Abstract
We establish necessary and sufficient conditions for a semigroup identity to hold in the monoid of nĂn 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 2Ă2 upper triangular tropical matrices are exactly the same as those which hold in the bicyclic monoid. Our results extend to a broader class of âchain structured tropical matrix semigroupsâ; we exhibit a faithful representation of the free monogenic inverse semigroup within such a semigroup, which leads also to a representation by 3Ă3 upper triangular tropical matrices.
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