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Green index in semigroups: generators, presentations, and automatic structures

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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∖T under the natural actions of T on S via right and left multiplication. This partitions the complement S∖T into T-relative -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 |S∖T| 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 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).

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Authors and Affiliations

  1. Centro de Matemática, Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    Alan J. Cain

  2. CAUL, Universidade de Lisboa, Avenida Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal

    Robert Gray

  3. School of Mathematics & Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, UK

    Nik Ruškuc

Authors
  1. Alan J. Cain
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  2. Robert Gray
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  3. Nik Ruškuc
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Correspondence to Alan J. Cain.

Additional information

Communicated by Mikhail V. Volkov.

The first author was supported by the project PTDC/MAT/69514/2006 ‘Semigroups and Languages’, funded by FCT and PIDDAC, and later by a FCT Ciência 2008 fellowship. At the time of the work described in this paper, the second author held an EPSRC Postdoctoral Fellowship at the University of St Andrews. We thank Simon Craik and Victor Maltcev for useful discussions during the preparation of this paper.

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Cain, A.J., Gray, R. & Ruškuc, N. Green index in semigroups: generators, presentations, and automatic structures. Semigroup Forum 85, 448–476 (2012). https://doi.org/10.1007/s00233-012-9406-2

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