Semidistributivity and Whitman Property in Implication Zroupoids

Abstract

In 2012, the second author introduced and studied the variety $\mathcal{I}$ of implication zroupoids that generalize De Morgan algebras and $\vee$-semilattices with $0$. An algebra $\mathbf{A}=⟨A,\to ,0⟩$, where $\to$ is binary and $0$ is a constant, is called an \emph{implication zroupoid} ($\mathcal{I}$-zroupoid, for short) if $\mathbf{A}$ satisfies: $(x\to y)\to z\approx [(z^{\prime }\to x)\to (y\to z{)}^{\prime }{]}^{\prime }$, where $x^{\prime }:=x\to 0$, and $0^{\prime \prime }\approx 0$. Let $\mathcal{I}$ denote the variety of implication zroupoids and $\mathbf{A}\in \mathcal{I}$. For $x,y\in \mathbf{A}$, let $x\wedge y:=(x\to y^{\prime }{)}^{\prime }$ and $x\vee y:=(x^{\prime }\wedge y^{\prime }{)}^{\prime }$. In an earlier paper we had proved that if $\mathbf{A}\in \mathcal{I}$, then the algebra ${\mathbf{A}}_{mj}=⟨A,\vee ,\wedge ⟩$ is a bisemigroup. In this paper we generalize the notion of semi-distributivity from lattices to bisemigroups and prove that, for every $\mathbf{A}\in \mathcal{I}$, the bisemigroup ${\mathbf{A}}_{mj}$ is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety $\mathcal{M}\mathcal{E}\mathcal{J}$ of $\mathcal{I}$, defined by the identity: $x\wedge y\approx x\vee y$, satisfies the Whitman Property.

11 pages