The rank of the endomorphism monoid of uniformly nested partition

TL;DR

This paper determines the minimal number of generators needed for the endomorphism monoid of a uniformly nested partition, revealing a direct relation between the partition's depth and the monoid's rank.

Contribution

It establishes that the rank of the endomorphism monoid of a uniformly nested partition of depth k is exactly 2k, providing a precise measure of its algebraic complexity.

Findings

The rank of the monoid is 2k for depth k.

The result links the depth of the partition to the algebraic structure of its endomorphisms.

Provides a clear formula for the minimal generating set size.

Abstract

We show that the rank of the endomorphism monoid of a unifromly nested partition of depth k is 2k.