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dc.contributor.authorCain, A.J.
dc.contributor.authorGray, R
dc.contributor.authorRuskuc, Nik
dc.identifier.citationCain , A J , Gray , R & Ruskuc , N 2012 , ' Green index in semigroups : generators, presentations and automatic structures ' Semigroup Forum , vol Online First . DOI: 10.1007/s00233-012-9406-2en
dc.identifier.otherPURE: 5158227
dc.identifier.otherPURE UUID: bd48078c-8bad-484a-b52a-288031114e6a
dc.identifier.otherScopus: 84871329719
dc.description.abstractThe 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).en
dc.relation.ispartofSemigroup Forumen
dc.rightsThis is an author version of this work. The original publication (c) Springer Science+Business Media, LLC 2012 is available at www.springerlink.comen
dc.subjectGreen indexen
dc.subjectAutomatic semigroupen
dc.subjectFiniteness conditionsen
dc.subjectQA Mathematicsen
dc.titleGreen index in semigroups : generators, presentations and automatic structuresen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews. Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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