Summary: In 2012, the second author [Sci. Math. Jpn. 75, No. 1, 21–50 (2012; Zbl 1279.06009)] introduced, and initiated the investigations into, the variety \(\mathcal{I}\) of implication zroupoids that generalize De Morgan algebras and \(\vee\)-semilattices with 0. An algebra \(\mathbf{A}=\langle A,\rightarrow,0\rangle\), where \(\rightarrow\) is binary and 0 is a constant, is called an implication zroupoid (\(\mathcal{I}\)-zroupoid, for short) if A satisfies: \((x\rightarrow y)\rightarrow z\approx [(z'\rightarrow x)\rightarrow (y\rightarrow z)']'\), where \(x':=x\rightarrow 0\), 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 \(x\wedge y:=(x\rightarrow y')'\) and \(x\vee y:=((x'\wedge y')'\). In an earlier paper, we had proved that if \(\mathbf{A}\in\mathcal{I}\), then the algebra \(\mathbf{A}_{mj}=\langle A,\vee,\wedge\rangle\) 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: \(x\wedge y \approx x\vee y\), satisfies the Whitman Property. We conclude the paper with two open problems.