Identities in upper triangular tropical matrix semigroups and the bicyclic monoid

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. ISSN 0021-8693

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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: Article
Additional Information: Funding Information: Work supported by the LIPA project, funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 683080).
Uncontrolled Keywords: bicyclic monoid,semigroup identities,upper triangular tropical matrices,algebra and number theory ,/dk/atira/pure/subjectarea/asjc/2600/2602
Faculty \ School: Faculty of Science > School of Computing Sciences
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Depositing User: LivePure Connector
Date Deposited: 07 Jun 2023 13:30
Last Modified: 07 Jun 2023 13:30
DOI: 10.1016/j.jalgebra.2017.12.032

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