On Liu algebras: a new composite structure of the BCL⁺ algebras and the semigroups

Abstract

This paper offers a new algebra, which is called the Liu algebra (which is named after author), because of its origin in BCL⁺ algebras, and connections between BCL⁺ algebras and semigroups, have more complex structures, or, saying a composite structure. While Liu algebras are dividing into two distinct parts that are structurally independent, we think there are good reasons to mash them up, can be enforced by algebraic operations on distributive laws. Here we introduce several new notions (i.e., ideal, funnel and deductive systems in Liu algebras). We show that if G and H be two algebras, if G ≅ H, then (L; *, •, 1) is an order isomorphism, and discuss some properties for Liu algebras.