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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).
Original language | English |
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Number of pages | 29 |
Journal | Semigroup Forum |
Volume | Online First |
Early online date | 23 May 2012 |
DOIs | |
Publication status | Published - 2012 |
Keywords
- Green index
- Presentations
- Automatic semigroup
- Finiteness conditions
Fingerprint
Dive into the research topics of 'Green index in semigroups: generators, presentations and automatic structures'. Together they form a unique fingerprint.Projects
- 2 Finished
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Automata Languages Decidability: Automata, Languages, Decidability in Algebra
Ruskuc, N. (PI) & Quick, M. (CoI)
1/03/10 → 31/05/14
Project: Standard
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Finiteness Conditions and Index: Finiteness Conditions and Index in Semigroups and Monoids
Gray, R. D. (PI)
1/02/08 → 31/01/11
Project: Standard