Remarks on skew Hilbert algebras and weak BCK*-algebras

Abstract Skew Hilbert algebras were recently introduced as a tool for a unified treatment of several ordered algebraic structures with implication important for mathematical logic. They were presented axiomatically as posets with implication, one of the axioms being stated in terms of lower and upper cones. In the paper, the important subclass of the so-called strong skew Hilbert algebras is shown to coincide with a class of weak BCK*-algebras that is axiomatized by simple quasi-equations. Several applications of skew Hilbert algebras are reconsidered in the context of weak BCK*-algebras, and the relevant classes of positive implicative and orthoimplicative weak BCK*-algebras are briefly reviewed.