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Confluence of the Chinese Monoid

  • Jörg EndrullisEmail author
  • Jan Willem Klop
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11760)

Abstract

The Chinese monoid, related to Knuth’s Plactic monoid, is of great interest in algebraic combinatorics. Both are ternary monoids, generated by relations between words of three symbols. The relations are, for a totally ordered alphabet, \(cba = cab = bca\) if \(a \le b \le c\). In this note we establish confluence by tiling for the Chinese monoid, with the consequence that every two words uv have extensions to a common word: \(\forall u, v. \;\exists x, y. \;ux = vy\).

Our proof is given using decreasing diagrams, a method for obtaining confluence that is central in abstract rewriting theory. Decreasing diagrams may also be applicable to various related monoid presentations.

We conclude with some open questions for the monoids considered.

References

  1. 1.
    Bezem, M., Klop, J.W., van Oostrom, V.: Diagram techniques for confluence. Inform. Comput. 141(2), 172–204 (1998)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Book, R.V., Otto, F.: String-Rewriting Systems. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  3. 3.
    Cain, A.J., Gray, R.D., Malheiro, A.: Finite Gröbner-Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids. J. Algebra 423, 37–53 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cassaigne, J., Espie, M., Krob, D., Novelli, J.C., Hivert, F.: The Chinese monoid. IJAC 11(3), 301–334 (2001)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Date, E., Jimbo, M., Miwa, T.: Representations of \(U_q(gl(n, C))\) at \(q = 0\) and the Robinson-Schensted correspondence. World Sci. 185–211 (1990)Google Scholar
  6. 6.
    de Bruijn, N.G.: A note on weak diamond properties. Memorandum 78–08, Eindhoven Uninversity of Technology (1978)Google Scholar
  7. 7.
    Dehornoy, P., Digne, F., Godelle, E., Krammer, D., Michel, J.: Foundations of Garside Theory. Tracts in Mathematics, vol. 22. European Mathematical Society, Zürich (2015)CrossRefGoogle Scholar
  8. 8.
    Endrullis, J., Klop, J.W.: De Bruijn’s weak diamond property revisited. Indagat. Math. 24(4), 1050–1072 (2013). In memory of N.G. (Dick) de Bruijn (1918–2012) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Endrullis, J., Klop, J.W.: Braids via term rewriting. Theor. Comp. Sci. 777, 260–295 (2019)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Endrullis, J., Klop, J.W., Overbeek, R.: Decreasing diagrams with two labels are complete for confluence of countable systems. In: Proceedings of the Conference on Formal Structures for Computation and Deduction (FSCD 2018), Leibniz International Proceedings in Informatics (LIPIcs), vol. 108, pp. 14:1–14:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018)Google Scholar
  11. 11.
    Guiraud, Y., Malbos, P., Mimram, S.: A homotopical completion procedure with applications to coherence of monoids. In: Proceedings of Conference on Rewriting Techniques and Applications (RTA 2013), LIPIcs, vol. 21, pp. 223–238. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)Google Scholar
  12. 12.
    Hage, N.: Finite convergent presentation of plactic monoid for type C. Int. J. Algebra Comput. 25(8), 1239–1264 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. J. ACM (JACM) 27(4), 797–821 (1980) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Karpuz, E.G.: Complete rewriting system for the Chinese monoid. Appl. Math. Sci. 4(22), 1081–1087 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Karpuz, E.G.: Finite derivation type property on the Chinese monoid. Appl. Math. Sci. 4, 1073–1080 (2010)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Knuth, D.E.: Permutations, matrices, and generalized young tableaux. Pacific J. Math. 34(3), 709–727 (1970)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lascoux, A., Leclerc, B., Thibon, J.: The plactic monoid. In: Algebraic Combinatorics on Words . Cambridge University Press (2001)Google Scholar
  18. 18.
    Lascoux, A., Schützenberger, M.P.: Le monoïde plaxique. Non-Commutative Struct. Algebra Geom. Combin. 109, 129–156 (1981)zbMATHGoogle Scholar
  19. 19.
    Lebed, V.: Plactic monoids: a braided approach. CoRR, abs/1612.05768 (2016)Google Scholar
  20. 20.
    Leclerc, B., Thibon, J.Y.: The robinson-schensted correspondence as the quantum straightening at q = 0. Electron. J. Combin. 3(2), R11 (1996)zbMATHGoogle Scholar
  21. 21.
    Schensted, C.: Longest increasing and decreasing subsequences. Can. J. Math. 13, 179–191 (1961)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)Google Scholar
  23. 23.
    van Oostrom, V.: Confluence by decreasing diagrams. Theoret. Comput. Sci. 126(2), 259–280 (1994)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Vrije Universiteit AmsterdamAmsterdamThe Netherlands
  2. 2.Centrum Wiskunde en InformaticaAmserdamThe Netherlands

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