Mathematics > Rings and Algebras
[Submitted on 22 Nov 2024]
Title:Transformation Semigroups Which Are Disjoint Union of Symmetric Groups
View PDF HTML (experimental)Abstract: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
$$
T_{E^*}(X)=\{\alpha\in T(X):\text{for all}\ x,y\in X, (x,y)\in E\Leftrightarrow (x\alpha,y\alpha)\in E\}.
$$
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
$$
Q_{E^*}(X)=\{\alpha\in T_{E^*}(X):|A\alpha|=1\ \text{and}\ A\cap X\alpha\neq\emptyset\ \text{for all}\ A\in X/E\}.
$$
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.
Submission history
From: Kritsada Sangkhanan [view email][v1] Fri, 22 Nov 2024 17:17:54 UTC (14 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.