On certain Semigroups of Transformations that preserve a partition

TL;DR

This paper investigates the structure and properties of semigroups of transformations that preserve a set partition, providing characterizations, enumerations, and cardinalities for various related semigroups and their elements.

Contribution

It offers new characterizations and enumerations of elements in semigroups of transformations preserving partitions, including idempotents and units, for both finite and arbitrary sets.

Findings

Characterization of elements in ( ext{X}, \u03a0)

Existence of nontrivial partitions for permutation groups

Cardinality formulas for semigroups and subsemigroups

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