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 ∨-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 ∈ I. For x, y ∈ A, let x ∧ y := (x → y')' and x ∨ y := (x' ∧ y')'. In an earlier paper, we had proved that if A ∈ I, then the algebra A_mj = ⟨ 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 every A ∈ I, the bisemigroup A_mj is 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.