Abstract

This paper shows that every Plactic algebra of finite rank admits a
finite Gröbner-Shirshov basis. The result is proved by using the
combinatorial properties of Young tableaux to construct a finite
complete rewriting system for the corresponding Plactic monoid, which
also yields the corollaries that Plactic monoids of finite rank have
finite derivation type and satisfy the homological finiteness properties
left- and right-${\rm FP}_\infty$. Also, answering a question of Zelmanov,
we apply this rewriting system and other techniques to show that Plactic monoids
of finite rank are biautomatic.