The Rank of the Endomorphism Monoid of a Partition

TL;DR

This paper determines the minimal number of generators needed for the endomorphism monoid of a uniform partition, solving an open problem by linking it to the rank of a wreath product of symmetric groups.

Contribution

It computes the rank of the endomorphism monoid of a uniform partition, revealing it as a wreath product of transformation semigroups and establishing the rank of such wreath products as two.

Findings

The rank of the endomorphism monoid of a uniform partition is explicitly calculated.

The rank of a wreath product of two symmetric groups is two.

The paper solves an open question in the theory of transformation semigroups.

Abstract

The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question.