Characterizing the largest commutative (full and partial) transformation semigroups of certain types
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912704974815232 |
|---|---|
| author | Paulista, Tânia |
| author_facet | Paulista, Tânia |
| contents | Let $X$ be a finite set. Let $\mathcal{T}(X)$ be the transformation semigroup on $X$ and let $\mathcal{P}(X)$ be the partial transformation semigroup on $X$. This paper is a contribution to the problem of characterizing the largest commutative subsemigroups of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$). In the process of looking for these semigroups, we also characterize the largest commutative subsemigroups of idempotents of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$); as well as the largest commutative subsemigroups of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$) that contain a unique idempotent. We also provide an alternative way to determine the largest commutative nilpotent subsemigroups of $\mathcal{T}(X)$ (which were previously characterized by Cain, Malheiro and the present author); and we describe the largest commutative nilpotent subsemigroups of $\mathcal{P}(X)$.
These results allow us to make conclusions regarding the clique numbers of the commuting graphs of $\mathcal{T}(X)$ and of $\mathcal{P}(X)$. We also determine their girths and knit degrees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing the largest commutative (full and partial) transformation semigroups of certain types Paulista, Tânia Combinatorics Group Theory Rings and Algebras 20M14, 20M20 (Primary), 05C05, 05C25 (Secondary) Let $X$ be a finite set. Let $\mathcal{T}(X)$ be the transformation semigroup on $X$ and let $\mathcal{P}(X)$ be the partial transformation semigroup on $X$. This paper is a contribution to the problem of characterizing the largest commutative subsemigroups of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$). In the process of looking for these semigroups, we also characterize the largest commutative subsemigroups of idempotents of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$); as well as the largest commutative subsemigroups of $\mathcal{T}(X)$ (respectively, $\mathcal{P}(X)$) that contain a unique idempotent. We also provide an alternative way to determine the largest commutative nilpotent subsemigroups of $\mathcal{T}(X)$ (which were previously characterized by Cain, Malheiro and the present author); and we describe the largest commutative nilpotent subsemigroups of $\mathcal{P}(X)$. These results allow us to make conclusions regarding the clique numbers of the commuting graphs of $\mathcal{T}(X)$ and of $\mathcal{P}(X)$. We also determine their girths and knit degrees. |
| title | Characterizing the largest commutative (full and partial) transformation semigroups of certain types |
| topic | Combinatorics Group Theory Rings and Algebras 20M14, 20M20 (Primary), 05C05, 05C25 (Secondary) |
| url | https://arxiv.org/abs/2511.09495 |