Raúl Ruiz Mora | Searcharxiv

arXiv2026

Riguet and Generalized Congruences on a Category: Relationships and Applications

We investigate Riguet congruences and generalized congruences on a category, focusing on their interrelations from both lattice-theoretic and category-theoretic perspectives. We also characterize functors that are full and surjective on objects in terms of regular epimorphisms, extremal epimorphisms and in terms of strong and regular generalized congruences. On the lattice-theoretic side, we prove that for a category $\mathsf{C}$, the set $\mathrm{RCgr}(\mathsf{C})$ of all Riguet congruences, ordered by inclusion, is a bounded directed-complete ordered set, while the set $\mathrm{GCgr}(\mathsf{C})$ of all generalized congruences is an algebraic lattice. We establish a bridge between these structures via a Scott continuous morphism. From a category-theoretic standpoint, we lift these results to relative adjunctions between the categories $\mathsf{RCgr}(\mathsf{C})$ and $\mathsf{GCgr}(\mathsf{C})$ associated to the above ordered sets, as well as between the categories $\mathsf{RCCat}$, of Riguet classified categories, and $\mathsf{GCCat}$, of generalized classified categories. Furthermore, within Manes' framework of categories of $\mathsf{K}$-objects with structure, we investigate the relationship between the wide subcategory $\mathsf{RCCat}_{\mathrm{full}}$ of $\mathsf{RCCat}$, whose morphisms are the full morphisms of $\mathsf{RCCat}$, and $\mathsf{GCCat}$, relating these constructions to the Grothendieck theory of fibrations. Finally, we present applications of Riguet congruences across various mathematical fields.

math.CT↗

arXiv2026

A Kleene theorem for free many-sorted algebras

In this work, we generalize Kleene's theorem from free single-sorted algebras to free many-sorted algebras. Our main result establishes that, under appropriate finitary assumptions, a language of a given sort in a free many-sorted algebra is recognizable if and only if it is regular.

cs.LO↗

arXiv2026

From higher-order rewriting systems to higher-order categorial algebras and higher-order Curry-Howard isomorphisms

This ongoing project aims to define and investigate, from the standpoint of category theory, order theory and universal algebra, the notions of higher-order many-sorted rewriting system and of higher-order many-sorted categorial algebra and their relationships, via the higher-order Curry-Howard isomorphisms. The ultimate goal, to be developed in future versions of this work, is to define and investigate the category of towers, whose objects will consist of families, indexed by $\mathbb{N}$, of higher-order many-sorted rewriting systems and of higher-order many-sorted categorial algebras, including higher-order Curry-Howard type results for the latter, together with an additional structure that intertwines such $\mathbb{N}$-families; and whose morphism from a tower to another will be families, indexed by $\mathbb{N}$, of morphisms between its higher-order many-sorted rewriting systems and of higher-order many-sorted categorial algebras compatible with their structures. All feedback is appreciated.

math.CT↗