Regular, Unit-regular, and Idempotent elements of semigroups of transformations that preserve a partition

Abstract

Let $X$ be a set and ${\mathcal{T}}_X$ be the full transformation semigroup on $X$. For a partition $\mathcal{P}$ of $X$, we consider semigroups $T(X,\mathcal{P})=\{f\in {\mathcal{T}}_X|(\forall X_i\in \mathcal{P})(\exists X_j\in \mathcal{P})X_{i}f\subseteq X_j\}$, $\Sigma (X,\mathcal{P})=\{f\in T(X,\mathcal{P})|(\forall X_i\in \mathcal{P})Xf\cap X_i\ne \varnothing \}$, and $\Gamma (X,\mathcal{P})=\{f\in {\mathcal{T}}_X|(\forall X_i\in \mathcal{P})(\exists X_j\in \mathcal{P})X_{i}f=X_j\}$. We characterize unit-regular elements of both $T(X,\mathcal{P})$ and $\Sigma (X,\mathcal{P})$ for finite $X$. We discuss set inclusion between $\Gamma (X,\mathcal{P})$ and certain semigroups of transformations preserving $\mathcal{P}$. We characterize and count regular elements and idempotents of $\Gamma (X,\mathcal{P})$. For finite $X$, we prove that every regular element of $\Gamma (X,\mathcal{P})$ is unit-regular and also calculate the size of $\Gamma (X,\mathcal{P})$.

16 pages

References in corpus (1)

• On certain Semigroups of Transformations that preserve a partition

Cited by in corpus (1)

• On unit-regular elements in various monoids of transformations