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

TL;DR

This paper characterizes and counts regular, idempotent, and unit-regular elements in semigroups of transformations that preserve a partition of a finite set, providing new insights into their algebraic structure.

Contribution

It offers new characterizations and counts of regular, idempotent, and unit-regular elements in semigroups of transformations preserving a partition, specifically for finite sets.

Findings

Characterization of unit-regular elements in T(X, P) and Σ(X, P) for finite X.

Counting of regular elements and idempotents in Γ(X, P).

Proof that all regular elements of Γ(X, P) are unit-regular for finite X.

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 $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X| (\forall X_i\in \mathcal{P}) (\exists X_j \in \mathcal{P})\;X_i f \subseteq X_j\}$, $\Sigma(X, \mathcal{P}) = \{f\in T(X, \mathcal{P})|(\forall X_i \in \mathcal{P})\; Xf \cap X_i \neq \emptyset\}$, 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 $\Gamma(X, \mathcal{P})$.