Hilbert Implication Algebra and Some Properties

Dejen Gerima Tefera

doi:10.1155/2021/3635185

Abstract

The concepts of Hilbert implication algebra and generalized Hilbert implication algebra are introduced. The comparison theorem of Hilbert implication algebra and generalized Hilbert implication algebra is proved. In addition, the idea of groupoid and commutative Hilbert implication algebras is investigated. Ideals and filters in Hilbert implication algebras are also discussed. In general, different theorems which show different properties are proved.

Highlights

Hilbert algebras are important in investigating certain algebraic logic, since they can be considered as a fragment of any propositional logic containing logical connective implications
The concepts of preimplication algebra and implication algebra based on orthosemilattice which generalize the concepts of implication algebra was discussed by Chajda in [3], and the notion of generalized Hilbert algebra with some other properties was investigated by Borzooei and Shohani in [4]
We introduced the concepts of Hilbert implication algebra and generalized Hilbert implication algebras are investigated

Summary

Introduction

Hilbert algebras are important in investigating certain algebraic logic, since they can be considered as a fragment of any propositional logic containing logical connective implications. The concept of a ⊗ −closed set and a ⊗ − homomorphism in lattice implication algebras and some of their properties is elaborated by Roh et al in [1]. The concepts of preimplication algebra and implication algebra based on orthosemilattice which generalize the concepts of implication algebra was discussed by Chajda in [3], and the notion of generalized Hilbert algebra with some other properties was investigated by Borzooei and Shohani in [4]. The concepts of Pseudo - Supplemented almost distributive fuzzy Lattice with some other propertis is introduced by Gerima Tefera in [6]. We introduced the concepts of Hilbert implication algebra and generalized Hilbert implication algebras are investigated. Groupoids in an implication algebra, commutative properties in a Hilbert implication algebra with some other properties are introduced. Throughout this paper, H represents Hilbert implication algebra unless otherwise mentioned

Preliminaries

Main Results

Conclusion