Andrzej Walendziak, Magdalena Wojciechowska-Rysiawa BIPARTITE PSEUDO-BL ALGEBRAS

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1 DEMONSTRATIO MATHEMATICA Vol. XLIII No Andrzej Walendziak, Magdalena Wojciechowska-Rysiawa BIPARTITE PSEUDO-BL ALGEBRAS Abstract. The class of bipartite pseudo-bl algebras (denoted by BP) and the class of strongly bipartite pseudo-bl algebras (denoted by BP 0 ) are investigated. We prove that the class BP 0 is a variety and show that BP is closed under subalgebras and arbitrary direct products but it is not a variety. We also study connections between bipartite pseudo-bl algebras and other classes of pseudo-bl algebras. 1. Introduction BL algebras were introduced by Hájek [9] in MV algebras introduced by Chang [1] are contained in the class of BL algebras. Georgescu and Iorgulescu [6] introduced pseudo-mv algebras as a noncommutative generalization of MV algebras. In 2000, in a natural way, there were introduced pseudo-bl algebras as a generalization of BL algebras and MV algebras. A pseudo-bl algebra is a pseudo-mv algebra if and only if the pseudo-double Negation condition (pdn, for short) is satisfied, that is, (x ) = (x ) = x for all x. Main properties of pseudo-bl algebras were studied in [2] and [3]. Pseudo-BL algebras correspond to a pseudo-basic fuzzy logic (see [10] and [11]). Bipartite MV algebras were defined and studied by Di Nola, Liguori and Sessa in [4]. Dymek [5] investigated bipartite pseudo-mv algebras. Georgescu and Leuştean [8] introduced the class BP of pseudo-bl algebras bipartite by some ultafilter and the subclass BP 0 of pseudo-bl algebras bipartite by all ultrafilters. In this paper we give some characterizations of bipartite and strongly bipartite pseudo-bl algebras. We prove that the class BP 0 is a variety and show that BP is closed under subalgebras and arbitrary direct products but it is not a variety. We also study connections between bipartite pseudo-bl algebras and other classes of pseudo-bl algebras Mathematics Subject Classification: 03G25, 06F05. Key words and phrases: pseudo-bl algebra, filter, ultrafilter, (strongly) bipartite pseudo-bl algebra.

2 488 A. Walendziak, M. Wojciechowska-Rysiawa 2. Preliminaries Definition 2.1. ([2]) Let (A,,,,,, 0, 1) be an algebra of type (2, 2, 2, 2, 2, 0, 0). The algebra A is called a pseudo-bl algebra if it satisfies the following axioms, for any x, y, z A : (C1) (A,,, 0, 1) is a bounded lattice, (C2) (A,, 1) is a monoid, (C3) x y z x y z y x z, (C4) x y = (x y) x = x (x y), (C5) (x y) (y x) = (x y) (y x) = 1. Throughout this paper A will denote a pseudo-bl algebra. For any x A and n = 0, 1,..., we put x 0 = 1 and x n+1 = x n x. Proposition 2.2. ([2])The following properties hold in A for all x, y A : (a) x y x y = 1 x y = 1, (b) x y x and x y y. Let us define x = x 0 and x = x 0 for all x A. Proposition 2.3. ([2]) The following properties hold in A for all x, y A: (a) x (x ) and x (x ), (b) x x = x x = 0, (c) x y implies y x and y x. Definition 2.4. A nonempty set F is called a filter of A if the following conditions hold: (F1) If x, y F, then x y F, (F2) if x F, y A, x y then y F. The filter F is called proper if F A. The set of all filters of A is denoted by Fil(A). For every subset X A, the smallest filter of A which contains X, that is the intersection of all filters F X, is said to be the filter generated by X and will be denoted by [X). Proposition 2.5. ([2]) If X A, then [X) = {y A : x 1 x n y for some n 1 and x 1,...,x n X}. Definition 2.6. Let F be a proper filter of A. (a) F is called prime iff for all x, y A, x y F implies x F or y F. (b) F is called maximal (or ultrafilter) iff whenever H is a filter such that F H A, then either H = F or H = A.

3 Bipartite pseudo-bl algebras 489 We denote by Max(A) the set of ultrafilters of A. Definition 2.7. A filter H of A is called normal if for every x, y A x y H x y H. Proposition 2.8. ([2]) Any ultrafilter of A is a prime filter of A. Proposition 2.9. ([2]) Any proper filter of A can be extended to an ultrafilter. and Following [8], for any F A, we define two sets F and F as follows: By Remark 1.13 of [8] we have F = {x A : x f for some f F } F = {x A : x f for some f F }. F = {x A : x F } and F = {x A : x F }. Lemma If F is a proper filter of A, then: (a) F F =, (b) F F =, (c) F A F, (d) F A F. Proof. (a) Suppose that x F F. Then x F and x f for some f F. Since F is a filter, from definition it follows that f F and f F. Using Proposition 2.3 (b) we have 0 = f f F. This contradicts the fact that F is proper. (b) Similar to (a). (c) Let x F. Then x F. Suppose that x F. Applying Proposition 2.3 (b) we obtain x x = 0 F. This is a contradiction, because F is proper. (d) Similar to (c). Proposition Let F be a proper filter of A. Then the following conditions are equivalent: (a) A = F F = F F, (b) F = F = A F, (c) x A (x F or (x F and x F)), (d) x A (x x, x x F) and F is prime. Proof. (a) (b). Follows easily from Lemma 2.10 (a) and (b). (b) (c). Let x A F. Therefore x F = F. Hence x F and x F. (c) (d). See Proposition 5.1 of [8].

4 490 A. Walendziak, M. Wojciechowska-Rysiawa (c) (a). Obvious. Proposition If F is a proper filter of A and one of the equivalent conditions of Proposition 2.11 holds, then F is an ultrafilter. Proof. Suppose that x / F and let U = [F {x}). We show that A = U. It suffices to prove that 0 U. Let x F and hence x F. Therefore x U. Consequently, 0 = x x U. Let h : A B be a homomorphism of pseudo-bl algebras. The set Ker(h) = {x A : h(x) = 1} is called the kernel of h. Proposition ([8]) Let h : A B be a homomorphism of pseudo-bl algebras. Then: (a) Ker(h) is a normal filter of A, (b) A/Ker(h) = B. Proposition ([8]) If H is a normal filter of A, then there is a bijection between the ultrafilters of A containing H and the ultrafilters of A/H. 3. Bipartite pseudo-bl algebras Definition 3.1. ([8]) A is called bipartite if A = F F = F F for some ultrafilter F. Define the class BP as follows: A BP A is bipartite. Let us denote by sup(a) the set {x x : x A} {x x : x A}. Proposition 3.2. ([8]) sup(a) = {x A : x x or x x }. Proposition 3.3. Let sup(a) be a proper filter. Then A BP. Proof. Suppose that sup(a) is a proper filter. By Proposition 2.9 there exists an ultrafilter F of A such that sup(a) F. From Proposition 2.8 we conclude that F is prime. Applying Propositions 2.11 and 2.12 we deduce that A BP. Proposition 3.4. A BP [sup(a)) A. Proof. : Assume that A BP and [sup(a)) = A. By Proposition 2.11, there exists an ultrafilter F of A such that x x, x x F for all x A. Then sup(a) F. Consequently, A = [sup(a)) F and hence A = F, a contradiction. : Suppose that [sup(a)) A. By Proposition 2.9, [sup(a)) can be extended to an ultrafilter F. From Proposition 2.11 we have A = F F = F F. Thus A BP. Proposition 3.5. If F = A {0} is an ultrafilter of A, then A is bipartite.

5 Bipartite pseudo-bl algebras 491 Proof. Let x A. Then x F or x = 0. If x = 0, then x = x = 1 F. By Proposition 2.11, A = F F = F F, and hence A is bipartite. Proposition 3.6. Any subalgebra of a bipartite pseudo-bl algebra is bipartite. Proof. Let A BP and suppose that B is a subalgebra of A. Let F be a proper filter of A satisfying the condition (d) of Proposition Then U = F B is a prime filter of B and supb U. By Propositions 2.11 and 2.12, U is an ultrafilter of B and B = U U = U U. Hence B is a bipartite pseudo-bl algebra. Proposition 3.7. Let A and A t (t T) be pseudo-bl algebras and A = t T A t. If A t0 is bipartite for some t 0 T, then A is bipartite. Proof. Let U t0 be a prime filter of A t0 such that sup(a t0 ) U t0. Let U = U t0 s =t 0 A s. It is obvious that U is a prime filter of A. For every x = (a t ) t T A, x x = (a t a t ) t T U and x x = (a t a t ) t T U. Therefore, A is bipartite. Corollary 3.8. Let A t (t T) be bipartite pseudo-bl algebras. Then A = t T A t is a bipartite pseudo-bl algebra. Proposition 3.9. A homomorphic image of a bipartite pseudo-bl algebra is not bipartite in general. Proof. Let A = A 1 A 2, where A 1 BP and A 2 / BP. We consider the projection map π 2 : A A 2. Obviously π 2 is a homomorphism from A onto A 2. From Proposition 3.7 we see that A is bipartite but, by assumption, A 2 is not bipartite. Corollary The class BP is not a variety. 4. Strongly bipartite pseudo-bl algebras We define the class BP 0 of pseudo-bl algebras as follows: A BP 0 iff A = F F = F F for any ultrafilter F of A. Algebras from the class BP 0 are called strongly bipartite. Of course, BP 0 BP. Proposition 4.1. The following conditions are equivalent: (a) (b) A is strongly bipartite, F Max(A) x A [x / F n N((x n ) F and (x n ) F)]. Proof. (a) (b). Let A BP 0 and let F be an ultrafilter. Suppose that x A F. By Proposition 2.11, x F and x F. Applying Propositions

6 492 A. Walendziak, M. Wojciechowska-Rysiawa 2.2 (b) and 2.3 (c) we have x (x n ) and x (x n ) for all n N. Then (x n ) F and (x n ) F. (b) (a). Let the condition (b) be satisfied and F be an ultrafilter of A. Suppose that x / F. Then (x n ) F and (x n ) F for n N. In particular, x F and x F. Thus the condition (c) of Proposition 2.11 holds. Consequently, A = F F = F F. Therefore, A is strongly bipartite. Proposition 4.2. ([8]) The following conditions are equivalent: (a) (b) A is strongly bipartite, sup(a) M(A), where M(A) = {F : F is an ultrafilter of A}. In [3], there were defined two sets: and U(A) := {x A : (x n ) x for all n N} V (A) := {x A : (x n ) x for all n N}. Proposition 4.3. ([3]) M(A) U(A) V (A). Proposition 4.4. U(A) V (A) sup(a). Proof. Let x U(A). Then (x n ) x for all n N. In particular, x x. By Proposition 3.2, x sup(a). Thus U(A) sup(a). Similarly, V (A) sup(a). From Propositions 4.3 and 4.4 we obtain Corrolary 4.5. M(A) sup(a). Theorem 4.6. The following are equivalent: (a) A BP 0, (b) sup(a) = U(A) = V (A) = M(A), (c) F Max(A) sup(a) F. Proof. (a) (b). We have U(A) sup(a) (by Proposition 4.4) M(A) (by Proposition 4.2) U(A) (by Proposition 4.3). Therefore sup(a) = U(A) = M(A). Similarly, sup(a) = V (A) = M(A). (b) (c). Obvious. (c) (a). Let (c) hold. Then sup(a) M(A) and hence, by Proposition 4.2, A is strongly bipartite. Proposition 4.7. Any subalgebra of strongly bipartite pseudo-bl algebra is strongly bipartite.

7 Bipartite pseudo-bl algebras 493 Proof. Let A be strongly bipartite and B be a subalgebra of A. Let F be an ultrafilter of B and F be the filter generated by F in A. Then F = {y A : y x for some x F } by Proposition 2.5. Suppose that 0 F. Hence 0 F. This contradicts the fact that F is proper. Then F is proper too. By Proposition 2.9, there is an ultrafilter U of A such that U F. It is easy to see that U B Fil(B) and U B F. Since F Max(B), it follows that U B = F. We obtain sup(b) sup(a) U, because A is strongly bipartite. As B is a subalgebra we have sup(b) B. Consequently, sup(b) U B = F. By Theorem 4.6, B is strongly bipartite. Proposition 4.8. The class BP 0 is closed under direct products. Proof. Let A = t T A t, and A t be bipartite for t T. Let F Max(A). Then there is t 0 T such that F = F t0 s =t 0 A s, where F t0 Max(A t0 ). Let x = (a t ) t T A. It is easily seen that x x = (a t a t ) t T F and x x = (a t a t ) t T F. Thus sup(a) F for each F Max(A), and therefore A BP 0 by Theorem 4.6. Proposition 4.9. Let A BP 0 and h : A B be a surjective homomorphism. Then B BP 0. Proof. Write H = Ker(h). By Proposition 2.13, H is a normal filter and B = A/H. From Proposition 2.14 it follows that every ultrafilter of A/H has a form F/H, where F is an ultrafilter of A containing H. We have sup(a/h) = {a/h (a/h) : a A} {a/h (a/h) : a A} = {a a /H : a A} {a a /H : a A} F/H, because sup(a) F. Consequently, B is strongly bipartite. Propositions yield Theorem The class BP 0 is a variety. Let B(A) denote the set of all complemented elements in the distributive lattice L(A) = (A,,, 0, 1) of a pseudo-bl algebra A. Proposition ([3]) The following are equivalent: (a) x B(A), (b) x x = 1, (c) x x = 1.

8 494 A. Walendziak, M. Wojciechowska-Rysiawa Write M n (A) = {F : F is a normal ultrafilter of A}. Recall that A is called semisimple iff M n (A) = {1}. Proposition Let A be a semisimple pseudo-bl algebra. Then A is strongly bipartite if and only if A = B(A). Proof. Let A BP 0. Then sup(a) M(A). It is easily seen that M(A) M n (A). Since A is semisimple, M n (A) = {1}. Consequently, sup(a) = {1}. Hence x x = 1 for all x A and by Proposition 4.11, B(A) = A. Assume now that x x = 1 for all x A. Therefore sup(a) = {1}. Hence sup(a) F for all F Max(A). From Theorem 4.6 it follows that A is strongly bipartite. A pseudo-bl algebra A is called good if it satisfies the following condition: (a ) = (a ) for all a A. We say that A is local if it has a unique ultrafilter. The order of a A, in symbols ord(a), is the smallest natural number n such that a n = 0. If no such n exists, then ord(a) =. A good pseudo-bl algebra A is called perfect if it is local and for any a A, Following [8], we define two sets: ord(a) < ord(a ) =. D(A) = {a A : ord(a) = } and D(A) = {a A : ord(a) < }. It is obvious that D(A) D(A) = and D(A) D(A) = A. Proposition ([8]) The following conditions are equivalent: (a) A is local; (b) D(A) is the unique ultrafilter of A. Proposition ([8]) Let A be a local good pseudo-bl algebra. The following are equivalent: (a) A is perfect, (b) D(A) = D(A) = D(A). Proposition Every perfect pseudo-bl algebra is strongly bipartite. Proof. Let A be perfect. Then it is local, and so, by Proposition 4.13, D(A) is the unique ultrafilter of A. We have A = D(A) D(A) and from Proposition 4.14 it follows that D(A) = D(A) = D(A). Consequently, A is strongly bipartite. Example ([13]) Let a, b, c, d R, where R is the set of all real numbers. We put by definition (a, b) (c, d) a < c or (a = c and b d).

9 Bipartite pseudo-bl algebras 495 For any x, y R, we define operations and as follows: x y = min{x, y} and x y = max{x, y}. The meet and the join are defined on R R component-wise. Let {( ) } 1 A = 2, b : b 0 {(a, b) : 1 < a < 1, b R} {(1, b) : b 0}. 2 For any (a, b), (c, d) A, we put: ( ) 1 (a, b) (c, d) = 2, 0 (ac, bc + d), ( ) [( 1 c (a, b) (c, d) = 2, 0 a, d b ) ] (1, 0), a ( ) [( ) ] 1 c (a, b) (c, d) = 2, 0 ad bc, (1, 0). a a Then (A,,,,,, ( 1 2, 0), (1, 0)) is a pseudo-bl algebra. Let (a, b) A. We have ( ) ( ) [( ) ] (a, b) = (a, b) 2, 0 = 2, 0 2a, b (1, 0) a and (a, b) = (a, b) It is easy to see that ( ) ( ) [( , 0 = 2, 0 2a, b ) ] (1, 0). 2a ((a, b) ) = (a, b) = ((a, b) ). Then A satisfies condition (pdn) and hence A is a good pseudo-bl algebra. (Moreover, A is a pseudo-mv algebra.) Let F = {(1, b) : b 0}. In [13], we proved that F is the unique ultrafilter of A. Consequently, A is local. Since F is normal (see [13]), we have M n (A) = {F } {(1, 0)}, and therefore A is not semisimple. Now we show that condition (c) of Proposition 2.11 is not satisfied. Indeed, let x = ( 3 4, 1). Then x / F and x = ( ) [( ) ] ( ) , 0 3, 4 (1, 0) = 3 3, 4 / F. 3 Therefore, A is not bipartite and obviously it is not strongly bipartite. Define {( ) } 1 B = 2, b R 2 : b 0 {(1, b) R 2 : b 0}. It is easy to see that (B,,,,,, ( 1 2, 0), (1, 0)) is a subalgebra of A. The subset F is also the unique ultrafilter of B and hence B is local. Now we

10 496 A. Walendziak, M. Wojciechowska-Rysiawa prove that B is perfect. By Proposition 4.13, D(B) = F. Let x = (a, b) B. We have ord(x) < x B F x F ord(x ) =. Thus B is perfect. From Proposition 4.15 it follows that B is strongly bipartite. Since A is not strongly bipartite, we see that A is not perfect by Proposition Acknowledgments. The authors are highly grateful to referee for her/his remarks and suggestions for improving the paper. References [1] C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), [2] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: Part I, Multiple- Valued Logic 8 (2002), [3] A. Di Nola, G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: Part II, Multiple- Valued Logic 8 (2002), [4] A. Di Nola, F. Liguori, S. Sessa, Using maximal ideals in the classification of MV algebras, Portugal. Math. 50 (1993), [5] G. Dymek, Bipartite pseudo-mv algebras, Discuss. Math., General Algebra and Applications 26 (2006), [6] G. Georgescu, A. Iorgulescu, Pseudo-MV algebras: a noncommutative extension of MV algebras, The Proceedings of the Fourth International Symposium on Economic Informatics, Bucharest, Romania, May 1999, [7] G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: a noncommutative extension of BL algebras, Abstracts of the Fifth International Conference FSTA 2000, Slovakia 2000, [8] G. Georgescu, L. Leuştean, Some classes of pseudo-bl algebras, J. Austral. Math. Soc. 73 (2002), [9] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Amsterdam, [10] P. Hájek, Fuzzy logics with noncommutative conjuctions, J. Logic Comput. 13 (2003), [11] P. Hájek, Observations on non-commutative fuzzy logic, Soft Computing 8 (2003), [12] J. Rachůnek, A non-commutative generalizations of MV algebras, Math. Slovaca 52 (2002), [13] A. Walendziak, M. Wojciechowska, Semisimple and semilocal pseudo-bl algebras, Demonstratio Math. 42 (2009), Andrzej Walendziak WARSAW SCHOOL OF INFORMATION TECHNOLOGY Newelska 6 PL WARSZAWA, POLAND walent@interia.pl Magdalena Wojciechowska-Rysiawa UNIVERSITY OF PODLASIE 3Maja54 PL SIEDLCE, POLAND magdawojciechowska6@wp.pl Received March 23, 2009; revised version August 5, 2009.