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Title: Green index in semigroups : generators, presentations and automatic structures
Authors: Cain, A.J.
Gray, R
Ruskuc, Nik
Keywords: Green index
Automatic semigroup
Finiteness conditions
QA Mathematics
Issue Date: 2012
Citation: Cain , A J , Gray , R & Ruskuc , N 2012 , ' Green index in semigroups : generators, presentations and automatic structures ' Semigroup Forum , vol Online First . , 10.1007/s00233-012-9406-2
Abstract: 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).
Version: Postprint
Status: Peer reviewed
ISSN: 0037-1912
Type: Journal article
Rights: This is an author version of this work. The original publication (c) Springer Science+Business Media, LLC 2012 is available at
Appears in Collections:Centre for Interdisciplinary Research in Computational Algebra (CIRCA) Research
University of St Andrews Research
Pure Mathematics Research

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