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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 Trelative 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 groupembeddable semigroups).
Original language  English 

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
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Dive into the research topics of 'Green index in semigroups: generators, presentations and automatic structures'. Together they form a unique fingerprint.Projects
 2 Finished

Automata Languages Decidability: Automata, Languages, Decidability in Algebra
1/03/10 → 31/05/14
Project: Standard

Finiteness Conditions and Index: Finiteness Conditions and Index in Semigroups and Monoids
Gray, R. D.
1/02/08 → 31/01/11
Project: Standard