On the structure and theory of McCarthy algebras

Abstract We provide a structural analysis for McCarthy algebras, the variety generated by the three-element algebra defining the logic of McCarthy (the non-commutative version of Kleene three-valued logics). Our analysis will be conducted in a very general algebraic setting by introducing McCarthy algebras as a subvariety of unital bands (idempotent monoids) equipped with an involutive (unary) $$\hbox {operation}^{~\prime }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>operation</mml:mtext> <mml:mrow> <mml:mspace/> <mml:mo>′</mml:mo> </mml:mrow> </mml:msup> </mml:math> satisfying $${x}^{\prime \prime }\approx x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo>″</mml:mo> </mml:msup> <mml:mo>≈</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> ; 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 main contributions consist in providing for McCarthy algebras: reduced and equivalent axiomatizations; a semilattice decomposition theorem; and representations as certain decorated posets from which the algebraic structure can be uniquely determined.