Transformation Semigroups Which Are Disjoint Union of Symmetric Groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910709125742592 |
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| author | Chaichompoo, Utsithon Sangkhanan, Kritsada |
| author_facet | Chaichompoo, Utsithon Sangkhanan, Kritsada |
| contents | Let $X$ be a nonempty set and $T(X)$ the full transformation semigroup on $X$. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by
$$
T_{E^*}(X)=\{α\in T(X):\text{for all}\ x,y\in X, (x,y)\in E\Leftrightarrow (xα,yα)\in E\}.
$$
We have the regular part of $T_{E^*}(X)$, denoted by $\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^*}(X)$. Defined the subsemigroup $Q_{E^*}(X)$ of $T_{E^*}(X)$ by
$$
Q_{E^*}(X)=\{α\in T_{E^*}(X):|Aα|=1\ \text{and}\ A\cap Xα\neq\emptyset\ \text{for all}\ A\in X/E\}.
$$
Then we can prove that this subsemigroup is the (unique) minimal ideal of $\mathrm{Reg}(T)$ which is called the kernel of $\mathrm{Reg}(T)$. In this paper, we will compute the rank of $Q_{E^*}(X)$ when $X$ is finite and prove an isomorphism theorem. Finally, we describe and count all maximal subsemigroups of $Q_{E^*}(X)$ where $X$ is a finite set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transformation Semigroups Which Are Disjoint Union of Symmetric Groups Chaichompoo, Utsithon Sangkhanan, Kritsada Rings and Algebras 20M17, 20M19, 20M20 Let $X$ be a nonempty set and $T(X)$ the full transformation semigroup on $X$. For any equivalence relation $E$ on $X$, define a subsemigroup $T_{E^*}(X)$ of $T(X)$ by $$ T_{E^*}(X)=\{α\in T(X):\text{for all}\ x,y\in X, (x,y)\in E\Leftrightarrow (xα,yα)\in E\}. $$ We have the regular part of $T_{E^*}(X)$, denoted by $\mathrm{Reg}(T)$, is the largest regular subsemigroup of $T_{E^*}(X)$. Defined the subsemigroup $Q_{E^*}(X)$ of $T_{E^*}(X)$ by $$ Q_{E^*}(X)=\{α\in T_{E^*}(X):|Aα|=1\ \text{and}\ A\cap Xα\neq\emptyset\ \text{for all}\ A\in X/E\}. $$ Then we can prove that this subsemigroup is the (unique) minimal ideal of $\mathrm{Reg}(T)$ which is called the kernel of $\mathrm{Reg}(T)$. In this paper, we will compute the rank of $Q_{E^*}(X)$ when $X$ is finite and prove an isomorphism theorem. Finally, we describe and count all maximal subsemigroups of $Q_{E^*}(X)$ where $X$ is a finite set. |
| title | Transformation Semigroups Which Are Disjoint Union of Symmetric Groups |
| topic | Rings and Algebras 20M17, 20M19, 20M20 |
| url | https://arxiv.org/abs/2411.15081 |