Green index in semigroups: generators, presentations and automatic structures

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.