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Tropical matrix groups

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Abstract

We study the structure of groups of finitary tropical matrices under multiplication. We show that the maximal groups of \(n \times n\) tropical matrices are precisely the groups of the form \(G \times \mathbb {R}\) where G is a group admitting a 2-closed permutation representation on n points. Each such maximal group is also naturally isomorphic to the full linear automorphism group of a related tropical polytope. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space.

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Notes

  1. Some authors use the term semimodule, to emphasise the non-invertibility of addition, but since no other kind of module exists over \(\mathbb {FT}\) we have preferred the more concise term.

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Authors and Affiliations

  1. Institute of Mathematics, University of Aberdeen, AB24 3UE, Aberdeen, UK

    Zur Izhakian

  2. School of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    Marianne Johnson & Mark Kambites

Authors
  1. Zur Izhakian
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  2. Marianne Johnson
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  3. Mark Kambites
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Correspondence to Zur Izhakian.

Additional information

Communicated by Jean-Eric Pin.

Zur Izhakian: Research supported by the Alexander von Humboldt Foundation. Marianne Johnson: Research supported by EPSRC Grant EP/H000801/1. Mark Kambites: Research supported by EPSRC Grant EP/H000801/1. Mark Kambites gratefully acknowledges the hospitality of Universität Bremen during a visit to Bremen.

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Izhakian, Z., Johnson, M. & Kambites, M. Tropical matrix groups. Semigroup Forum 96, 178–196 (2018). https://doi.org/10.1007/s00233-017-9894-1

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