊗ Ao ) is of the following form quiver (Q K K K ··· ↓ (βj , j) K → K ··· ↓ ↓ .. .. . . ↓ ↓ K → K ··· ↓ ↓ (βi , j) (i′ ,αo )

(i′ ,αoi+1 )

(i′ ,αoj )

K −→i K −→ K → · · · → K → K ↓ .. K →K . . . .. . . . . . .

··· . .

But the commutativity relations in I¯B⊗K Ao implies that the linear map on the arrow

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141

(i′ , αjo ) is nonzero. Moreover, the composition of the linear maps on the arrows (i′ , αio ), . . . , o (i′ , αj−1 ) is also nonzero. Then the composition of the linear maps on the arrows (i′ , αio ), o . . . , (i′ , αj−1 ), (i′ , αjo ) is nonzero which contradicts the fact that (i′ , αio ) · · · (i′ , αjo ) ∈ I¯B⊗K Ao . Consequently, ω ∈ I¯B⊗K Ao , and so βj · · · βi ∈ I ′ . Dual arguments show that if βj · · · βi is a generator of I ′ then αio · · · αjo ∈ I o . Therefore I = I ′ , and so A ∼ ¤ = B. Proof (of Theorem 2). Let A, B be two Nakayama K-algebras which are factors of hereditary algebras. Assume that A and B are stably equivalent of Morita type. Then there are basic K-algebras A1 and B1 such that A and A1 (resp. B and B1 ) are Morita equivalent. Thus A1 and B1 are triangular K-algebras which are stably equivalent of Morita type. Furthermore, we infer by the main result in [7] that the numbers of right simple A-modules and B-modules coincide. Moreover, we deduce from the proof of Theorem ?? in [11] that there is a left-sided quasi-invertible A1 -B1 -bimodule X. Thus we infer by Lemma 5.3 that A1 , B1 are isomorphic. Consequently, A, B are Morita equivalent. If A, B are Morita equivalent then it is clear that they are stably equivalent of Morita type. ¤

References [1] M. Auslander and I. Reiten: “On a Theorem of E. Green on the Dual of the Transpose”, Proc. ICRA V, CMS Conf. Proc., Vol. 11, (1991), pp. 53–65. [2] M. Brou´e: “Equivalences of Blocks of Group Algebras”, In: V. Dlab and L.L. Scott (Eds.): Finite Dimensional Algebras and Related Topics, NATO ASI Series C, Vol. 424, Kluwer Academic Press, Dodrecht, 1992, pp. 1–26. [3] P. Gabriel: Auslander-Reiten sequences and representation-finite algebras, Springer Lecture Notes in Math., Vol. 831, Berlin, 1980, pp. 1–71. [4] D. Happel: “Hochschild cohomology of finite-dimensional algebras”, In: Seminaire d’Algebre P. Dubriel et M-P. Maliavin, Lecture Notes in Math., Vol. 1404, Springer, Berlin, 1989, pp. 108–126. [5] S. MacLane: Homology, Springer-Verlag, Berlin, 1963. [6] K. Morita: “Duality for modules and its applications to the theory of rings with minimum condition”, Sci. Rep. Tokyo Kyoiku Daigaku Sec. A, Vol. 6, (1958), pp. 83–142. [7] Z. PogorzaÃly: “Algebras stably equivalent to selfinjective special biserial algebras”, Comm. Algebra, Vol. 22, (1994), pp. 1127–1160. [8] Z. PogorzaÃly: “Left-right projective bimodules and stable equivalence of Morita type”, Colloq. Math., Vol. 88(2), (2001), pp. 243–255. [9] Z. PogorzaÃly and A. Skowro´ nski: “On algebras whose indecomposable modules are multiplicity-free”, Proc. London Math. Soc., Vol. 47, (1983), pp. 463–479. [10] J. Rickard “Derived equivalences as derived functors”, J. London Math. Soc., Vol. 2(43), (1991), pp. 37–48.

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[11] J. Rickard: “Some Recent Advances in Modular Representation Theory”, Proc. ICRA VIII, CMS Conf. Proc., Vol. 23, (1998), pp. 157–178. [12] C.M. Ringel: Tame Algebras and Integral Quadratic Forms, Springer Lecture Notes in Math., Vol. 1099, Berlin, 1984. [13] D. Simson: Linear Representations of Partially Ordered Sets and Vector Space Categories, Gordon & Breach Science Publishers, Amsterdam, 1992.

CEJM 3(1) 2005 143–154

On the Two-Point Boundary-Value Problem for the Riccati Matrix Differential Equation V.N. Laptinsky1∗ , I.I. Makovetsky† Institute of Applied Optics of NAS of Belarus, 11, B. Biruli St., 212793 Mogilev, Belarus

Received 18 May 2004; revised 29 November 2004 Abstract: Constructive sufficient conditions for univalent resolvability of a two-point boundary value problem for nonlinear Riccati equation are obtained. An illustrative example is given. c Central European Science Journals. All rights reserved. ° Keywords: two-point boundary value problem, matrix differential equation MSC (2000): 34B10

1

Introduction

Consider a Riccati equation of the following form: dX = A(t)X + XB(t) + XQ(t)X + F(t), dt

(1)

where A, B, Q, F ∈ C(I , Rn×n ), I = [0; ω], ω > 0. We study a two-point boundary-value problem for (1) in case of MX(0) + NX(ω) = 0,

(2)

where M and N are real n × n matrices. This equation is prominent in differential equation theory and its applications. Similar problems were considered with the aid of qualitative methods in [1], [2] and [4]-[7] and on the basis of constructive methods in [3], [8] and [9]. The present work is a continuation of [10] and deals with a constructive analysis of problem (1), (2) on the basis of the method presented in [3]. ∗ †

E-mail: [email protected] E-mail: [email protected]

144

2

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

Theorem of existence and uniqueness

We introduce the following notations: Dρ = {(t, X) : t ∈ I, kXk ≤ ρ} , α = max kA(t)k , β = max kB(t)k , δ = max kQ(t)k , t

t

t

° ° ° ° ˜ h = max kF(t)k , γ = °Φ−1 ° , m = °N−1 (M + N)° , H(ω) ≡ t

Zω

H(τ )dτ,

0

˜ ˜ R = N (M + N + NA(ω)), S = −B(ω), ϕ(ρ) = a0 ρ2 + a1 ρ + a2 , kHkC = max kH(t)k , −1

t

where ρ > 0, t ∈ I, k · k is a multiplicative norm of matrices, for example, any of norms given in [11, page 21],Φ is the linear operator, ΦH = RH − HS, H ∈ C(I , R n×n ), n×n C = C(I, ¶ space of continuous µ n × n matrices, ¶ µ R ) is the Banach 1 1 a0 = γωδ m + (α + β)ω + 1 , a1 = γω(α + β) m + ω(α + β) , and 2 2 µ ¶ 1 a2 = γωh m + ω(α + β) + 1 . 2 Theorem. Let the following conditions be fulfilled: (A1) det N 6= 0 (A2) matrices R and S have no common eigenvalues, (A3) ϕ(ρ) ≤ ρ, (A4) ϕ0 (ρ) < 1. Then a unique solution of problem (1), (2) exists in the region Dρ . Proof. Using a procedure presented in [3], we first obtain a matrix integral equation, which is equivalent to problem (1), (2). Let X = X(t) be a solution of this problem. Then it follows from (1) that X(t) = X(t0 ) +

Zt

[A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] dτ,

(3)

t0

where t0 ∈ I. From (3) and (2) we get (M + N)X(t) = (M + N)

Zt

[A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] dτ − (4)

0

−N

Zω

[A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] dτ .

0

Using in (4) the following relationships:     Zω Z t Zτ Zω Zω ˜ ˙ )dτ +  A(σ)dσ  X(τ ˙ )dτ. A(τ )X(τ )dτ = A(ω)X(t) −  A(σ)dσ  X(τ 0

0

0

t

τ

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

Zω

˜ X(τ )B(τ )dτ = X(t)B(ω) −

0

Zt 0



˙ ) X(τ

Zτ 0



A(σ)dσ  dτ +

˙ ) = dX(τ )/dτ , we get by virtue of (A1) that where X(τ h i ˜ ˜ M + N + NA(ω) X(t) + NX(t)B(ω) = = (M + N) +N

Zt

Zt

Zω t



˙ ) X(τ

Zω τ

145



A(σ)dσ  dτ,

[A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] dτ +

0

{A1 (τ ) [A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] +

0

+ [A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] B1 (τ )} dτ − Zω −N {A2 (τ ) [A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] +

(5)

t

+ [A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] B2 (τ )} dτ − Zω −N (X(τ )Q(τ )X(τ ) + F(τ ))dτ , 0

where H1 (τ ) ≡

Zτ

H(σ)dσ, H2 (τ ) ≡

0

Zω

H(σ)dσ.

τ

Using condition (A1), we premultiply equation (5) through by N−1 and obtain h i ˜ ˜ N−1 M + N + NA(ω) X(t) + X(t)B(ω) = =N

+

−1

Zt

(M + N)

Zt

[A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] dτ +

0

{A1 (τ ) [A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] +

0

+ [A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] B1 (τ )} dτ − Zω − {A2 (τ ) [A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] + t

+ [A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )] B2 (τ )} dτ − Zω − (X(τ )Q(τ )X(τ ) + F(τ ))dτ , 0

(6)

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V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

Since matrices R and S have no common eigenvalues (see condition (A2)), the operator Φ is invertible in accordance with [12, p.207], the operator Φ−1 being linear and restricted. For reasons given, it follows from (6) that

X (t) =   Zt   = Φ−1 N−1 (M + N) (A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ ))dτ +   0  ω Z −1 +Φ [KA (t, τ ) (A(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )) + (7)  0

+ (A(τ )X(τ ) + X(τ )B(τ ) + X(τ)Q(τ )X(τ ) + F(τ )) KB (t, τ )] dτ − Zω  − (X(τ )Q(τ )X(τ ) + F(τ ))dτ ,  0

where

KH (t, τ ) =

   H1 (τ ), 0 ≤ τ ≤ t ≤ ω,

  −H2 (τ ), 0 ≤ t < τ ≤ ω.

With the aid of simple calculations one can show the converse: any continuous solution of the matrix integral equation (7) is a solution of problem (1), (2). To construct the solution of equation (7), we use the following algorithm: Xk+1 (t) = = Φ−1 N−1 (M + N)

Zt

[A(τ )Xk (τ ) + Xk (τ )B(τ ) + Xk (τ )Q(τ )Xk (τ )) + F(τ )] dτ +

0

+Φ−1

 ω Z 

[KA (t, τ ) (A(τ )Xk (τ ) + Xk (τ )B(τ ) + Xk (τ )Q(τ )Xk (τ ) + F(τ )) + (8)

0

+ (A(τ )Xk (τ ) + Xk (τ )B(τ ) + Xk (τ )Q(τ )Xk (τ ) + F(τ )) KB (t, τ )] dτ − ω Z  − (Xk (τ )Q(τ )Xk (τ ) + F(τ ))dτ ,  0

where X0 (t) is the arbitrary C(I, Rn×n ) - class matrix, which belongs to the sphere kX0 kC ≤ ρ. Using condition (A3) and induction on k, one can show readily that matrix functions Xm (t), m = 1, 2, . . . , obtained from (8) belong to the sphere kXkC ≤ ρ. Now

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

147

we study the problem of convergence of this algorithm. From (8) we have: Xm+1 (t) −Xm (t) =  Zt  −1 −1 = Φ N (M + N) [A(τ )(Xm (τ ) − Xm−1 (τ ))+  0

+(Xm (τ ) − Xm−1 (τ ))B(τ ) + Xm Q(τ )Xm − Xm−1 Q(τ )Xm−1 ]  ω Z −1 +Φ [KA (t, τ ) (A(τ )(Xm (τ ) − Xm−1 (τ ))+ 

o

dτ + (9)

0

+(Xm (τ ) − Xm−1 (τ ))B(τ ) + Xm Q(τ )Xm − Xm−1 Q(τ )Xm−1 ) +

+ (A(τ )(Xm (τ ) − Xm−1 (τ )) + (Xm (τ ) − Xm−1 (τ ))B(τ ) +

+Xm (τ )Q(τ )Xm (τ ) − Xm−1 (τ )Q(τ )Xm−1 (τ )) KB (t, τ )] dτ −  Zω  − (Xm (τ )Q(τ )Xm (τ ) − Xm−1 (τ )Q(τ )Xm−1 (τ ))dτ ,  0

m = 1, 2, . . .

Having performed estimations on the norm in (9), we obtain the recurrent estimation kXm+1 − Xm kC ≤ q kXm − Xm−1 kC

(m = 1, 2, . . .),

(10)

where q = ϕ0 (ρ). On the basis of (10) we have the explicit estimation kXm+1 − Xm kC ≤ q m kX1 − X0 kC

(m = 1, 2, . . .).

(11)

Using (11) and conditions (A3) and (A4), the sequence {Xk }∞ 0 constructed on the basis of algorithm 8 can be proved to converge uniformly on t ∈ I to the matrix function X(t) ∈ Dρ , which is a solution of the integral equation (7). The uniqueness of the solution X = X(t) is obvious within the framework of the contraction mapping principle [13, p.605]) since the integrated operator defined with the right side of (7) is a contraction due to condition (A4). Then from (11) we have the estimation describing an accuracy of calculations by algorithm (8): kX − Xk kC ≤

qk kX1 − X0 kC 1−q

(k = 0, 1, 2, . . .).

(12)

Using (12), kXkC can be estimated: for k = 0 and X0 = 0 we have kXkC ≤

kX1 kC . 1−q

(13)

148

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

We deduce the estimation for kX1 kC . From (8) for k = 0, X0 = 0 we have   Zt   −1 −1 N (M + N) F(τ )dτ + X1 (t) = Φ   0  ω  Zω Z  [KA (t, τ ) F(τ ) + F(τ )KB (t, τ )] dτ − F(τ )dτ . +Φ−1   0

(14)

0

Estimating on the norm in (14), we get sequentially ° ° kX1 (t)k ≤ °Φ−1 N−1 (M + N)°

Zt

kF(τ )k dτ +

0

  ω Zω Z   ° −1 ° ° ° + Φ [kKA (t, τ )k kF(τ )k + kF(τ )k kKB (t, τ )k] dτ + kF(τ )k dτ ≤   0

0

µ

1 1 ≤ γmht + γh(α + β)(t2 + (ω − t)2 ) + γωh ≤ γωh m + (α + β)ω + 1 2 2

thus obtaining the following estimation

¶

kX1 kC ≤ a2 .

= a2 ,

(15)

Then it follows from (15) and (13) that kX(t)kC ≤

a2 . 1 − q(ρ)

(16)

A similar estimation can be obtained from equation (7). Actually, considering (7) as an identity (for X = X(t)), we can make the following estimations: kX(t)k ≤ γm

Zt

kA(τ )X(τ ) + X(τ )B(τ ) + X(τ )Q(τ )X(τ ) + F(τ )k dτ +

0

+γ

 ω Z 

[kKA (t, τ )k + kKB (t, τ )k] kA(τ )X(τ ) + X(τ )B(τ )+

0

+X(τ )Q(τ )X(τ ) + F(τ )kdτ +

Zω

(kX(τ )Q(τ )X(τ ) + F(τ )k)dτ

0

¤ ≤ γmt (α + β) kX(τ )k + δ kX(τ )k2 + h + ¢ ¡ 1 + γ (α + β) t2 + (ω − t)2 [(α + β) kX(τ )k] + 2 ¡ ¢ +δ kX(τ )k2 + h + γω δ kX(τ )k2 + h ≤ £

  

≤

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

149

µ ¶ 1 ≤ γωδ m + (α + β)ω + 1 kXk2C + 2 µ ¶ µ ¶ 1 1 +γω(α + β) m + ω(α + β) kXkC + γωh m + ω(α + β) + 1 . 2 2 From this we have the inequality ¶ ½ µ 1 kXkC ≤ γω δ m + (α + β)ω + 1 kXk2C + 2 µ ¶ µ ¶¾ 1 1 +(α + β) m + ω(α + β) kXkC +h m + ω(α + β) + 1 . 2 2

(17)

Inequality (17) can be rewritten in the following form: a0 kXk2C − (1 − a1 ) kXkC + a2 ≥ 0. After elementary calculations we have the estimation kXkC ≤ ρ1 ,

(18)

where ρ1 is the minimum root of the equation a0 ρ2 − (1 − a1 )ρ + a2 = 0, i.e. ρ1 =

1 − a1 −

q

(1 − a1 )2 − 4a0 a2 2a0

.

Estimation (18) is more preferable than estimation (16) since ρ1 <

a2 . 1 − q(ρ1 )

(19)

The validity of this non-evident inequality can be shown with the aid of simple calculations. Actually, inequality (19) is equivalent to the following inequality: 1 − a1 −

p

(1 − a1 )2 − 4a0 a2 a2

Assuming that condition (A4) holds, consequently we get equivalent inequalities (1 − a1 )2

p

(1 − a1 )2 − 4a0 a2 < (1 − a1 )2 − 2a0 a2 ,

that is, inequality (19) is involved.

4a20 a22 > 0,

150

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

References [1] N.P. Erugin: Reading Book on the General Course of Differential Equations, Nauka&Technica, Minsk, 1979. [2] V.I. Zubov: Lectures on the Control Theory, Nauka, Moscow, 1975. [3] V.N.Laptinsky: Constructive Analysis of Controlled Oscillating Systems, Institute of Mathematics, NAS of Belarus, Minsk, 1998. [4] V.B. Larin: Control of Walking Apparatuses, Naukova Dumka, Kiev, 1980. [5] V.I. Mironenko: Linear Dependence of Functions along Solutions of Differential Equations, Belarussian State University, Minsk, 1981. [6] Yu.I. Paraev: Lyapunov and Riccati Equations, State University, Tomsk, 1989. [7] Ia. N. Roytenberg: Automatic Control, Nauka, Moscow, 1978. [8] A.M. Samoilenko, V.N. Laptinsky, K.K. Kenjebaev: Constructive Research Approaches of Periodic and Multipoint Boundary Value Problems, Institute of Mathematics, NAS of Ukraine, Kiev, 1999. [9] V.N. Laptinsky and V.V. Pugin: Report Theses of Math. Conf. in memory of Prof. S.G. Kondratenia. Brest, (1998), pp. 23. [10] V.N. Laptinsky and I.I. Makovetsky: “On Solvability of the Two-Point BoundaryValue Problem for the Nonlinear Lyapunov Equation”, Herald of the Mogilev State University, Vol. 15, (2003), pp. 176–181. [11] B.P. Demidovich: Lectures on the Mathematical Theory of Stability, Nauka, Moscow, 1967. [12] F.R. Gantmakher: The Matrix Theory, Nauka, Moscow, 1967. [13] L.V.Kantorovich and G.P.Akilov: Functional Analysis, Nauka, Moscow, 1977.

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

ti

X11,1 (t)

X12,1 (t)

X21,1 (t)

X22,1 (t)

0

2, 318 · 10−4

1, 60 · 10−5

5, 26 · 10−5

−7, 6 · 10−6

π/5

8, 171 · 10−4

2, 077 · 10−4

2, 426 · 10−4

−5, 953 · 10−4

2π/5

1, 1766 · 10−3

7, 067 · 10−4

7, 371 · 10−4

−9, 575 · 10−4

3π/5

1, 1765 · 10−3

1, 3217 · 10−3

1, 3425 · 10−3

−9, 544 · 10−4

4π/5

8, 134 · 10−4

1, 8207 · 10−3

1, 8216 · 10−3

−5, 872 · 10−4

π

2, 182 · 10−4

2, 024 · 10−3

1, 9840 · 10−3

2, 3 · 10−6

6π/5

−3, 857 · 10−4

1, 8184 · 10−3

1, 7608 · 10−3

5, 883 · 10−4

7π/5

−7, 650 · 10−4

1, 3073 · 10−3

1, 2321 · 10−3

9, 477 · 10−4

8π/5

−7, 713 · 10−4

6, 740 · 10−4

5, 953 · 10−4

9, 446 · 10−4

9π/5

−4, 056 · 10−4

1, 629 · 10−4

8, 76 · 10−5

5, 802 · 10−4

2π

1, 847 · 10−4

−3, 11 · 10−5

−1, 045 · 10−4

−7, 6 · 10−6

Table 1

151

152

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

ti

X11,2 (t)

X12,2 (t)

X21,2 (t)

X22,2 (t)

0

−2, 8 · 10−6

−4 · 10−7

−2, 3 · 10−6

−1 · 10−7

π/5

5, 870 · 10−4

1, 907 · 10−4

1, 893 · 10−4

−5, 879 · 10−4

2π/5

9, 513 · 10−4

6, 907 · 10−4

6, 906 · 10−4

−9, 511 · 10−4

3π/5

9, 514 · 10−4

1, 3088 · 10−3

1, 3102 · 10−3

−9, 511 · 10−4

4π/5

5, 874 · 10−4

1, 8088 · 10−3

1, 8116 · 10−3

−5, 877 · 10−4

π

−1, 2 · 10−6

1, 998 · 10−3

2, 0030 · 10−3

0

6π/5

−5, 896 · 10−4

1, 8089 · 10−3

1, 8111 · 10−3

5, 879 · 10−4

7π/5

−9, 532 · 10−4

1, 3088 · 10−3

1, 3092 · 10−3

9, 511 · 10−4

8π/5

−9, 532 · 10−4

6, 9072 · 10−4

6, 889 · 10−4

9, 511 · 10−4

9π/5

−5, 893 · 10−4

1, 905 · 10−4

1, 872 · 10−4

5, 879 · 10−4

2π

−6 · 10−7

−7 · 10−7

−4, 6 · 10−6

2 · 10−7

Table 2

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

153

ti

|X11,1 (t) − X11 (t)|

|X12,1 (t) − X12 (t)|

|X21,1 (t) − X21 (t)|

|X22,1 (t) − X22 (t)|

0

2, 318 · 10−4

1, 59 · 10−5

5, 25 · 10−5

7, 6 · 10−6

π/5

2, 293 · 10−4

1, 67 · 10−5

5, 16 · 10−5

7, 5 · 10−6

2π/5

2, 255 · 10−4

1, 56 · 10−5

4, 61 · 10−5

6, 4 · 10−6

3π/5

2, 254 · 10−4

1, 27 · 10−5

3, 35 · 10−5

3, 3 · 10−6

4π/5

2, 256 · 10−4

1, 16 · 10−5

1, 26 · 10−5

5 · 10−7

π

2, 182 · 10−4

1, 24 · 10−5

1, 59 · 10−5

2, 3 · 10−6

6π/5

2, 020 · 10−4

9, 3 · 10−6

4, 81 · 10−5

5 · 10−7

7π/5

1, 860 · 10−4

1, 6 · 10−6

7, 68 · 10−5

3, 3 · 10−6

8π/5

1, 797 · 10−4

1, 70 · 10−5

9, 56 · 10−5

6, 4 · 10−6

9π/5

1, 822 · 10−4

2, 80 · 10−5

1, 034 · 10−4

7, 5 · 10−6

2π

1, 846 · 10−4

3, 11 · 10−5

1, 045 · 10−4

7, 6 · 10−6

Table 3

154

V.N. Laptinsky, I.I. Makovetsky / Central European Journal of Mathematics 3(1) 2005 143–154

ti

|X11,2 (t) − X11 (t)|

|X12,2 (t) − X12 (t)|

|X21,2 (t) − X21 (t)|

|X22,2 (t) − X22 (t)|

0

2, 8 · 10−6

3 · 10−7

2, 2 · 10−6

1 · 10−7

π/5

7 · 10−7

3 · 10−7

1, 7 · 10−7

1 · 10−7

2π/5

3 · 10−7

2 · 10−7

4 · 10−7

0

3π/5

3 · 10−7

1 · 10−7

1, 2 · 10−6

0

4π/5

3 · 10−7

1 · 10−7

2, 6 · 10−6

0

π

1, 1 · 10−6

1 · 10−7

3, 0 · 10−6

0

6π/5

1, 8 · 10−6

1 · 10−7

2, 1 · 10−6

1 · 10−7

7π/5

2, 1 · 10−6

1 · 10−7

1 · 10−7

0

8π/5

2, 1 · 10−6

2 · 10−7

2, 1 · 10−6

0

9π/5

1, 5 · 10−6

4 · 10−7

3, 8 · 10−6

1 · 10−7

2π

5 · 10−7

7 · 10−7

4, 6 · 10−6

2 · 10−7

Table 4