On Implicative and Positive Implicative GE Algebras

Andrzej Walendziak

doi:10.18778/0138-0680.2023.21

Authors

Andrzej Walendziak Siedlce University of Natural Sciences and Humanities, Faculty of Exact and Natural Sciences, Institute of Mathematics Author https://orcid.org/0000-0002-9866-0781

DOI:

https://doi.org/10.18778/0138-0680.2023.21

Keywords:

GE algebra, tGE algebra, BCK algebra, Hilbert algebra, (positive) implicativity

Abstract

GE algebras (generalized exchange algebras), transitive GE algebras (tGE algebras, for short) and aGE algebras (that is, GE algebras
verifying the antisymmetry) are a generalization of Hilbert algebras. Here some properties and characterizations of these algebras are investigated. Connections between GE algebras and other classes of algebras of logic are studied. The implicative and positive implicative properties are discussed. It is shown that the class of positive implicative GE algebras (resp. the class of implicative aGE algebras) coincides with the class of generalized Tarski algebras (resp. the class of Tarski algebras). It is proved that for any aGE algebra the property of implicativity is equivalent to the commutative property. Moreover, several examples to illustrate the results are given. Finally, the interrelationships between some classes of implicative and positive implicative algebras are presented.

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