Abstract:We exhibit a faithful representation of the plactic monoid of every finite rank as a monoid of upper triangular matrices over the tropical semiring. This answers a question first posed by Izhakian and subsequently studied by several authors. A consequence is a proof of a conjecture of Kubat and Okniński that every plactic monoid of finite rank satisfies a non-trivial semigroup identity. In the converse direction, we show that every identity satisfied by the plactic monoid of rank is satisfied by the monoid of upper triangular tropical matrices. In particular this implies that the variety generated by the upper triangular tropical matrices coincides with that generated by the plactic monoid of rank , answering another question of Izhakian.

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中文翻译：

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实用类半体动物的热带表征和身份

摘要：我们真实地表示了每个有限等级的实用半形体，作为热带半环上的上三角矩阵的半体形。这回答了伊扎克先提出的一个问题，随后又由几位作者进行了研究。结果就是对Kubat和Okniński的猜想的证明，即有限等级的所有实用半形体都满足非平凡的半群身份。在相反的方向上，我们表明，等级的实用齐半体满足的每个恒等式都由上三角热带矩阵的齐半体满足。特别是，这暗示着上三角热带矩阵所产生的多样性与等级的实用等分群所产生的多样性相吻合，从而回答了以扎金人的另一个问题。

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