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 $\mathcal{P}=\{X_i|i\in I\}$ 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 $S(X,\mathcal{P})$.

13 pages

Cited by in corpus (2)

• Regular, Unit-regular, and Idempotent elements of semigroups of transformations that preserve a partition
• On unit-regular elements in various monoids of transformations