On the structure and theory of McCarthy algebras

We provide a structural analysis for McCarthy algebras, the variety generated by the three-element algebra defining the logic ofMcCarthy(thenon-commutativeversionof Kleene three-valued logics). Ouranalysis will beconductedinaverygeneralalgebraic setting by introducing McCarthy algebras as a subvariety of unital bands (idempotent monoids) equipped with an involutive (unary) operation ′ satisfying x′′ ≈ x; herein referred to as i-ubands. Prominent (commutative) subvarieties of i-ubands include Boolean algebras, ortholattices, Kleene algebras, and involutive bisemilattices, hence i-ubands provide an algebraic common ground for several non-classical logics. Our maincontributions consistinproviding forMcCarthyalgebras:reducedandequivalent axiomatizations; a semilattice decomposition theorem; and representations as certain decorated posets from which the algebraic structure can be uniquely determined.