On implication in MV-algebras

Abstract

In [1], the authors introduced the notion of a weak implication algebra, which reflects properties of implication in MV-algebras, and demonstrated that the class of weak implication algebras is definitionally equivalent to the class of upper semilattices whose principal filters are compatible MV-algebras. It is easily seen that weak implication algebras are just duals of commutative BCK-algebras. We show here that most results of [1] are, in fact, immediate consequences of two well-known facts: (i) a bounded commutative BCKalgebra possesses a natural upper semilattice structure, (ii) the class of MV-algebras and that of bounded commutative BCK-algebras are definitionally equivalent.

Triple Representation Theorem for orthocomplete homogeneous effect algebras

“…Recall that Proposition 3.12 immediately yields (using the same considerations as in [13, Proposition 19]) that an orthocomplete generalized effect algebra of meager elements of a sharply dominating homogeneous effect algebra is a commutative BCK-algebra with the relative cancellation property. Hence, by the result of J. Cīrulis (see [4]) it is the dual of a weak implication algebra introduced in [2]. Proposition 3.13.…”