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

Abstract In a previous paper, we recast Morgado hyperlattices and Sette implicative hyperlattices (IHLs) 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_\omega $ 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_\omega $, induce a Kalman-style functor between the category of Sette IHLs and the category of enriched hyperalgebras for $C_\omega $. Specifically, we exhibit an equivalence of categories between Sette IHLs and their enriched hyperalgebraic counterparts using Kalman and forgetful functors. Similar results are extended to two axiomatic extensions of $C_\omega $.