Semidistributivity and Whitman Property in Implication Zroupoids

JUAN MANUEL CORNEJO; HANAMANTAGOUDA P. SANKAPPANAVAR

MATHEMATICA SLOVACA

VERSITA

Lugar: Varsovia; Año: 2021 vol. 71 p. 1329 - 1338

0139-9918

In 2012, the second author introduced and initiated the investigations into the variety $mathcal{I}$ of implication zroupoids that generalize De Morgan algebras and $lor$-semilattices with $0$.An algebra $mathbf A = langle A, o, 0 angle$, where $o$ is binary and $0$ is a constant, is called an emph{implication zroupoid} ($mathcal{I}$-zroupoid, for short) if $mathbf A$ satisfies: $(x o y) o z approx [(z´ o x) o (y o z)´]´$, where $x´ : = x o 0$, and $ 0´´ approx 0$. Let $mathcal{I}$ denote the variety of implication zroupoids and $mathbf A in mathcal{I}$. For $x,y in mathbf A$, let $x land y := (x o y´)´$ and $x lor y := (x´ land y´)´$. In an earlier paper we had proved that if $mathbf A in mathcal{I}$, then the algebra $mathbf A_{mj} = langle A, lor, land angle$ 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 $mathbf A in mathcal{I}$, the bisemigroup $mathbf A_{mj}$ is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety $mathcal{MEJ}$ of $mathcal I$, defined by the identity: $x land y approx x lor y$, satisfies the Whitman Property. We conclude the paper with two open problems.