arXiv subjects

Víctor Marín

Publications and source records attributed to Víctor Marín.

8 recordsLinked to original sources

arXiv2025

A P-theorem for Inverse Semigroupoids through Ordered Globalizations

We prove that every ordered partial action of an inverse semigroupoid on a partially ordered set admits a globalization. This result is used to establish a connection between ordered partial actions of groupoids and a multi-object analogue of McAlister triples. As a consequence, we obtain a multi-object version of the P-theorem: every E-unitary inverse semigroupoid is isomorphic to a semidirect product arising from an ordered partial action of a groupoid on a multi-object version of a semilattice.

math.CT↗

arXiv2024

Partial Groupoid Actions on Smooth Manifolds

Given a smooth partial action $α$ of a Lie groupoid $G$ on a smooth manifold $M,$ we provide necessary and sufficient conditions for $α$ to be globalizable with smooth globalization. As an application, we provide results on the differentiable structure of orbit and stabilizer spaces induced by $α,$ which leads to other criteria for its globalization in terms of its orbit maps in the case that $α$ is free and transitive. Further, under the assumption that $α$ is free and proper, we prove that there exists exactly one differentiable structure on the quotient structure of the orbit equivalence space $M/G$ such that the quotient map $π:M\to M/G$ is a submersion

math.DG↗

arXiv2021

Ring theoretic properties of partial skew groupoid rings with applications to Leavitt path algebras

Let $α=(A_g,α_g)_{g\in G}$ be a group-type partial action of a connected groupoid $G$ on a ring $A=\bigoplus_{z\in G_0}A_z$ and $B=A\star_αG$ the corresponding partial skew groupoid ring. In the first part of this paper we investigate the relation of several ring theoretic properties between $A$ and $B$. For the second part, using that every Leavitt path algebra is isomorphic to a partial skew groupoid ring obtained from a partial groupoid action $λ$, we characterize when $λ$ is group-type. In such a case, we obtain ring theoretic properties of Leavitt path algebras from the results on general partial skew groupoid rings. Several examples that illustrate the results on Leavitt path algebras are presented.

math.RA↗

arXiv2020

On direct product, semidirect product of groupoids and partial actions

We present some constructions of groupoids as: direct product, semidirect product, and we give necessary and sufficient conditions for a groupoid to be embedded into a direct product of groupoids. Also, we establish necessary and sufficient conditions to determine when a semidirect product is direct. Moreover, we establish an equivalence between the category of strict partial groupoid actions and the category of star injective functors. Finally, we give a relation of categorical type between the actions groupoids (G, X) and (G, Y), being Y a universal globalization of X.

math.CT↗

arXiv2020

The commutative inverse semigroup of partial abelian extensions

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action $α_G$ of a finite group $G$ on an algebra $S$ such that $S$ is an $α_G$-partial Galois extension of $S^{α_G}$ and a normal subgroup $H$ of $G$, we prove that $α_G$ induces a unital partial action $α_{G/H}$ of $G/H$ on the subalgebra of invariants $S^{α_H}$ of $S$ such that $S^{α_H}$ is an $α_{G/H}$-partial Galois extension of $S^{α_G}$. Second, assuming that $G$ is abelian, we construct a commutative inverse semigroup $T_{par}(G,R)$, whose elements are equivalence classes of $α_G$-partial abelian extensions of a commutative algebra $R$. We also prove that there exists a group isomorphism between $T_{par}(G,R)/ρ$ and $T(G,A)$, where $ρ$ is a congruence on $T_{par}(G,R)$ and $T(G,A)$ is the classical Harrison group of the $G$-isomorphism classes of the abelian extensions of a commutative ring $A$. It is shown that the study of $T_{par}(G,R)$ reduces to the case where $G$ is cyclic. The set of idempotents of $T_{par}(G,R)$ is also investigated.

math.RA↗

arXiv2020

On partial Galois abelian extensions

In this article we construct the inverse semigroup of equivalence classes of partial Galois abelian extensions of a commutative ring R with same group G, called the Harrison partial inverse semigroup.

math.RA↗

arXiv2020

Isomorphism Theorems for Groupoids and Some Applications

Using an algebraic point of view we present an introduction to the groupoid theory, that is, we give fundamental properties of groupoids as, uniqueness of inverses and properties of the identities, and study subgroupoids, wide subgroupoids and normal subgroupoids. We also present the isomorphism theorems for groupoids and as an applications we obtain the corresponding version of Zassenhaus Lemma and Jordan-Hölder Theorem for groupoids. Finally inspired by the Ehresmann-Schein-Nambooripad Theorem we improve a result of R. Exel concerning a one to one correspondence between partial actions of groups and actions of inverse semigroups.

math.GR↗

arXiv2019

On normal subgroupoids

In this paper we present some algebraic properties of subgroupoids and normal subgroupoids. We define the normalizer of a wide subgroupoid $\mathcal{H}$ and show that, as in the case of groups, the normalizer is the greatest wide subgroupoid of the groupoid $\mathcal{G}$ in which $\mathcal{H}$ is normal. Furthermore, we give the definition of center and commutator and prove that both are normal subgroupoids, the first one of the union of all the isotropy groups of $\mathcal{G}$ and the second one of $\mathcal{G}$. Finally, we introduce the concept of inner isomorphism of $\mathcal{G}$ and show that the set of all the inner isomorphisms of $\mathcal{G}$ is a normal subgroupoid, which is isomorphic to the quotient groupoid of $\mathcal{G}$ by its center $\mathcal{Z}(\mathcal{G})$, which extends to groupoids a well-known result in groups.

math.GR↗