Hyper Swap Structures: The Case Study of LFIs and Hyper Boolean Algebras
arXiv:2606.28672
The paper defines hyper Boolean algebras using Morgado hyperlattices, shows they provide semantics for several Logics of Formal Inconsistency via hyper swap structures, and constructs a Kalman‑style functor giving a categorical equivalence between hyper Boolean algebras and corresponding hyper 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.
21 pages. Minor typos and corrections were made