arXiv subjects

Ramón H. Ruiz-Medina

Publications and source records attributed to Ramón H. Ruiz-Medina.

7 recordsLinked to original sources

arXiv2026

The Burnside ring of simple $\mathcal{C}$-sets

The Burnside ring of a finite category, introduced by Webb, generalizes the classical Burnside ring of a finite group. However, unlike the classical case, the Burnside ring of a finite category has finite rank if and only if the category is equivalent to a groupoid. In this article, we introduce a new invariant associated with a finite category $\mathcal{C}$, called the \emph{simple Burnside ring} of $\mathcal{C}$ and denoted by $B^S(\mathcal{C})$. This construction is obtained from simple $\mathcal{C}$-sets and generalizes the classical Burnside ring of a finite group. Moreover, the ring $B^S(\mathcal{C})$ always has finite rank. We develop the basic theory of simple $\mathcal{C}$-sets and study several structural properties of the ring $B^S(\mathcal{C})$. In particular, we determine all ring homomorphisms from $B^S(\mathcal{C})$ to $\mathbb{Z}$, describe its prime spectrum, and obtain a decomposition theorem expressing $B^S(\mathcal{C})$ as a product of simple Burnside rings of strongly connected subcategories.

math.CT↗

arXiv2026

A note on simple actions of small monoids

These notes aim to exemplify the concept of action and partial action of a finite monoid $M$, identifying among the so-called indecomposable actions those that are also simple. We present all the simplified examples of monoids (non-groups) of order two and three, explicitly visualizing which are simple and which are not. In addition, we prove three general lemmas relating simplicity to indecomposability in the context of partial actions, and establish a result on the relation between sub-actions of total actions and partial actions.

math.GR↗

arXiv2026

Finite generating sets for monoids of $G$-equivariant functions

Given the action of a group $G$ on a set $X$, the set of all $G$-equivariant functions, i.e., those satisfying $f(g\cdot x)=g\cdot f(x)$ for all $g\in G$ and $x\in X$, forms a monoid under composition. In this work we study their generating sets. First, we propose bounds for the cardinalities of the generating sets of their group of units, denoted by $\operatorname{Aut}_{G}(X)$. Subsequently, using so-called orbital infiltrations, certain transformations that turn out to be indispensable and provide relevant structural information about the monoid, we determine conditions on the group $G$, the set $X$, and the action that prevent the whole monoid $\operatorname{End}_{G}(X)$ from admitting a finite generating set.

math.GR↗

arXiv2025

Characterization of groupoid categories in terms of its category of $\mathcal{C}$-sets

A $\mathcal{C}$-set is a functor from the category $\mathcal{C}$ to the category of finite sets and functions. The category of $\mathcal{C}$-sets, $\mathcal{C} - \operatorname*{set}$, is defined as the category whose objects are $\mathcal{C}$-sets, and whose morphisms are natural transformations between them. In this document we provide some concise characterizations of groupoids in terms of their category of $\mathcal{C}$-sets.

math.CT↗

arXiv2025

Cardinalities in finite monoids of $G$-equivariant functions

A set with a group action is referred to as a $G$-set, and the set of functions that commute with this action forms a monoid under function composition. This paper examines the case where the $G$-set is finite, which implies that the monoid of $G$-equivariant functions is also finite. The document provides formulas for calculating the cardinality of this monoid, its group of units, and explores special cases of $G$-equivariant functions, known as fixing elementary collapsings. All of these results are expressed in terms of specific properties of the $G$-set, including the number of orbits and certain indices of the subgroups acting as stabilizers.

math.GR↗

arXiv2024

El monoide de endomorfismos de $G$-conjuntos: estructuras y otras propiedades algebraicas

Given the action of a group $G$ on a set $X$, an endomorphism of $X$ is a function $f:X \rightarrow X$ which is $G$-equivariant, that is, it commutes with the action, i.e., $f(g\cdot x)= g\cdot f(x)$, for all $x\in X$. The set of endomorphisms of a $G$-set $X$ is a monoid, with the composition of functions , which we will denote $\mathrm{End}_{G}(X)$. Given subsets $U,N\subseteq M$, we say that $U$ generates $M$ modulo $N$ if it is satisfied that $M= \langle U \cup N \rangle$. The relative rank of M modulo N is the minimum cardinality of a set $U$ to generate $M$ modulo $N$. In this work we address the particular case in which $G$ and $X$ are finite to calculate the relative rank of the endomorphism monoid $\mathrm{End}_{G}(X)$ modulo its group of units, denoted by $\mathrm{Aut}_{G}(X)$. We also address structure situations, such as isomorphisms of $\mathrm{Aut}_{G}(X)$ and $\mathrm{End}_{G}(X)$ with other known structures.

math.GR↗

arXiv2023

The relative rank of the endomorphism monoid of a finite $G$-set

For a group $G$ acting on a set $X$, let $\text{End}_G(X)$ be the monoid of all $G$-equivariant transformations, or $G$-endomorphisms, of $X$, and let $\text{Aut}_G(X)$ be its group of units. After discussing few basic results in a general setting, we focus on the case when $G$ and $X$ are both finite in order to determine the smallest cardinality of a set $W \subseteq \text{End}_G(X)$ such that $W \cup \text{Aut}_G(X)$ generates $\text{End}_G(X)$; this is known in semigroup theory as the relative rank of $\text{End}_G(X)$ modulo $\text{Aut}_G(X)$.

math.GR↗