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In 2012, the second author introduced, and initiated the investigations into, the variety  of implication zroupoids that generalize De Morgan algebras and ∨-semilattices with 0. An algebra  A  = 〈  A , →, 0 〉, where → is binary and 0 is a constant, is called an  implication zroupoid  (-zroupoid, for short) if  A  satisfies: ( x  →  y ) →  z  ≈ [( z ′ →  x ) → ( y  →  z )′]′, where  x ′ :=  x  → 0, and 0″ ≈ 0. Let  denote the variety of implication zroupoids and  A  ∈ . For  x ,  y  ∈  A , let  x  ∧  y  := ( x  →  y ′)′ and  x  ∨  y  := ( x ′ ∧  y ′)′. In an earlier paper, we had proved that if  A  ∈ , then the algebra  A   mj   = 〈 A , ∨, ∧〉 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  A  ∈ , the bisemigroup  A   mj   is semidistributive. Secondly, we generalize the Whitman Property from lattices to bisemigroups and prove that the subvariety  of , defined by the identity:  x  ∧  y  ≈  x  ∨  y , satisfies the Whitman Property. We conclude the paper with two open problems.

Semidistributivity and Whitman Property in implication zroupoids