Semidistributivity and Whitman Property in implication zroupoids

Abstract

In 2012, the second author introduced, and initiated the investigations into, the variety I of implication zroupoids that generalize De Morgan algebras and V-semilattices with 0. An algebra A = {A, →, 0}, where → is binary and 0 is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies: (x → y) → z ≈ [(z′ → x) → (y → z)′]′, where x′:= x → 0, and 0″ ≈ 0. Let I denote the variety of implication zroupoids and A ϵ I. For x, y ϵ A, let x Λ y:= (x → y′)′ and x V y:= (x′ Λ y′)′. In an earlier paper, we had proved that if A ϵ I, then the algebra Amj = {A, V, Λ} is a bisemigroup. The purpose of this paper is two-fold: First, we generalize the notion of semidistributivity from lattices to bisemigroups and prove that, for every A ϵ I, the bisemigroup Amj is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety M ϵJ of I, defined by the identity: x Λ y ≈ x V y, satisfies the Whitman Property. We conclude the paper with two open problems.