The minimal number of generators of a finite semigroup

TL;DR

This paper derives a formula for the minimal number of generators needed for finite Rees matrix semigroups over groups, and applies it to analyze the ranks of transformation semigroups and related structures.

Contribution

It provides a new explicit formula for the rank of Rees matrix semigroups over groups, linking structure matrix dimensions and subgroup ranks.

Findings

Formula for the rank of Rees matrix semigroups over groups.

Application to minimal generating sets of transformation semigroups.

Results on maximum rank of subsemigroups of transformation monoids.

Abstract

The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (non necessarily regular) Rees matrix semigroup over a group. The formula is expressed in terms of the dimensions of the structure matrix, and the relative rank of a certain subset of the structure group obtained from subgroups generated by entries in the structure matrix, which is assumed to be in Graham normal form. This formula is then applied to answer questions about minimal generating sets of certain natural families of transformation semigroups. In particular, the problem of determining the maximum rank of a subsemigroup of the full transformation monoid (and of the symmetric inverse semigroup) is considered.