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Juan Climent Vidal

Publications and source records attributed to Juan Climent Vidal.

9 recordsLinked to original sources
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

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↗
arXiv2025

Lallement functor is a weak right multiadjoint

For a plural signature $Σ$ and with regard to the category $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$, of naturally preordered idempotent $Σ$-algebras and surjective homomorphisms, we define a contravariant functor $\mathrm{Lsys}_Σ$ from $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ to $\mathsf{Cat}$, the category of categories, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ the category $\mathbf{I}$-$\mathsf{LAlg}(Σ)$, of $\mathbf{I}$-semi-inductive Lallement systems of $Σ$-algebras, and a covariant functor $(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \cdot)$ from $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ to $\mathsf{Cat}$, that assigns to $\mathbf{I}$ in $\mathsf{NPIAlg}(Σ)_{\mathsf{s}}$ the category $(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \mathbf{I})$, of the coverings of $\mathbf{I}$, i.e., the ordered pairs $(\mathbf{A},f)$ in which $\mathbf{A}$ is a $Σ$-algebra and $f\colon \mathbf{A}\longrightarrow \mathbf{I}$ a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories $\int^{\mathsf{NPIAlg}(Σ)_{\mathsf{s}}}\mathrm{Lsys}_Σ$ and $\int_{\mathsf{NPIAlg}(Σ)_{\mathsf{s}}}(\mathsf{Alg}(Σ)\,{\downarrow_{\mathsf{s}}}\, \cdot)$; define a functor $\mathfrak{L}_Σ$ from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor.

math.CT↗
arXiv2023

Płonka Adjunction

For a signature $Σ$ and its subsignature $Σ^{\neq 0}$ without $0$-ary operation symbols, we prove (1) that there are strong Lawvere adjoint cylinders between the category $\mathsf{Ssl}$, of sup-semilattices, and the categories $\int^{\mathsf{Ssl}}\mathrm{Isys}_Σ$, of sup-semilattice inductive systems of $Σ$-algebras, and $\int^{\mathsf{Ssl}}\mathrm{Isys}_{Σ^{\neq 0}}$, of sup-semilattice inductive systems of $Σ^{\neq 0}$-algebras; (2) that there exists an adjunction between $\mathsf{Ssl}$ and the category $\mathsf{Alg}(Σ^{\neq 0})$, of $Σ^{\neq 0}$-algebras; (3) that there exists an adjunction between the categories $\mathsf{Ssl}$ and $\mathsf{Lnb}$, the category of left normal bands; (4) after defining and stating several technical results on the category $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$, of Płonka $Σ^{\neq 0}$-algebras, and defining functors $J_{Σ^{\neq 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to $\mathsf{Alg}(Σ^{\neq 0})\otimes\mathsf{Lnb}$, the tensor product of $\mathsf{Alg}(Σ^{\neq 0})$ and $\mathsf{Lnb}$, and $P_{Σ^{\neq 0}}$ from $\mathsf{Alg}(Σ^{\neq 0})\otimes\mathsf{Lnb}$ to $\mathsf{Alg}(Σ^{\neq 0})$, we prove that $P_{Σ^{\neq 0}}\circ J_{Σ^{\neq 0}}$ has a left adjoint; finally, (5) after defining a functor $\mathrm{Is}_{Σ^{\neq 0}}$ from $\mbox{\sffamily{\upshape{PłAlg}}}(Σ^{\neq 0})$ to $\int^{\mathsf{Ssl}}\mathrm{Isys}_{Σ^{\neq 0}}$ we prove the main result of this paper: that $\mathrm{Is}_{Σ^{\neq 0}}$ has a left adjoint $\mbox{\upshape{Pł}}_{Σ^{\neq 0}}$, which is the Płonka sum.

math.CT↗
arXiv2017

A characterization of the $n$-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem

A theorem of single-sorted algebra states that, for a closure space $(A,J)$ and a natural number $n$, the closure operator $J$ on the set $A$ is $n$-ary if, and only if, there exists a single-sorted signature $Σ$ and a $Σ$-algebra $\mathbf{A}$ such that every operation of $\mathbf{A}$ is of an arity $\leq n$ and $J = \mathrm{Sg}_{\mathbf{A}}$, where $\mathrm{Sg}_{\mathbf{A}}$ is the subalgebra generating operator on $A$ determined by $\mathbf{A}$. On the other hand, a theorem of Tarski asserts that if $J$ is an $n$-ary closure operator on a set $A$ with $n\geq 2$, and if $i<j$ with $i$, $j\in \mathrm{IrB}(A,J)$, where $\mathrm{IrB}(A,J)$ is the set of all natural numbers $n$ such that $(A,J)$ has an irredundant basis ($\equiv$ minimal generating set) of $n$ elements, such that $\{i+1,\ldots, j-1\}\cap \mathrm{IrB}(A,J) = \varnothing$, then $j-i\leq n-1$. In this article we state and prove the many-sorted counterparts of the above theorems. But, we remark, regarding the first one under an additional condition: the uniformity of the many-sorted closure operator.

math.LO↗
arXiv2017

When are profinite many-sorted algebras retracts of ultraproducts of finite many-sorted algebras?

For a set of sorts $S$ and an $S$-sorted signature $Σ$ we prove that a profinite $Σ$-algebra, i.e., a projective limit of a projective system of finite $Σ$-algebras, is a retract of an ultraproduct of finite $Σ$-algebras if the family consisting of the finite $Σ$-algebras underlying the projective system is with constant support. In addition, we provide a categorial rendering of the above result. Specifically, after obtaining a category where the objects are the pairs formed by a nonempty upward directed preordered set and by an ultrafilter containing the filter of the final sections of it, we show that there exists a functor from the just mentioned category whose object mapping assigns to an object a natural transformation which is a retraction.

math.CT↗
arXiv2016

Eilenberg theorems for many-sorted formations

A theorem of Eilenberg establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. In this article after defining, for a fixed set of sorts $S$ and a fixed $S$-sorted signature $Σ$, the concepts of formation of congruences with respect to $Σ$ and of formation of $Σ$-algebras, we prove that the algebraic lattices of all $Σ$-congruence formations and of all $Σ$-algebra formations are isomorphic, which is an Eilenberg's type theorem. Moreover, under a suitable condition on the free $Σ$-algebras and after defining the concepts of formation of congruences of finite index with respect to $Σ$, of formation of finite $Σ$-algebras, and of formation of regular languages with respect to $Σ$, we prove that the algebraic lattices of all $Σ$-finite index congruence formations, of all $Σ$-finite algebra formations, and of all $Σ$-regular language formations are isomorphic, which is also an Eilenberg's type theorem.

cs.FL↗
arXiv2012

On the Morphisms and Transformations of Tsuyoshi Fujiwara (as a concretion of a bidimensional many-sorted general algebra and its application to the equivalence between many-sorted clones and algebraic theories)

For single-sorted algebras, Fujiwara defined, through the concept of family of basic mapping-formulas, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras and an equivalence relation, the conjugation, on the families of basic mapping-formulas, which corresponds to the relation of inner isomorphism for algebras. In this paper we extend the theory of Fujiwara about morphisms to the many-sorted algebras, by defining the concept of polyderivor between many-sorted signatures, which assigns to basic sorts, words and to formal operations, families of derived terms, and under which the standard signature morphisms, the basic mapping-formulas of Fujiwara, and the derivors of Goguen-Thatcher-Wagner are subsumed. Then, by means of the homomorphisms between Bénabou algebras, which are the algebraic counterpart of the finitary many-sorted algebraic theories of Bénabou, we define the composition of polyderivors from which we get a corresponding category, isomorphic to the category of Kleisli for a monad on the standard category of many-sorted signatures. Next, by defining the notion of transformation between polyderivors, we endow the category of many-sorted signatures and polyderivors with a structure of 2-category. From this we get a derived 2-category of many-sorted specifications in which we prove the equivalence of the many-sorted specifications of Hall and Bénabou, and deduce the equivalence of the categories of Hall and Bénabou algebras. Besides, by defining corresponding categories of generalized many-sorted terms, we prove that the realization of these terms in the many-sorted algebras is invariant under polyderivors and compatible with the transformations between polyderivors, and from this we get an example, among others, of the new concept of 2-institution, itself an strict generalization of that of institution by Goguen and Burstall.

math.CT↗