Boolean and Central Elements and Cantor-Bernstein Theorem in Bounded Pseudo-BCK-Algebras?

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 〈→, , 1〉-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 〈→, , 0, 1〉-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 Jakubı́k [5] for orthogonally σ-complete pseudo-MV-algebras.