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arxiv: 2606.28672 · v1 · pith:Z2OJ5S67new · submitted 2026-06-27 · 🧮 math.LO

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

Marcelo E. Coniglio , Kaique M. A. Roberto , Ana C. Golzio This is my paper

Pith reviewed 2026-06-30 08:58 UTC · model grok-4.3

classification 🧮 math.LO
keywords hyper swap structureshyper Boolean algebrasMorgado hyperlatticesLogics of Formal Inconsistencyparaconsistent logicsKalman functorhyperalgebras
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The pith

Hyper Boolean algebras based on Morgado hyperlattices generate hyper swap structures that characterize several logics of formal inconsistency.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines hyper Boolean algebras from Morgado hyperlattices and shows they generate hyper swap structures. These structures supply semantics for multiple paraconsistent logics inside the LFI hierarchy. Kalman-style functors are then built that produce categorical equivalences between the category of hyper Boolean algebras and the category of hyper algebras for each such logic, with the hyper swap structures serving as the representative objects on one side.

Core claim

Hyper Boolean algebras based on Morgado hyperlattices generate hyper swap structures whose semantics naturally characterize several paraconsistent logics from the hierarchy of Logics of Formal Inconsistency, and for each such logic a Kalman-style functor establishes a categorical equivalence between the hyper Boolean algebras and the hyper algebras for the logic.

What carries the argument

Hyper swap structures generated by hyper Boolean algebras, serving as the representative objects that mediate the Kalman-style categorical equivalences.

Load-bearing premise

The hyper swap structures obtained from the new hyper Boolean algebras correctly validate the axioms and rules of the targeted LFIs.

What would settle it

An explicit counter-model in one of the LFIs where a hyper swap structure over a hyper Boolean algebra fails to satisfy an axiom or inference rule of that logic.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces hyper Boolean algebras constructed from Morgado hyperlattices, proves basic properties of these algebras, and shows that hyper swap structures generated from them provide semantics for several logics in the LFI hierarchy. It further defines Kalman-style functors establishing categorical equivalences between the category of hyper Boolean algebras and categories of hyper algebras for the corresponding LFIs, with the hyper swap structures serving as representative objects.

Significance. If the constructions and proofs are correct, the work extends prior swap-structure semantics to the hyperalgebra setting for paraconsistent logics and supplies categorical equivalences that could support further developments in algebraic semantics for LFIs.

major comments (1)
  1. Abstract: the central claims (characterizations of LFIs via hyper swap structures and the Kalman-style categorical equivalences) are stated without any theorems, equations, proof sketches, or verification details, so soundness cannot be assessed from the provided material.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review and feedback on our manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: Abstract: the central claims (characterizations of LFIs via hyper swap structures and the Kalman-style categorical equivalences) are stated without any theorems, equations, proof sketches, or verification details, so soundness cannot be assessed from the provided material.

    Authors: Abstracts are conventionally limited to a high-level summary of contributions and do not contain theorems, equations or proof sketches. The full manuscript supplies these details: hyper Boolean algebras and their basic properties appear in Section 2; the characterizations of the relevant LFIs via hyper swap structures, including explicit constructions and soundness results, are developed in Section 3; the Kalman-style functors and the proofs of the categorical equivalences are given in Section 4. Soundness is therefore assessable from the complete text. If the editor wishes, we can expand the abstract with brief references to the main theorems. revision: partial

Circularity Check

0 steps flagged

Minor self-citation to prior definition of hyper swap structures; new definitions and proofs provide independent content.

full rationale

The paper cites its own prior work solely to introduce the notion of hyper swap structures, then defines hyper Boolean algebras from Morgado hyperlattices, proves basic properties, and establishes the LFI characterizations plus Kalman-style equivalences as new results. This matches the pattern of a single non-load-bearing self-citation (score 2) with no self-definitional reductions, fitted predictions, uniqueness theorems, or ansatzes smuggled via citation. The derivation chain is self-contained against the stated external benchmarks of LFIs and categorical equivalences.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The claims rest on newly introduced algebraic structures (hyper Boolean algebras) and functors whose properties are asserted without external benchmarks or independent evidence mentioned.

axioms (1)
  • domain assumption Morgado hyperlattices possess the structural properties needed to define hyper Boolean algebras that generate suitable hyper swap structures
    Invoked as the foundation for the new algebras in the abstract.
invented entities (2)
  • hyper Boolean algebras no independent evidence
    purpose: To generate hyper swap structures for LFI semantics
    Newly introduced concept based on Morgado hyperlattices.
  • hyper swap structures no independent evidence
    purpose: To provide semantics for LFIs
    Generalization introduced in authors' previous paper.

pith-pipeline@v0.9.1-grok · 5654 in / 1183 out tokens · 43772 ms · 2026-06-30T08:58:03.601773+00:00 · methodology

discussion (0)

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Reference graph

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