Semigroup identities and varieties of plactic monoids

In this paper, we study the semigroup identities satisfied by finite rank plactic monoids. We find a new set of semigroup identities of the plactic monoid of rank [Formula: see text] for [Formula: see text], which are shorter than those previously known when [Formula: see text]. Using these semigroup identities we show that for all [Formula: see text], the plactic monoid of rank [Formula: see text] satisfies a semigroup identity not satisfied by the semigroup of [Formula: see text] upper triangular tropical matrices. We then prove that the plactic monoid of rank [Formula: see text] generates a different semigroup variety for each rank [Formula: see text].