Relation Between Be-Algebras and G-Hilbert Algebras

Abstract

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.

Keywords

• (heyting
• implication
• (g-)hilbert) algebra
• be/ci-algebra
• dual (s/q/bck)-algebra
• gi-algebra
• implication groupoid
• pre-logic
• mv - algebra
• 06d20
• 06f35
• 03g25