Green index in semigroups: generators, presentations, and automatic structures

Abstract

The Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S∖T under the natural actions of T on S via right and left multiplication. This partitions the complement S∖T into T-relative -classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schutzenberger group. If the Rees index |S∖T| is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows 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 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 (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups).