Tropical plactic algebra, the cloaktic monoid, and semigroup representations

A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetic, encapsulating the ubiquitous plactic monoid P_n. This algebra manifests a natural framework for accommodating representations of P_n, and equivalently of Young tableaux, and its moderate coarsening — the cloaktic monoid K_n and the co-cloaktic monoid (Figure presented.). The faithful linear representations of K_n and (Figure presented.) by tropical matrices, which constitute a tropical plactic algebra, are shown to provide linear representations of the plactic monoid. To this end the paper develops a special type of configuration tableaux, corresponding bijectively to semi-standard Young tableaux. These special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways, including algorithmic benefits. The interplay between these algebraic-combinatorial structures establishes a profound machinery for exploring semigroup attributes, in particular satisfying of semigroup identities. This machinery is utilized here to prove that K_n and (Figure presented.) admit all the semigroup identities satisfied by n×n triangular tropical matrices, which holds also for P₃.