Reaching the minimum ideal in a finite semigroup

TL;DR

This paper introduces depth parameters for finite semigroups to measure the complexity of generating elements in the minimum ideal, providing bounds for various semigroup families and product constructions.

Contribution

It defines new depth parameters for finite semigroups and establishes bounds for wreath and direct product semigroups, advancing understanding of their generating complexities.

Findings

Estimated depth parameters for specific semigroup families

Provided upper bounds for wreath products

Analyzed depth in direct product of monoids

Abstract

We introduce the depth parameters of a finite semigroup, which measure how hard it is to produce an element in the minimum ideal when we consider generating sets satisfying some minimality conditions. We estimate such parameters for some families of finite semigroups, and we obtain an upper bound for wreath products and direct products of two finite (transformation) monoids. Keywords: semigroup, generating set, minimum ideal, A-depth of a semigroup