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<p>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) <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\hbox {operation}^{~\prime }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>operation</mtext> <mrow> <mspace width="3.33333pt" /> <mo>′</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({x}^{\prime \prime }\approx x\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mo>″</mo> </msup> <mo>≈</mo> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>; 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.</p>

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On the structure and theory of McCarthy algebras

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月16日

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  • Stefano Bonzio,
  • Gavin St. John

摘要

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) \(\hbox {operation}^{~\prime }\) operation ′ satisfying \({x}^{\prime \prime }\approx x\) 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 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.