Finite Gröbner–Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids

Alan J. Cain, Robert Gray, António Malheiro

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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 FP∞FP∞. Also, answering a question of Zelmanov, we apply this rewriting system and other techniques to show that Plactic monoids of finite rank are biautomatic.
Original languageEnglish
Pages (from-to)37–53
Number of pages17
JournalJournal of Algebra
Publication statusPublished - 1 Feb 2015


  • Plactic algebra
  • Plactic monoid
  • Gröbner–Shirshov basis
  • Complete rewriting system
  • Young tableau
  • Automatic monoids

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