On the structure and theory of McCarthy algebras

TL;DR

The paper develops a comprehensive algebraic framework for McCarthy algebras by introducing i-ubands (unital bands with involution) and MK-algebras, showing McCarthy algebras are generated by the 3-element algebra M3 and are fully axiomatized by Konikowska’s postulates. It establishes a reduced and equivalent axiomatization, proves a semilattice decomposition of McCarthy algebras into a direct system of Boolean fibers, and provides a representation of algebras as decorated posets that uniquely determine the algebra. The authors also analyze weak/distributive/absorption properties, compare non-commutative McCarthy algebras with Kleene logics, and explore the subvariety lattice of i-ubands, including how Boolean algebras arise as a subvariety and the existence of op-variants. Collectively, the work offers structural, representational, and axiomatic tools for a wide class of non-classical logics within a unified algebraic setting, with potential applications to computation and logic semantics.

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 $x''\approx x$; herein referred to as i-ubands. Prominent (commutative) subvarieties of i-ubands include Boolean algebras, ortholattices, Kleene algebras, and involutive bisemilattices, hence i-ubands 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.

Figures (9)

• Figure 1: The tables of the 3-element algebra $\mathbf{M_3} = \langle \{0,1,\varepsilon\}, +,\cdot,{}', 0,1 \rangle$. The operations $\cdot,+$ denote McCarthy conjunction and disjunction, respectively.
• Figure 2: The four idempotent monoids of cardinality 3, up to isomorphism.
• Figure 3: The ten non-isomorphic i- ubands of cardinality 3.
• Figure 4: The four subclassical i- ubands of cardinality 3 over the set $X=\{1,0,\varepsilon \}$ and signature $\langle \cdot,{}',1 \rangle$.
• Figure 5: The Hasse diagrams (left-right) for posets $\langle \mathbf{WK},\leq_{\vee_{\mathsf{wk}}} \rangle$, $\langle \mathbf{SK},\leq_{\vee_{\mathsf{sk}}} \rangle$, and $\langle \mathbf{M_3},\leq_{+_{\mathsf{m}}} \rangle$ ($\cong \langle \mathbf{M_3}^\mathsf{op},\leq_{+_{\mathsf{m}}} \rangle$).

Theorems & Definitions (133)

• Remark 2.1
• Proposition 2.2
• proof
• Proposition 3.1: De Morgan Dual Equivalence
• Definition 3.2
• Remark 3.3
• Remark 3.4
• Proposition 3.5
• proof
• Corollary 3.6