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A new tropical plactic algebra is introduced in which the Knuth relations areinferred from the underlying semiring arithmetics, encapsulating the ubiquitousplactic monoid $\mathcal{P}_n$. This algebra manifests a natural framework foraccommodating representations of $\mathcal{P}_n$, or equivalently of Youngtableaux, and its moderate coarsening -- the cloaktic monoid $\mathcal{K}_n$and the co-cloaktic $ ^{\operatorname{co}}\mathcal{K}_n$. The faithful linearrepresentations of $\mathcal{K}_n$ and $\, ^{\operatorname{co}} \mathcal{K}_n$by tropical matrices, which constitute a tropical plactic algebra, are shown toprovide linear representations of the plactic monoid. To this end the paperdevelops a special type of configuration tableaux, corresponding bijectively tosemi-standard Young tableaux. These special tableaux allow a systematicencoding of combinatorial properties in numerical algebraic ways, includingalgorithmic benefits. The interplay between these algebraic-combinatorialstructures establishes a profound machinery for exploring semigroup attributes,in particular satisfying of semigroup identities. This machinery is utilizedhere to prove that $\mathcal{K}_n$ and $\, ^{\operatorname{co}} \mathcal{K}_n$admit all the semigroup identities satisfied by $n \times n$ triangulartropical matrices, which holds also for $\mathcal{P}_3$.

Tropical plactic algebra, the cloaktic monoid, and semigroup representations