## Abstract

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 P_{3}.

Original language | English |
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Pages (from-to) | 290-366 |

Number of pages | 77 |

Journal | Journal of Algebra |

Volume | 524 |

DOIs | |

State | Published - 15 Apr 2019 |

Externally published | Yes |

## Keywords

- Cloaktic monoid
- Colored weighted digraphs
- Configuration tableaux
- Forward semigroup
- Idempotent semirings
- Plactic monoid
- Semigroup identities
- Semigroup representations
- Symmetric group
- Tropical matrix algebra
- Tropical plactic algebra
- Young tableaux