Idempotent generation in the endomorphism monoid of a uniform partition

TL;DR

This paper studies the structure and enumeration of idempotents in the endomorphism monoid of a uniform partition, providing explicit descriptions, ranks, and classifications of minimal generating sets.

Contribution

It introduces a detailed analysis of idempotent generation in the endomorphism monoid of a uniform partition, including enumeration, structural decomposition, and minimal generating set classification.

Findings

The subsemigroup generated by idempotents decomposes into a direct product and a wreath product.

The rank and idempotent rank of the generated subsemigroup are equal.

All minimal idempotent generating sets are classified and enumerated.

Abstract

Denote by $\mathcal T_n$ and $\mathcal S_n$ the full transformation semigroup and the symmetric group on the set $\{1,\ldots,n\}$, and $\mathcal E_n=\{1\}\cup(\mathcal T_n\setminus \mathcal S_n)$. Let $\mathcal T(X,\mathcal P)$ denote the set of all transformations of the finite set $X$ preserving a uniform partition $\mathcal P$ of $X$ into $m$ subsets of size $n$, where $m,n\geq2$. We enumerate the idempotents of $\mathcal T(X,\mathcal P)$, and describe the subsemigroup $S=\langle E\rangle$ generated by the idempotents $E=E(\mathcal T(X,\mathcal P))$. We show that $S=S_1\cup S_2$, where $S_1$ is a direct product of $m$ copies of $\mathcal E_n$, and $S_2$ is a wreath product of $\mathcal T_n$ with $\mathcal T_m\setminus \mathcal S_m$. We calculate the rank and idempotent rank of $S$, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of $\mathcal E_n$.