On certain semigroups of transformations that preserve a partition

Abstract

Let X be a nonempty set, and let ${\mathcal{T}}_X$ be the full transformation semigroup on X. For a partition $\mathcal{P}=\left\{X_i\right|i\in I\}$ of X, we consider the semigroup $T(X,\mathcal{P})=\{f\in {\mathcal{T}}_X|\forall X_i\exists X_j,X_{i}f\subseteq X_j\},$ the subsemigroup $\Sigma (X,\mathcal{P})=\{f\in T(X,\mathcal{P})|Xf\cap X_i\ne \varnothing \forall X_i\},$ and the group of units $S(X,\mathcal{P})$ of $T(X,\mathcal{P}).$ In this paper, we first characterize the elements of $\Sigma (X,\mathcal{P}).$ For a permutation f of finite X, we next observe whether there exists a nontrivial partition $\mathcal{P}$ of X such that $f\in S(X,\mathcal{P}).$ We then characterize and enumerate the idempotents in the semigroup $\Sigma (X,\mathcal{P})$ for arbitrary and finite X, respectively. We also characterize the elements of $S(X,\mathcal{P}).$ For finite X, we finally calculate the cardinality of $T(X,\mathcal{P}),\Sigma (X,\mathcal{P}),$ and $S(X,\mathcal{P}).$

Keywords:

• Idempotents
• partitions
• permutation groups
• transformation semigroups

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20M15
• 20M20

Acknowledgment

We are thankful to the referee for bringing the paper [1] to our notice regarding results in the section 4, and helpful suggestions to improve the manuscript.