arXiv subjects

Héctor Pinedo

Publications and source records attributed to Héctor Pinedo.

At least 19 recordsLinked to original sources

arXiv2026

Very good gradings on structural matrix rings

Let $R$ be a nonzero associative unital ring, let $G$ be a group, and let $ρ$ be a preorder on $\{1,\ldots,n\}$. A $G$-grading on $ρ$ induces a very good $G$-grading on the structural matrix ring $M_n(ρ,R)$. We show that, for each of the properties trivial, symmetric, epsilon-strong and strong, the grading on $ρ$ has the property if and only if the induced ring grading does. The epsilon-crossed product and crossed product properties pass from $ρ$ to the ring, but the converses fail in general. We also give a concrete criterion for epsilon-strongness and show that a very good $G$-grading on $M_n(ρ,R)$ that is strong satisfies $|G|\leq n$. When $ρ$ is an equivalence relation and the neutral component is diagonal, very good gradings correspond bijectively to free partial actions of $G$ on $\{1,\ldots,n\}$ with orbit relation $ρ$. These gradings are epsilon-crossed products, and over a field the correspondence gives a classification up to graded algebra isomorphism.

math.RA↗

arXiv2026

On Γ-embeddings and partial actions of function spaces

This paper deals with the extension of partial actions of topological groups on topological spaces. Within this framework, we introduce a class of topological embeddings defined via the inverse semigroup of homeomorphisms between open subsets of a topological space. We describe several embeddings of this type, referred to as $Γ$- embeddings, and we place particular emphasis on one of them. In particular, we prove that every topological space $Y$ admits a $Γ$-embedding into the space of continuous functions $C(X, Y )$, equipped with the compact-open topology, where $X$ is a compact space. Consequently, any partial action $θ$ of a topological group $G$ on $ Y$ naturally induces a partial action $\hatθ$ on $C(X, Y ).$ Throughout the paper, we investigate various relationships between these actions, as well as between their corresponding globalizations and enveloping spaces.

math.GN↗

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↗

arXiv2024

Twisted products: Enveloping actions and equivariant absolute neighborhood extensors

The classical notion of twisted product is studied in the context of partial actions, in particular, we show that the globalization of a partial action is a twisted product. In addition, we establish conditions for the metrizability of twisted products, and some homotopy and categorical properties are proved. Furthermore, sufficient conditions for the enveloping space to be an equivariant absolute neighborhood extensor are also studied.

math.GN↗

arXiv2023

Very good gradings on matrix rings are epsilon-strong

We investigate properties of group gradings on matrix rings $M_n(R)$, where $R$ is an associative unital ring and $n$ is a positive integer. More precisely, we introduce very good gradings and show that any very good grading on $M_n(R)$ is necessarily epsilon-strong. We also identify a condition that is sufficient to guarantee that $M_n(R)$ is an epsilon-crossed product, i.e. isomorphic to a crossed product associated with a unital twisted partial action. In the case where $R$ has IBN, we are able to provide a characterization of when $M_n(R)$ is an epsilon-crossed product. Our results are illustrated by several examples.

math.RA↗

arXiv2023

Partial actions of groups on generalized matrix rings

Let $n$ be a positive integer and $R=(M_{ij})_{1\leq i,j\leq n}$ be a generalized matrix ring. For each $1\leq i,j\leq n$, let $I_i$ be an ideal of the ring $R_i:=M_{ii}$ and denote $I_{ij}=I_iM_{ij}+M_{ij}I_j$. We give sufficient conditions for the subset $I=(I_{ij})_{1\leq i,j\leq n}$ of $R$ to be an ideal of $R$. Also, suppose that $α^{(i)}$ is a partial action of a group $\mathtt{G}$ on $R_i$, for all $1\leq i\leq n$. We construct, under certain conditions, a partial action $γ$ of $\mathtt{G}$ on $R$ such that $γ$ restricted to $R_i$ coincides with $α^{(i)}$. We study the relation between this construction and the notion of Morita equivalent partial group action given in [1]. Moreover, we investigate properties related to Galois theory for the extension $R^γ\subset R$. Some examples to illustrate the results are considered in the last part of the paper.

math.RA↗

arXiv2023

The graded structure of Leavittt path algebras viewed as partial skew group rings

Let $E$ be a directed graph, $\mathbb K$ be a field, and $\mathbb F$ be the free group on the edges of $E$. In this work, we use the isomorphism between Leavitt path algebras and partial skew group rings to endow $L_{\mathbb K}(E)$ with an $\mathbb F$-gradation and study some algebraic properties of this gradation. More precisely, we show that graded cleanness, graded unit-regularity, and strong gradeness of $L_{\mathbb K}(E)$ are all equivalent.

math.RA↗

arXiv2023

Partial actions of groups on profinite spaces

We show that for a partial action $η$ with closed domain of a compact group $G$ on a profinite space $X$ the space of orbits $X/\!\sim_G$ is profinite, this leads to the fact that when $G$ is profinite the enveloping space $X_G$ is also profinite. Moreover, we provide conditions for the induced quotient map $π_G: X\to X/\!\sim_G$ of $η$ to have a continuous section. Relations between continuous sections of $π_G$ and continuous sections of the quotient map induced by the enveloping action of $η$ are also considered. At the end of this work we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

math.GN↗

arXiv2022

Epsilon-strongly graded rings: Azumaya algebras and partial crossed products

The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let $G$ be a group, $A=\oplus_{g\in G}\,A_g$ an epsilon-strongly graded ring and ${\bf pic}{R}$ the Picard semigroup of $R:=A_1$. We prove that the isomorphism class $[A_g]$ is an element of ${\bf pic}{R}$, for all $g\in G$. Thus, the association $g\mapsto [A_g]$ determines a partial representation of $G$ on ${\bf pic}{R}$ which induces a partial action $γ$ of $G$ on the center $Z(R)$ of $R$. Sufficient conditions for $A$ to be an Azumaya $R^γ$-algebra are presented in the case that $R$ is commutative. We study when $B$ is a partial crossed product in the following cases: $B=\operatorname{M}_n(A)$ is the ring of matrices with entries in $A$, or $B={\bf grm}{M}=\bigoplus_{l \in G}{\bf Mor}_A(M,M)_l$ is the direct sum of graded endomorphisms of left graded $A$-module $M$ with degree $l$, or $B={\bf grm}{M}$ where $M=A\otimes_{R}N$ is the induced module of a left $R$-module $N$. Finally, assuming that $R$ is semiperfect, we prove that there exists an epsilon-strongly graded subring of $A$ which is graded equivalent to a partial crossed product.

math.RA↗

arXiv2021

A topological correspondence between partial actions of groups and inverse semigroup actions

We present some generalizations of the well-known correspondence, found by R. Exel, between partial actions of a group $G$ on a set $X$ and semigroup homomorphism of $S(G)$ on the semigroup $I(X)$ of partial bijections of $X,$ being $S(G)$ an inverse monoid introduced by Exel. We show that any unital premorphism $θ:G\to S$, where $S$ is an inverse monoid, can be extended to a semigroup homomorphism $θ^*:T\to S$ for any inverse semigroup $T$ with $S(G)\subseteq T\subseteq P^*(G)\times G,$ being $P^*(G)$ the semigroup of non-empty subset of $G$, and such that $E(S)$ satisfies some lattice theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on $Γ(X)$, the inverse semigroup of partial homeomorphism between open subsets of a locally compact Hausdorff space $X$.

math.GN↗

arXiv2021

Object-unital groupoid graded modules

In a previous article (see \cite{CNP}), we introduced and analyzed ring-theoretic properties of object unital $\mathcal{G}$-graded rings $R$, where $\mathcal{G}$ is a groupoid. In the present article, we analyze the category $\grmod$ of unitary $\G$-graded modules over such rings. Following ideas developed earlier by one of the authors in \cite{lundstrom2004}, we analyze the forgetful functor $U \colon \grmod \to \rmod$ and aim to determine properties $\mathcal{P}$ for which the following implications are valid for modules $M$ in $\grmod$: $M$ is $\mathcal{P}$ $\Rightarrow$ $U(M)$ is $\mathcal{P}$; $U(M)$ is $\mathcal{P}$ $\Rightarrow$ $M$ is $\mathcal{P}$. Here we treat the cases when $\mathcal{P}$ is any of the properties: direct summand, projective, injective, free, simple and semisimple. Moreover, graded versions of results concerning classical module theory are established, as well as some structural properties related to the category $\grmod$.

math.RA↗

arXiv2021

Partial actions of groups on hyperspaces

Let $X$ be a compact Hausdorff space. In this work we translate partial actions of $X$ to partial actions on some hyperspaces determined by $X,$ this gives an endofunctor $2^{-}$ in the category of partial actions on compact Hausdorff spaces which generates a monad in this category. Moreover, structural relations between partial actions $θ$ on $X$ and partial determined by $2^θ$ are established.

math.GN↗

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

Restriction and Extension of Partial Actions

Given a partial action $α=(A_g,α_g)_{g\in \mathcal{G}}$ of a connected groupoid $\mathcal{G}$ on a ring $A$ and an object $x$ of $\mathcal{G}$, the isotropy group $\mathcal{G}(x)$ acts partially on the ideal $A_x$ of $A$ by the restriction of $α$. In this paper we investigate the following reverse question: under what conditions a partial group action of $\mathcal{G}(x)$ on an ideal of $A$ can be extended to a partial groupoid action of $\mathcal{G}$ on $A$? The globalization problem and some applications to the Morita and Galois theories are also considered, as extensions of similar results from the group actions case.

math.RA↗

arXiv2020

On the structure of nearly epsilon and epsilon-strongly graded rings

In this work we study the classes of epsilon and nearly epsilon-strongly graded rings by a group $G$. In particular, we extend Dade's theorem to the realm of nearly epsilon-strongly graded rings. Moreover, we introduce the category SIM$S-$gr of symmetrically graded modules and use it to present a new characterization of strongly graded rings. A functorial approach is used to obtain a characterization of epsilon-strongly graded rings. Finally, we determine conditions for which an epsilon-strongly graded ring can be written as a direct sum of strongly graded rings and a trivially graded ring.

math.RA↗

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

Object-unital groupoid graded rings, crossed products and separability

We extend the classical construction by Noether of crossed product algebras, defined by finite Galois field extensions, to cover the case of separable (but not necessarily finite or normal) field extensions. This leads us naturally to consider non-unital groupoid graded rings of a particular type that we call object unital. We determine when such rings are strongly graded, crossed products, skew groupoid rings and twisted groupoid rings. We also obtain necessary and sufficient criteria for when object unital groupoid graded rings are separable over their principal component, thereby generalizing previous results from the unital case to a non-unital situation.

math.RA↗