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Identities in upper triangular tropical matrix semigroups and the bicyclic monoid

Daviaud, L., Johnson, M. and Kambites, M. (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

Abstract

We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of n x 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 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 upper triangular tropical matrices.

Publication Type: Article
Additional Information: © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Publisher Keywords: Semigroup identities, Upper triangular tropical matrices, Bicyclic monoid
Departments: School of Mathematics, Computer Science & Engineering > Computer Science
URI: http://openaccess.city.ac.uk/id/eprint/21291
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