Γ-Implication Algebras

This paper introduces the concepts of [Formula: see text]-Implication algebras (Hilbert algebras, KU algebras, JU algebras, BE algebras, and CI algebras) each serving as a generalization of their respective counterparts: Implication algebras, Hilbert algebras, KU-algebras, BE-algebras, and CI-algebras. It is established that every commutative, self-distributive [Formula: see text]-BE algebra is a [Formula: see text]-Hilbert algebra, and that any [Formula: see text]-Implication algebra is [Formula: see text]-BE algebras. Furthermore, every [Formula: see text]-Hilbert algebra is shown to be a self-distributive [Formula: see text]-BE algebra. We prove that [Formula: see text]-Implication algebras are [Formula: see text]-Hilbert algebras(KU-algebras, BE-algebras). We study some properties of [Formula: see text]-Implication algebras (Hilbert algebras, KU-algebras, JU-algebras, BE algebras, and CI algebras). We establish that every commutative and self-distributive [Formula: see text]-BE algebra is a [Formula: see text]-Hilbert algebra.