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) operation satisfying ; herein referred to as i-ubands. Prominent (commutative) subvarieties of i-ubands include Boolean algebras, ortholattices, Kleene algebras, and involutive bisemilattices, hence i-ubands provides 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.
Cite
@article{arxiv.2503.10816,
title = {On the structure and theory of McCarthy algebras},
author = {Stefano Bonzio and Gavin St. John},
journal= {arXiv preprint arXiv:2503.10816},
year = {2025}
}
Comments
This version incorporates a proper citation to the 1990 article of Guzman and Squier, as well the addition Section 6