Semigroup identities in the monoid of two-by-two tropical matrices

The main result of the paper proves that the monoid of all 2 x 2 tropical matrices satisfies a non-trivial semigroup identity, unlike the case of matrices over infinite fields. We exploit a strong connection between the monoid of all 2 x 2 tropical matrices and the bicyclic monoid to show that a variant of Adian's identity for the bicyclic monoid holds in our case.We use generic matrices of tropical polynomials to prove our result.