Regularity of the semigroup of transformations preserving a length
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913375181602816 |
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| author | Sommanee, Worachead |
| author_facet | Sommanee, Worachead |
| contents | Let $X_n = \{1,2,\dots,n\}$ be a finite set $(n\geq 2)$ and $T_n$ the full transformation semigroup on $X_n$. For a positive integer $l\leq n-1$, we define $$T_n(l) = \{α\in T_n \colon \forall x,y\in X_n,\, |x-y| = l \;\Rightarrow\; |xα- yα| = l\}$$ and $$T^*_n(l) = \{α\in T_n \colon \forall x,y\in X_n,\, |x-y| = l \;\Leftrightarrow\; |xα- yα| = l\}.$$ Then $T_n(l)$ and $T^*_n(l)$ are subsemigroups of $T_n$. In this paper, we give a necessary and sufficient condition for $T_n(l)$ to be regular. Moreover, we prove that $T^*_n(l)$ is a regular semigroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01015 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularity of the semigroup of transformations preserving a length Sommanee, Worachead Group Theory 20M20 Let $X_n = \{1,2,\dots,n\}$ be a finite set $(n\geq 2)$ and $T_n$ the full transformation semigroup on $X_n$. For a positive integer $l\leq n-1$, we define $$T_n(l) = \{α\in T_n \colon \forall x,y\in X_n,\, |x-y| = l \;\Rightarrow\; |xα- yα| = l\}$$ and $$T^*_n(l) = \{α\in T_n \colon \forall x,y\in X_n,\, |x-y| = l \;\Leftrightarrow\; |xα- yα| = l\}.$$ Then $T_n(l)$ and $T^*_n(l)$ are subsemigroups of $T_n$. In this paper, we give a necessary and sufficient condition for $T_n(l)$ to be regular. Moreover, we prove that $T^*_n(l)$ is a regular semigroup. |
| title | Regularity of the semigroup of transformations preserving a length |
| topic | Group Theory 20M20 |
| url | https://arxiv.org/abs/2406.01015 |