Idempotent rank in the endomorphism monoid of a non-uniform partition

TL;DR

This paper determines the minimal number of idempotent transformations needed to generate the semigroup of transformations preserving a non-uniform partition, extending previous results from the uniform case.

Contribution

It calculates the rank and idempotent rank of the semigroup of transformations preserving a non-uniform partition and classifies minimal generating sets, extending prior uniform case results.

Findings

Calculated the rank and idempotent rank of the semigroup

Classified and enumerated minimal idempotent generating sets

Extended results from the uniform to the non-uniform case

Abstract

We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.