Implication Zroupoids and Birkhoff Systems

Abstract

An algebra $mathbf A = langle A, to, 0 rangle$, where $to$ is binary and $0$ is a constant, is called an implication zroupoid ($mathcal{I}$-zroupoid, for short) if $mathbf{A}$ satisfies the identities: $(x to y) to z approx [(z' to x) to (y to z)']'$, where $x' : = x to 0$, and $ 0'' approx 0$. These algebras generalize De Morgan algebras and $vee$-semilattices with zero. Let $mathcal{I}$ denote the variety of implication zroupoids. The investigations into the structure of $mathcal{I}$ and of the lattice of subvarieties of $mathcal{I}$, begun in 2012, have continued in several papers (see the Bibliography at the end of the paper). The present paper is a sequel to that series of papers and is devoted to making further contributions to the theory of implication zroupoids. The main purpose of this paper is to prove that if $mathbf{A}$ is an algebra in the variety $mathcal{I}$, then the derived algebra $mathbf{A}_{mj} := langle A; wedge, vee rangle$, where $a land b := (a to b')'$ and $a lor b := (a' land b')'$, satisfies the Birkhoff's identity (BR): $x land (x lor y) approx x lor (x land y)$. As a consequence, the implication zroupoids $mathbf A$ whose derived algebras $mathbf{A}_{mj}$ are Birkhoff systems are characterized. Another interesting consequence of the main result is that there are bisemigroups that are not bisemilattices but satisfy the Birkhoff's identity, which leads us naturally to define the variety of "Birkhoff bisemigroups'' as bisemigroups satisfying the Birkhoff identity, as a generalization of Birkhoff systems. The paper concludes with some open problems.