Identities of tropical matrix semigroups and the plactic monoid of rank 4

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 UTn(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, P4, we find a new set of identities satisfied by P4 shorter than those previously known, and show that the semigroup variety generated by P4 is strictly contained in the variety generated by UT5(T).