Relational depth of transformation semigroups and their ideals

TL;DR

This paper introduces the concept of relational depth in finite semigroups, analyzing how the ideal structure influences the complexity of defining the semigroup.

Contribution

It formally defines relational depth for finite semigroups with chain-structured J-classes and computes this depth for key transformation semigroups.

Findings

Relational depth is explicitly determined for ideals in full transformation monoids.

Relational depth measures the complexity of defining semigroups via generators and relations.

The concept applies to semigroups with chain-structured J-classes.

Abstract

We introduce the concept of relational depth of a finite semigroup $S$ whose $J$-classes form a chain. It captures how far down in the ideal structure one is obliged to go in order to define the semigroup by generators and defining relations. We determine the exact value for the relational depth of an arbitrary ideal in the full transformation monoid, symmetric inverse monoid and in the partial transformation monoid.