COMMENTATIONES MATHEMATICAE Vol. 51, No. 2 (2011), 121-124
Andrzej Walendziak, Magdalena Wojciechowska-Rysiawa
Some characterizations of pseudo-BL-chains
Abstract.Pseudo-BL-chains are linearly ordered pseudo-BL-algebras. Characteriza- tions of them in terms of concepts of lattice theory are given.
2000 Mathematics Subject Classification: 03G25, 06F05.
Key words and phrases: Pseudo-BL-algebra (-chain), Lattice, Join irreducible ele- ment, Stone lattice.
BL-algebras were introduced by H´ajek [8] in 1998. Chang [1] introduced MV- algebras, which are contained in the class of BL-algebras. A noncommutative exten- tion of MV-algebras, called pseudo-MV-algebras, were introduced by Georgescu and Iorgulescu [5] and independently by Rach˚unek [12]. A concept of pseudo-BL-algebras were firstly introduced by Georgescu and Iorgulescu in [6] as noncommutative gene- ralization of BL-algebras and pseudo-MV-algebras. In [2] and [3], there were given basic properties of pseudo-BL-algebras. The pseudo-BL-algebras correspond to a pseudo-basic fuzzy logic (see [9] and [10]).
K¨uhr [11] proved that a pseudo-BL-algebra A is linearly ordered if and only if every proper filter of A is prime. Dvureˇcenskij [4] showed that every pseudo-BL- chain admits a state and is good. In [13], Rach˚unek and ˇSalounova characterized pseudo-BL-chains by means of fuzzy filters. Recently, the second author defined in [14] anti fuzzy filters and using them gave a characterization of a pseudo-BL-chain.
In this paper, we characterize pseudo-BL-chains in terms of concepts of lattice theory.
Definition 1 Let A = (A; ∧, ∨, , →, ;, 0, 1) be an algebra of type (2, 2, 2, 2, 2, 0, 0). A is called a pseudo-BL-algebra if it satisfies the following axioms, for any a, b, c∈ A :
(C1) (A; ∧, ∨, 0, 1) is a bounded lattice, (C2) (A; , 1) is a monoid,
(C3) a b ¬ c ⇔ a ¬ b → c ⇔ b ¬ a ; c, (C4) a ∧ b = (a → b) a = a (a ; b),
122 Some characterizations of pseudo-BL-chains
(C5) (a → b) ∨ (b → a) = (a ; b) ∨ (b ; a) = 1.
Let A be a pseudo-BL-algebra. A is called trivial if 0 = 1. The following property holds in A and will be used in the sequel:
(1) a6 b ⇔ a → b = 1 ⇔ a ; b = 1.
For any pseudo-BL-algebra A, the reduct L(A) = (A; ∧, ∨, 0, 1) is a bounded distri- butive lattice. A pseudo-BL-chain is a pseudo-BL-algebra such that its lattice order is linear. The basic properties of pseudo-BL-algebras were studied in [2] and [3].
Now we recall some notions of lattice theory. Let L = (L; ∧, ∨) be a lattice. An element u ∈ L is called join irreducible if u = a ∨ b implies u = a or u = b. By J(L) we denote the set of all join irreducible elements of L. For any a ∈ L, we set:
(a] = {x ∈ L : x 6 a} and [a) = {x ∈ L : x > a}. The dual of L is the lattice L∂ with the same underlying set, but with a6 b in L∂ if and only if b6 a in L.
Let L = (L; ∧, ∨, 0) be a lattice with 0 and let a ∈ L. An element a∗ is a pseudocomplement of a if and only if a ∧ a∗ = 0 and a ∧ x = 0 implies that x 6 a∗. An element can have at most one pseudocomplement. L is said to be pseudocomplemented if each element of L has a pseudocomplement. A Stone lattice is a bounded distributive lattice L in which every element a has a pseudocomplement a∗ and the identity
a∗∨ a∗∗= 1
holds in L. If L∂ is a Stone lattice, then we say that L is a dual Stone lattice. L is a double Stone lattice if L and L∂ are Stone lattices.
Let L = (L; ∧, ∨, 0, 1) be a bounded lattice. By B(L) we denote the set of all complemented elements in L. For any pseudo-BL-algebra A, we put B(A) = B(L(A)).
Proposition 2 Every bounded chain is a double Stone lattice.
Proof Let L be a chain with 0 and 1. We set a∗= 0 for a ∈ L − {0} and 0∗= 1.
Obviously, a∗∨ a∗∗ = 1 for all a ∈ L. Then L is a Stone lattice. It is easy to see that L∂ is also bounded chain and therefore L∂ is a Stone lattice.
Proposition 3 Let L be a bounded distributive lattice. Then L is a Stone lattice if and only if for every a ∈ L there is b ∈ B(L) such that {x ∈ L : a ∧ x = 0} = (b].
Proof Let L be a Stone lattice and a ∈ L. Observe that a∗ ∈ B(L). Indeed, a∗∧ a∗∗= 0 and a∗∨ a∗∗= 1. Next we prove that Ia:= {x ∈ L : a ∧ x = 0} = (a∗].
By the definition of a pseudocomplement, a ∧ x = 0 implies that x 6 a∗. Therefore, Ia⊆ (a∗]. Now let x6 a∗. Then a∧x 6 a∧a∗= 0 and hence x ∈ Ia. Thus (a∗] ⊆ Ia and consequently, Ia = (a∗].
Conversely, let Ia = (b] for some b ∈ B(L). Hence a ∧ b = 0 and if a ∧ x = 0, then x 6 b. From this we have b = a∗. Therefore, every element of L has a
A. Walendziak, M. Wojciechowska-Rysiawa 123
pseudocomplement. Since b ∈ B(L), there is b0 ∈ L such that b ∧ b0 = 0 and b∨ b0= 1. Then b06 b∗and 1 = b ∨ b06 b ∨ b∗. It follows that b ∨ b∗= 1. But b = a∗ and hence a∗∨ a∗∗= 1. Consequently, L is a Stone lattice.
From the Duality Principle we have
Corollary 4Let L be a bounded distributive lattice. Then L is a dual Stone lattice if and only if for every a ∈ L there is b ∈ B(L) such that {x ∈ L : a ∨ x = 1} = [b).
A pseudo-BL-algebra A is called directly indecomposable if whenever A ∼= A1× A2, then eitherA1or A2is trivial.
Proposition 5 ([7])A pseudo-BL-algebra A is directly indecomposable if and only if B(A) = {0, 1}.
Proposition 6 ([7]) Any pseudo-BL-chain is directly indecomposable.
Theorem 7 Let A be a pseudo-BL-algebra. The following are equivalent:
(a) A is a pseudo-BL-chain;
(b) For every a, b ∈ A, a ∨ b = 1 implies a = 1 or b = 1;
(c) 1 ∈ J(L(A)).
Proof The implications (a) ⇒ (b) and (b) ⇒ (c) are obvious.
(c) ⇒ (a): Let a, b ∈ A. From (C5) we have (a → b) ∨ (b → a) = 1. By assumption, a → b = 1 or b → a = 1. Applying (1) we get a 6 b or b 6 a. Thus A
is a pseudo-BL-chain.
Theorem 8 Let A be a pseudo-BL-algebra. The following are equivalent:
(a) A is a pseudo-BL-chain;
(b) A is directly indecomposable and L(A) is a dual Stone lattice.
Proof (a) ⇒ (b): Follows from Propositions 2 and 6.
(b) ⇒ (a): Let a, b ∈ A such that a ∨ b = 1. By Corollary 4, {x ∈ L : a ∨ x = 1} = [e) for some e ∈ B(A). Applying Proposition 5 we get e = 0 or e = 1. Let e = 0. Hence 0 ∈ A = {x ∈ L : a ∨ x = 1} and therefore a = 1. If e = 1, then b∈ {x ∈ L : a ∨ x = 1} = {1}, so b = 1. By Theorem 7, A is a pseudo-BL-chain.
References
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124 Some characterizations of pseudo-BL-chains
[4] A. Dvureˇcenskij, Every linear pseudo BL-algebra admits a state, Soft Comput.11 (2007), 495-501.
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[13] J. Rach˚unek and D. ˇSalounova, Fuzzy filters and fuzzy prime filters of bounded R`-monoids and pseudo BL-algebras, Information Sciences178 (2008), 3474-3481.
[14] M. Wojciechowska-Rysiawa, Anti fuzzy filters of pseudo-BL-algebras, submitted.
Andrzej Walendziak
Warsaw School of Information Technology Newelska 6, PL-01447 Warszawa, Poland E-mail: walent@interia.pl
Magdalena Wojciechowska-Rysiawa
Institute of Mathematics and Physics, Siedlce University of Natural Sciences and Humanities 3 Maja 54, PL-08110 Siedlce, Poland
E-mail: magdawojciechowska6@wp.pl
(Received: 27.02.2011)
