On Some Classes of Commutative Weak BCK-Algebras

Abstract

Formally, a description of weak BCK-algebras can be obtained by replacing (in the standard axiom set by K. Iseki and S. Tanaka) the first BCK axiom $${(x - y) - (x - z) \le z - y}$$(x-y)-(x-z)≤z-y by its weakening $${z \le y \Rightarrow x - y \le x - z}$$z≤y?x-y≤x-z . It is known that every weak BCK-algebra is completely determined by the structure of its initial segments (sections). We consider weak BCK-algebras with De Morgan complemented, orthocomplemented and orthomodular sections, as well as those where sections satisfy a certain compatibility condition, and characterize each of these classes of algebras by an equation or quasi-equation. For instance, those weak BCK-algebras in which all initial segments are De Morgan complemented are just commutative weak BCK-algebras.