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Hyper swap structures and Kalman functors: the case study of da Costa logic $C_ω$

Marcelo E. Coniglio, Kaique Roberto, Ana Claudia Golzio

TL;DR

This work develops hyper swap structures, a hyperalgebraic generalization of Kalman twist structures, to semantically model the paraconsistent logic $C_ω$ and its extensions. It establishes a Kalman-style correspondence between the category of Sette implicative hyperlattices and enriched hyper $C_ω$ algebras via functors $\mathsf S$ and $\mathsf U$, proving an equivalence of categories that represents swap-structure semantics entirely within hyperalgebraic terms. The authors extend the framework to two axiomatic extensions, $C_{min}$ and $C_ω^+$, obtaining analogous equivalences (with $C_ω^+$ featuring deterministic negation in swap semantics) and showing that $C_ω^+$ cannot be captured by any finite Nmatrix. These results provide a unified, algebraic perspective on non-classical logics through hyperswap constructions and enriched hyperalgebras, enabling semantic transfers from base lattices to their enriched hypercounterparts. The work also outlines future directions, including intrinsic axiomatizations of enriched hyper algebras and applications to broader families of LFIs.

Abstract

In a previous paper, we recast Morgado hyperlattices and Sette implicative hyperlattices in lattice-theoretic terms. By utilizing swap structures induced by implicative lattices, we obtained a direct proof of soundness and completeness for da Costa's paraconsistent logic $C_ω$ with respect to Sette's hyperalgebraic semantics. Inspired by Kalman functors in the context of twist structures, we introduce the notion of hyper swap structures, a novel class of hyperalgebras that naturally generalize swap structure semantics. We prove that these hyperalgebras, besides providing another class of hyperalgebraic models for $C_ω$, induce a Kalman-style functor between the category of Sette implicative hyperlattices and the category of enriched hyperalgebras for $C_ω$. Specifically, we exhibit an equivalence of categories between Sette implicative hyperlattices and their enriched hyperalgebraic counterparts using Kalman and forgetful functors. Similar results are extended to two axiomatic extensions of $C_ω$.

Hyper swap structures and Kalman functors: the case study of da Costa logic $C_ω$

TL;DR

This work develops hyper swap structures, a hyperalgebraic generalization of Kalman twist structures, to semantically model the paraconsistent logic and its extensions. It establishes a Kalman-style correspondence between the category of Sette implicative hyperlattices and enriched hyper algebras via functors and , proving an equivalence of categories that represents swap-structure semantics entirely within hyperalgebraic terms. The authors extend the framework to two axiomatic extensions, and , obtaining analogous equivalences (with featuring deterministic negation in swap semantics) and showing that cannot be captured by any finite Nmatrix. These results provide a unified, algebraic perspective on non-classical logics through hyperswap constructions and enriched hyperalgebras, enabling semantic transfers from base lattices to their enriched hypercounterparts. The work also outlines future directions, including intrinsic axiomatizations of enriched hyper algebras and applications to broader families of LFIs.

Abstract

In a previous paper, we recast Morgado hyperlattices and Sette implicative hyperlattices in lattice-theoretic terms. By utilizing swap structures induced by implicative lattices, we obtained a direct proof of soundness and completeness for da Costa's paraconsistent logic with respect to Sette's hyperalgebraic semantics. Inspired by Kalman functors in the context of twist structures, we introduce the notion of hyper swap structures, a novel class of hyperalgebras that naturally generalize swap structure semantics. We prove that these hyperalgebras, besides providing another class of hyperalgebraic models for , induce a Kalman-style functor between the category of Sette implicative hyperlattices and the category of enriched hyperalgebras for . Specifically, we exhibit an equivalence of categories between Sette implicative hyperlattices and their enriched hyperalgebraic counterparts using Kalman and forgetful functors. Similar results are extended to two axiomatic extensions of .

Paper Structure

This paper contains 9 sections, 25 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Morgado hyperlattices and Sette hyperalgebras
  3. The logic $C_\omega$
  4. Hyper Swap structures for $C_\omega$
  5. An Equivalence of Categories between IHL and EHC$_\omega$A
  6. Extending the results to axiomatic extensions of $C_\omega$
  7. The logic $C_{min}$
  8. The logic $C_\omega^+$
  9. Final Remarks

Key Result

Proposition 2.8

Let $A,B,C\subseteq L$ be stable sets, and let $\#,\#' \in\{\curlywedge, \curlyvee\}$. Then, $A \# B$ is stable and, for all $a\in A$, $b\in B$ and $c\in C$:

Theorems & Definitions (71)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Prosets
  • Definition 2.5
  • Definition 2.6: Morgado hyperlattices, morgado1962introduccao
  • Definition 2.7: Stable sets, coniglio2020non
  • Proposition 2.8
  • Definition 2.9
  • Definition 2.10: Sette implicative hyperlattices, sette1971algebras
  • ...and 61 more