Boolean and central elements and Cantor-Bernstein theorem in bounded pseudo-BCK-algebras Jan K¨uhr Katedra algebry a geometrie Pˇr´ırodovˇedeck´a fakulta, Univerzita Palack´eho v Olomouci Tomkova 40, CZ-77900 Olomouc, Czech Republic E-mail:
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A pseudo-BCK-algebra is an algebra (A, →, axioms:
, 1) of type h2, 2, 0i satisfying the following
(x → y) ((y → z) (x → z)) = 1, (x y) → ((y z) → (x z)) = 1, 1→x=x=1 x, x → 1 = 1, x → y = 1 & y → x = 1 ⇒ x = y. Georgescu and Iorgulescu [3] introduced pseudo-BCK-algebras (in a slightly different way) as a non-commutative generalization of BCK-algebras, in the sense that if → = , then the algebra (A, →, 1) is a BCK-algebra. Pseudo-BCK-algebras relate to (non-commutative) residuated lattices as BCK-algebras do to commutative residuated lattices; specifically, by [6], pseudoBCK-algebras are just the h→, , 1i-subreducts of (non-commutative) integral residuated lattices. For every pseudo-BCK-algebra (A, →, , 1), the relation 6 defined by a 6 b if and only if a → b = 1 (or, equivalently, a b = 1) is a partial order on A such that 1 is the greatest element of A. By a bounded pseudo-BCK-algebra we mean an algebra (A, →, , 0, 1) where (A, →, , 1) is a pseudo-BCK-algebra with least element 0 (with respect to 6). Bounded pseudoBCK-algebras arise as the h→, , 0, 1i-subreducts of bounded integral residuated lattices. One of the most prominent examples of bounded pseudo-BCK-algebras are pseudo-MValgebras [2] (also called GMV-algebras [7]), which are term equivalent to bounded pseudoBCK-algebras that satisfy the identity (x y) → y = (y → x) x. The standard operations ⊕,− ,∼ are given by a ⊕ b := (a 0) → b = (b → 0) a, a− := a → 0, and a∼ := a 0. It is known that there is a one-one correspondence between direct product decompositions ϕ : A → A1 × A2 of a (pseudo-)MV-algebra A and those elements a ∈ A which have a complement in the underlying lattice of A. Likewise, these boolean elements coincide with the ⊕-idempotents and form a subalgebra of A which is a boolean algebra in its own right. Boolean elements are a basic tool used in Cantor-Bernstein theorems proved by De Simone, Mundici and Navara [1] for σ-complete MV-algebars and by Jakub´ık [5] for orthogonally σ-complete pseudo-MV-algebras. 1
The present paper has two parts. The first one is devoted to boolean and central elements (in a sense, this is a continuation of a recent paper by Gispert and Torrens [4] where the topic is studied within BCK-algebras). In the second part, a Cantor-Bernstein type theorem for certain bounded pseudo-BCK-algebras is established. Let (A, →, , 0, 1) be a bounded pseudo-BCK-algebra. We shall say that a ∈ A is a boolean element (in A) if a− = a∼ , a−∼ = a∼− and a ∨ a− = 1. The set of all boolean elements of A will be denoted by B(A). Let us agree to write a0 instead of a− = a∼ if a ∈ B(A). Theorem 1. Let A be a bounded pseudo-BCK-algebra. Then B(A) is the largest boolean subalgebra of A (here, a boolean subalgebra is a subalgebra satisfying the equations x → y = x y, (x → y) → y = (y → x) → x and x → (x → y) = x → y). If a ∈ B(A), then the mapping ϕ : x 7→ (a0 → x, a → x) = (a ∨ x, a0 ∨ x) is a subdirect embedding of A into the direct product [a, 1] × [a0 , 1] and the following are equivalent: (i) ϕ is an isomorphism; (ii) for every x ∈ [a, 1] and y ∈ [a0 , 1] there exists z ∈ A such that a0 → z = x and a → z = y; in this case, z = x ∧ y. We shall say that a ∈ A is a central element in A if a = ψ−1 (0, 1) or a = ψ−1 (1, 0) for some direct product decomposition ψ : A → A1 × A2 . The set of all central elements of A is denoted by C (A). Proposition 2. For every bounded pseudo-BCK-algebra A, C (A) is a subalgebra of B(A). In order for a boolean element a ∈ A to be central in A, it is necessary and sufficient that a satisfies either of the above conditions (i) and (ii). In general, C (A) and B(A) are distinct, but, for instance, if A satisfies the identities x− = x∼ and x−∼ = x, then B(A) = C (A). Proposition 3. Let A be a bounded pseudo-BCK-algebra, a ∈ C (A), b ∈ A and a 6 b. Then b ∈ C (A) if and only if b ∈ C ([a, 1]). In the following theorem of Cantor-Bernstein type (Theorem 4) we shall consider two versions of the condition (P), one of which is a generalization of orthogonal σ-completeness of pseudo-MV-algebras, the other one is a modification of σ-completeness of pseudo-MValgebras: Condition (P1) (P1a) A is orthogonally σ-complete, i.e., for every countable set {xi | i ∈ I} ⊆ A such that V xi ∨ x j = 1 for all i , j, there exists i∈I xi = inf A {xi | i ∈ I}. (P1b) For every countable set {ai | i ∈ I} of central elements of A such that ai ∨ a j = 1 for all W i , j, there exists i∈I a0i = supA {a0i | i ∈ I} belonging to C (A). Condition (P2) (P2a) C (A) is closed under countable suprema, i.e., whenever {ai | i ∈ I} is a countable set of W central elements, then i∈I ai exists and belongs to C (A). W (P2b) If {ai | i ∈ I} is a countable set of central elements with i∈I a0i = 1, then every set V {xi | i ∈ I} ⊆ A such that xi > ai for all i ∈ I has the infimum i∈I xi in A. 2
Theorem 4. Let A and B be bounded pseudo-BCK-algebras satisfying the condition (P). If A [b, 1] where b ∈ C (B), and B [a, 1] where a ∈ C (A), then A B. The proof of Theorem 4 splits into two lemmata. Let A be a bounded pseudo-BCK-algebra satisfying (P). We then have: Lemma 5. Given a sequence a0 , a1 , a2 , . . . of central elements of A with ai ∨ a j = 1 for all i , j, W let aω := n
