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On Some Classes of Commutative Weak BCK-Algebras

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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 \({(x - y) - (x - z) \le z - y}\) by its weakening \({z \le y \Rightarrow x - y \le x - z}\) . 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.

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Correspondence to Jānis Cīrulis.

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Presented by Jacek Malinowski

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Cīrulis, J. On Some Classes of Commutative Weak BCK-Algebras. Stud Logica 103, 479–490 (2015). https://doi.org/10.1007/s11225-014-9575-y

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