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Identities of tropical matrices and plactic monoids

We study semigroup varieties generated by full and upper triangular tropical matrix semigroups and the plactic monoid of rank 4. We prove that the upper triangular tropical matrix semigroup $UT_n(\mathbb{T})$ generates a different semigroup variety for each dimension $n$. We show a weaker version of this fact for the full matrix semigroup: full tropical matrix semigroups of different prime dimensions generate different semigroup varieties. For the plactic monoid of rank 4, $\mathbb{P}_4$, we find a new set of identities satisfied by $\mathbb{P}_4$ shorter than those previously known, and show that the semigroup variety generated by $\mathbb{P}_4$ is strictly contained in the variety generated by $UT_5(\mathbb{T})$.