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Green index in semigroups : generators, presentations and automatic structures
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dc.contributor.author | Cain, A.J. | |
dc.contributor.author | Gray, R | |
dc.contributor.author | Ruskuc, Nik | |
dc.date.accessioned | 2012-06-13T11:31:01Z | |
dc.date.available | 2012-06-13T11:31:01Z | |
dc.date.issued | 2012 | |
dc.identifier | 5158227 | |
dc.identifier | bd48078c-8bad-484a-b52a-288031114e6a | |
dc.identifier | 84871329719 | |
dc.identifier.citation | Cain , A J , Gray , R & Ruskuc , N 2012 , ' Green index in semigroups : generators, presentations and automatic structures ' , Semigroup Forum , vol. Online First . https://doi.org/10.1007/s00233-012-9406-2 | en |
dc.identifier.issn | 0037-1912 | |
dc.identifier.other | ORCID: /0000-0003-2415-9334/work/73702084 | |
dc.identifier.uri | https://hdl.handle.net/10023/2760 | |
dc.description.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). | |
dc.format.extent | 29 | |
dc.format.extent | 261782 | |
dc.language.iso | eng | |
dc.relation.ispartof | Semigroup Forum | en |
dc.subject | Green index | en |
dc.subject | Presentations | en |
dc.subject | Automatic semigroup | en |
dc.subject | Finiteness conditions | en |
dc.subject | QA Mathematics | en |
dc.subject.lcc | QA | en |
dc.title | Green index in semigroups : generators, presentations and automatic structures | en |
dc.type | Journal article | en |
dc.contributor.sponsor | EPSRC | en |
dc.contributor.sponsor | EPSRC | en |
dc.contributor.institution | University of St Andrews.Pure Mathematics | en |
dc.contributor.institution | University of St Andrews.Centre for Interdisciplinary Research in Computational Algebra | en |
dc.identifier.doi | 10.1007/s00233-012-9406-2 | |
dc.description.status | Peer reviewed | en |
dc.identifier.grantnumber | EP/H011978/1 | en |
dc.identifier.grantnumber | EP/E043194/1 | en |
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