Hyper Swap Structures: The Case Study of LFIs and Hyper Boolean Algebras

Abstract: In a previous paper, we introduced the notion of hyper swap structures, a novel class of hyperalgebras that naturally generalizes swap structures semantics. In this paper we introduce the concept of hyper Boolean algebras based on Morgado hyperlattices, proving some basic properties. From this, we show that several paraconsistent logics in the hierarchy of Logics of Formal Inconsistency (LFIs) can be naturally characterized in terms of hyper swap structures semantics generated by hyper Boolean algebras. Finally, for each of these LFIs we obtain a Kalman-style functor which establishes an equivalence between the category of hyper Boolean algebras and a category of hyper algebras for the corresponding LFI having the hyper swap structures as representative objects.

• The paper introduces hyper swap structures and hyper Boolean algebras as a categorical foundation for LFIs, overcoming the limitations of traditional matrix semantics.
• It establishes categorical equivalence between enriched hyper mbC algebras and HBAs through functorial constructions, ensuring semantic robustness and precision.
• The study extends the framework to a complete LFI hierarchy, providing modular completeness proofs and unifying diverse semantic approaches.

Hyper Swap Structures, Logics of Formal Inconsistency, and Hyper Boolean Algebras

Introduction and Context

The paper "Hyper Swap Structures: The Case Study of LFIs and Hyper Boolean Algebras" (2606.28672) provides a categorical and algebraic foundation for the semantics of Logics of Formal Inconsistency (LFIs) using hyperalgebras, specifically hyper Boolean algebras (HBAs), and generalizes swap structure semantics for non-deterministic logics. The main contribution is a detailed development of hyper swap structures and the introduction of HBAs, together with rigorous categorical equivalence results connecting HBAs to semantic hyperalgebras for several LFIs.

The investigation is situated within the abstract algebraic logic (AAL) tradition, focusing on the limitations of standard algebraic techniques for non-deterministic logics, particularly those that are not algebraizable by finite logical matrices or standard twist-structure methods. The central theme is the extension of categorical dualities and representation results that are well-known in deterministic settings to the non-deterministic—and specifically paraconsistent—domain.

Background: From Twist to Hyper Swap Structures

Twist structure semantics, particularly via Kalman functors, underlie the algebraization of many non-classical logics, offering equivalences between certain lattices and enriched algebraic categories (e.g., centered Kleene, Nelson, or Heyting algebras). These constructions, however, are generally inapplicable to logics such as LFIs, which resist finite matrix semantics and standard truth-functionality due to their paraconsistency.

Swap structures, first introduced for LFIs, generalize the twist approach by organizing truth values as tuples subjected to non-deterministic hyperoperations, interpreted via Nmatrices. These structures accommodate the non-truth-functional, multi-valued behavior necessary for LFIs, supporting the semantics of connectives like paraconsistent negation and consistency operators.

The gap identified in prior work is that the Kalman-style functors, when applied in the hyperalgebraic setting (i.e., from a base algebra to a hyperalgebra), do not in general admit left adjoints or categorical equivalence, due to the mismatch between deterministic and non-deterministic operations and the "forgetfulness" not yielding an algebra. The present work overcomes this by replacing the base with a suitable hyperalgebraic category—hyper Boolean algebras—thus attaining a symmetric categorical structure analogous to that of twist constructions in deterministic settings.

Hyper Boolean Algebras: Definition and Properties

HBAs are introduced as a class of hyperalgebras based on bounded distributive hyperlattices, equipped with a hypernegation satisfying classical Boolean properties up to the similarity relation induced by the hyperlattice preorder. Specifically, the negation is involutive ($x \in --x$), satisfies the law of excluded middle ($x \curlyvee -x \equiv \top$), and the law of non-contradiction ($x \curlywedge -x \equiv \bot$); the similarity condition ensures that negation respects equivalence classes modulo the induced symmetric preorder.

HBAs generalize Boolean algebras to the hyperalgebraic, non-deterministic setting and operate as the "base" objects over which hyper swap structures are defined. The paper establishes foundational properties and characterizes the relationship between HBAs and Sette implicative hyperlattices, further showing that every HBA induces an implicative hyperlattice structure with implication defined as $a \multimap b = -a \curlyvee b$.

Hyper Swap Structures for LFIs

Hyper swap structures are constructed over a given HBA as triples of elements constrained by natural analogues of the LFI semantic conditions: $z_1 \curlyvee z_2 \equiv \top$ (excluded middle) and $z_1 \curlywedge z_2 \curlywedge z_3 \equiv \bot$ (gentle explosion law). The hyperoperators on these structures strictly generalize those found in earlier swap structure semantics: conjunction, disjunction, and implication are coordinate-wise derived from the base HBA, while the paraconsistent negation and consistency operations are implemented as coordinate swaps.

The soundness and completeness theorem is established: for the logic mbC, syntactic provability, semantic entailment over the class of all hyper mbC algebras (HmbCAs), and semantic entailment over the class of hyper swap structures all coincide. This result rigorously affirms the semantic adequacy and faithfulness of the hyper swap structure approach.

Categorical Equivalence via Kalman Functors

A central technical development is the categorical analysis of these semantics. The paper defines the functor $S^{\mathsf{mbC}}$ from the category of HBAs to a category of enriched hyper mbC algebras (EHmbCAs), and the corresponding "reduct" functor $U^{\mathsf{mbC}}$ in the opposite direction. By refining the notion of morphisms and enriching the target category with additional constraints (including stability and uniqueness up to similarity), the authors establish a categorical equivalence between HBAs and EHmbCAs, natural in both objects and morphisms.

This equivalence generalizes prior representation theorems in the twist structure tradition and provides a functorial account of the hyper swap construction. The categorical approach extends seamlessly to a hierarchy of LFIs: the paper demonstrates how each stepwise stronger system (e.g., mbCciw, mbCci, Ci, Cie, Cia) admits a corresponding subcategory of hyperalgebras and a refinement of the swap structure semantics, preserving the categorical duality.

Extension to the Entire LFI Hierarchy

By imposing additional algebraic conditions—mirroring the axiomatic extensions for various LFIs—on the hyper swap structures (e.g., relating the consistency operator to hyper-negation, or enforcing more stringent equivalence conditions), the framework is generalized to semantics for mbCciw, mbCci, Ci, Cie, and Cia.

Correspondingly, soundness and completeness theorems are provided for each logic, with each logic's class of hyperalgebras and associated hyper swap structures forming a categorical equivalence with HBAs. This unifies the semantics of a broad class of LFIs and demonstrates the modularity and adaptability of the hyperalgebraic framework.

Implications and Future Directions

The categorical equivalence and explicit construction of hyper swap structures over HBAs yield several implications:

• Theoretical Unification: The work extends duality-theoretic and algebraic logic methodologies to non-deterministic and paraconsistent logics, illustrating that such systems can be semantically characterized without resorting to purely matrix-based or non-structural approaches.
• Semantic Robustness: The functorial construction establishes that hyper swap semantics are not ad hoc, but canonically derived and preserved under categorical equivalence, bolstering their relevance for larger classes of non-classical logics.
• Modularity: The framework demonstrates that further extensions (as in Ivlev-like modal logics or LETs) may be accommodated once suitable base hyperalgebraic structures are specified, opening directions for new semantic analyses and completeness theorems.
• Algebraizability: The setting invites deeper exploration into order-algebraizability and the interface between hyperalgebraic approaches and more general frameworks such as Blok-Pigozzi algebraizability.
• Categorical Characterization: An open theoretical problem is posed: to find a purely axiomatic, non-constructive description of the categories of enriched hyperalgebras corresponding to these semantics—a question likely to have significant impact on further logical and categorical research.

Conclusion

This paper rigorously develops the theory of hyper swap structures and hyper Boolean algebras as a categorical and hyperalgebraic foundation for the semantics of LFIs. By bridging the gap between non-deterministic logic and categorical algebra, the results systematically generalize known twist structure semantics to the field of paraconsistency, establish soundness and completeness for a wide class of LFIs, and provide full categorical equivalence between their semantic and abstract algebraic incarnations. The methodological and conceptual advances set a robust agenda for further research in non-classical logic, algebraic logic, and categorical duality in non-deterministic settings.