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

Abstract

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 [Formula: see text] generates a different semigroup variety for each dimension [Formula: see text]. 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, [Formula: see text], we find a new set of identities satisfied by [Formula: see text] shorter than those previously known, and show that the semigroup variety generated by [Formula: see text] is strictly contained in the variety generated by [Formula: see text].