  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 $UT_5(\mathbb{T})$.
