On Some Classes of Commutative Weak BCK-Algebras

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 by its weakening . 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.