Green index in semigroups: generators, presentations and automatic structures

A.J. Cain, R Gray, Nik Ruskuc

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6 Citations (Scopus)
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The Green index of a subsemigroup T of a semigroup S is given by counting
strong orbits in the complement S n T under the natural actions of T on S via right and left multiplication. This partitions the complement S nT into T-relative H -classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index ΙS n TΙ 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 Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger 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).

Original languageEnglish
Number of pages29
JournalSemigroup Forum
VolumeOnline First
Early online date23 May 2012
Publication statusPublished - 2012


  • Green index
  • Presentations
  • Automatic semigroup
  • Finiteness conditions


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