Implication Zroupoids and Birkhoff Systems

An algebra A=⟨A,→,0⟩, where → is binary and 0 is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies the identities: (x→y)→z≈((z ′→x)→(y→z)′)′, where x′:=x→0, and 0′′≈0. These algebras generalize De Morgan algebras and ∨-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∧(x∨y)≈x∨(x∧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 Amj:=⟨A;∧,∨⟩, where a∧b:=(a→b′)′ and a∨b:=(a′∧b′)′, satisfies the Birkhoff s identity. As a consequence, we characterize the implication zroupoids A whose derived algebras Amj 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.