On pre-Hilbert and positive implicative pre-Hilbert algebras

Andrzej Walendziak

doi:10.18778/0138-0680.2024.07

Authors

Andrzej Walendziak Institute of Mathematics and Physics, Siedlce University Sieldlce, Poland https://orcid.org/0000-0002-9866-0781

DOI:

https://doi.org/10.18778/0138-0680.2024.07

Keywords:

Hilbert algebra, pre-Hilbert algebra, BCK-algebra, BCC-algebra, BE-algebra, positive implicativity

Abstract

In the paper, pre-Hilbert algebras are defined as a generalization of Hilbert algebras (namely, a Hilbert algebra is just a pre-Hilbert algebra satisfying the property of antisymmetry). Pre-Hilbert algebras have been inspired by Henkin's Positive Implicative Logic. Their properties and characterizations are investigated. Some important results and examples are given. Moreover, positive implicative pre-Hilbert algebras are introduced and studied, their connections with some algebras of logic are presented. The hierarchies existing between the classes of algebras considered here are shown.

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