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Identities in upper triangular tropical matrix semigroups and the bicyclic monoid
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Daviaud, Laure, Johnson, Marianne and Kambites, Mark (2018) Identities in upper triangular tropical matrix semigroups and the bicyclic monoid. Journal of Algebra, 501 . pp. 503-525. doi:10.1016/j.jalgebra.2017.12.032 ISSN 0021-8693.
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Official URL: https://doi.org/10.1016/j.jalgebra.2017.12.032
Abstract
We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of n×n upper triangular tropical matrices, in terms of equivalence of certain tropical polynomials. This leads to an algorithm for checking whether such an identity holds, in time polynomial in the length of the identity and size of the alphabet. It also allows us to answer a question of Izhakian and Margolis, by showing that the identities which hold in the monoid of 2×2 upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid. Our results extend to a broader class of “chain structured tropical matrix semigroups”; we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup, which leads also to a representation by 3×3 upper triangular tropical matrices.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | ||||||||
| Library of Congress Subject Headings (LCSH): | Semigroups, Monoids | ||||||||
| Journal or Publication Title: | Journal of Algebra | ||||||||
| Publisher: | Academic Press Inc Elsevier Science | ||||||||
| ISSN: | 0021-8693 | ||||||||
| Official Date: | 1 May 2018 | ||||||||
| Dates: |
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| Volume: | 501 | ||||||||
| Page Range: | pp. 503-525 | ||||||||
| DOI: | 10.1016/j.jalgebra.2017.12.032 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 17 January 2018 | ||||||||
| Date of first compliant Open Access: | 6 March 2019 | ||||||||
| RIOXX Funder/Project Grant: |
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