Relation Between Be-Algebras and G-Hilbert Algebras

• Main Authors: Rezaei Akbar, Saeid Arsham Borumand
• Format: Article
• Language: English
• Published: Sciendo 2018-06-01
• Series: Discussiones Mathematicae - General Algebra and Applications
• Subjects: (heyting; implication; (g-)hilbert) algebra; be/ci-algebra; dual (s/q/bck)-algebra; gi-algebra; implication groupoid; pre-logic; mv - algebra; 06d20; 06f35; 03g25
• Online Access: https://doi.org/10.7151/dmgaa.1285

Hilbert algebras are important tools for certain investigations in algebraic logic since they can be considered as fragments of any propositional logic containing a logical connective implication and the constant 1 which is considered as the logical value “true” and as a generalization of this was defined the notion of g-Hilbert algebra. In this paper, we investigate the relationship between g-Hilbert algebras, gi-algebras, implication gruopoid and BE-algebras. In fact, we show that every g-Hilbert algebra is a self distributive BE-algebras and conversely. We show cannot remove the condition self distributivity. Therefore we show that any self distributive commutative BE-algebras is a gi-algebra and any gi-algebra is strong and transitive if and only if it is a commutative BE-algebra. We prove that the MV -algebra is equivalent to the bounded commutative BE-algebra.