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We establish necessary and sufficient conditions for a semigroup identity tohold in the monoid of $n\times n$ upper triangular tropical matrices, in termsof equivalence of certain tropical polynomials. This leads to an algorithm forchecking whether such an identity holds, in time polynomial in the length ofthe identity and size of the alphabet. It also allows us to answer a questionof Izhakian and Margolis, by showing that the identities which hold in themonoid of $2\times 2$ upper triangular tropical matrices are exactly the sameas those which hold in the bicyclic monoid. Our results extend to a broaderclass of "chain structured tropical matrix semigroups"; we exhibit a faithfulrepresentation of the free monogenic inverse semigroup within such a semigroup,which leads also to a representation by $3\times 3$ upper triangular matrixsemigroups, and a new proof of the fact that this semigroup satisfies the sameidentities as the bicyclic monoid.

Identities in Upper Triangular Tropical Matrix Semigroups and the Bicyclic Monoid