El monoide de endomorfismos de $G$-conjuntos: estructuras y otras propiedades algebraicas
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Ruiz-Medina, Ramón H. |
| author_facet | Ruiz-Medina, Ramón H. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10209 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | El monoide de endomorfismos de $G$-conjuntos: estructuras y otras propiedades algebraicas Ruiz-Medina, Ramón H. Group Theory 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. |
| title | El monoide de endomorfismos de $G$-conjuntos: estructuras y otras propiedades algebraicas |
| topic | Group Theory |
| url | https://arxiv.org/abs/2408.10209 |