  An algebra $A = \langle A, \to, 0 \rangle$, where $\to$ is binary and $0$ is
a constant, is called an implication zroupoid (I-zroupoid, for short) if 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 $\lor$-semilattices with zero. Let I denote the variety
of implication zroupoids. For details on the motivation leading to these
algebras, we refer the reader to [San12] (or the relevant papers mentioned at
the end of this paper). The investigations into the structure of the lattice of
subvarieties of I, begun in [San12], have continued in [CS16a, CS16b, CS17a,
CS17b, CS18a, CS18b, CS19] and [GSV19]. The present paper is a sequel to this
series of papers and is devoted to making further contributions to the theory
of implication zroupoids.
  The identity (BR): $x \land (x \lor y) \approx x \lor (x \land y)$ is called
the Birkhoff's identity. The main purpose of this paper is to prove that if A
is an algebra in the variety I, then the derived algebra $A_{mj} := \langle A;
\land, \lor \rangle$, where $a \land b := (a \to b')'$ and $a \lor b := (a'
\land b')'$, satisfies the Birkhoff's identity. As a consequence, we
characterize the implication zroupoids A whose derived algebras $A_{mj}$ are
Birkhoff systems. It also follows from the main result that there are
bisemigroups that are not bisemilattices but satisfy the Birkhoff's identity,
which suggests a more general notion, than Birkhoff systems, of "Birkhoff
bisemigroups" as bisemigroups satisfying the Birkhoff's identity. The paper
concludes with an open problem on Birkhoff bisemigroups.
