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

TL;DR

This paper studies the algebraic structure of endomorphisms of G-sets, focusing on their monoid properties, relative rank, and relationships with automorphism groups, especially in finite cases.

Contribution

It calculates the relative rank of the endomorphism monoid modulo automorphisms for finite G-sets and explores structural isomorphisms with known algebraic structures.

Findings

Computed the relative rank of End_G(X) modulo Aut_G(X) for finite G-sets.

Identified conditions under which End_G(X) and Aut_G(X) are isomorphic to other algebraic structures.

Provided structural insights into the algebraic properties of G-set endomorphisms.

Abstract

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.