Identities in upper triangular tropical matrix semigroups and the bicyclic monoid

Laure Daviaud, Marianne Johnson, Mark Kambites

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19 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)503-525
Number of pages23
JournalJournal of Algebra
Early online date6 Mar 2018
Publication statusPublished - 1 May 2018


  • Bicyclic monoid
  • Semigroup identities
  • Upper triangular tropical matrices

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