Semidistributivity and Whitman Property in implication zroupoids

Abstract

AbstractIn 2012, the second author introduced, and initiated the investigations into, the variety 𝓘 of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. An algebraA= 〈A, →, 0 〉, where → is binary and 0 is a constant, is called animplication zroupoid(𝓘-zroupoid, for short) ifAsatisfies: (x→y) →z≈ [(z′ →x) → (y→z)′]′, wherex′ :=x→ 0, and 0″ ≈ 0. Let 𝓘 denote the variety of implication zroupoids andA∈ 𝓘. Forx,y∈A, letx∧y:= (x→y′)′ andx∨y:= (x′ ∧y′)′. In an earlier paper, we had proved that ifA∈ 𝓘, then the algebraAmj= 〈A, ∨, ∧〉 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 everyA∈ 𝓘, the bisemigroupAmjis semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety 𝓜𝓔𝓙 of 𝓘, defined by the identity:x∧y≈x∨y, satisfies the Whitman Property. We conclude the paper with two open problems.