Implication Zroupoids and Birkhoff Systems

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