Green index in semigroups: generators, presentations and automatic structures

TL;DR

This paper explores the structure of semigroups with finite Green index subsemigroups, providing methods to derive generators and presentations, and demonstrating the preservation of key properties like automaticity and finite presentability.

Contribution

It introduces a framework for constructing presentations and analyzing properties of semigroups with finite Green index subsemigroups, generalizing classical group theory results.

Findings

Methods to obtain generators for subsemigroups and Schutzenberger groups

Construction of semigroup presentations from substructures

Preservation of properties like automaticity and finite presentability

Abstract

Let S be a semigroup and let T be a subsemigroup of S. Then T acts on S by left- and by right multiplication. This gives rise to a partition of the complement of T in S, and to each equivalence class of this partition we naturally associate a relative Schutzenberger group. We show how generating sets for S may be used to obtain generating sets for T and the Schutzenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schutzenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity, finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups). These results provide common generalisations of several classical results from group theory and Rees index results from semigroup theory.