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 $f:X\to X$ 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 ${\mathrm{E}\mathrm{n}\mathrm{d}}_G(X)$. Given subsets $U,N\subseteq M$, we say that $U$ generates $M$ modulo $N$ if it is satisfied that $M=⟨U\cup N⟩$. 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 ${\mathrm{E}\mathrm{n}\mathrm{d}}_G(X)$ modulo its group of units, denoted by ${\mathrm{A}\mathrm{u}\mathrm{t}}_G(X)$. We also address structure situations, such as isomorphisms of ${\mathrm{A}\mathrm{u}\mathrm{t}}_G(X)$ and ${\mathrm{E}\mathrm{n}\mathrm{d}}_G(X)$ with other known structures.