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 u, v 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.
Dedication. Our paper is dedicated in friendship to Catuscia Palamidessi for her 60th anniversary, with fond memories of the second author of cooperations during her stays around 1990 at CWI Amsterdam; with admiration for her work and accomplishments.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Already two labels suffice for proving confluence using decreasing diagrams of every countable abstract reduction system [10]. The completeness for uncountable systems remains a long-standing open problem [23]. For proving commutation (a generalisation of confluence involving two relations) decreasing diagrams are not complete, as established in [8].
References
Bezem, M., Klop, J.W., van Oostrom, V.: Diagram techniques for confluence. Inform. Comput. 141(2), 172–204 (1998)
Book, R.V., Otto, F.: String-Rewriting Systems. Springer, Heidelberg (1993)
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)
Cassaigne, J., Espie, M., Krob, D., Novelli, J.C., Hivert, F.: The Chinese monoid. IJAC 11(3), 301–334 (2001)
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)
de Bruijn, N.G.: A note on weak diamond properties. Memorandum 78–08, Eindhoven Uninversity of Technology (1978)
Dehornoy, P., Digne, F., Godelle, E., Krammer, D., Michel, J.: Foundations of Garside Theory. Tracts in Mathematics, vol. 22. European Mathematical Society, Zürich (2015)
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)
Endrullis, J., Klop, J.W.: Braids via term rewriting. Theor. Comp. Sci. 777, 260–295 (2019)
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)
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)
Hage, N.: Finite convergent presentation of plactic monoid for type C. Int. J. Algebra Comput. 25(8), 1239–1264 (2015)
Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. J. ACM (JACM) 27(4), 797–821 (1980)
Karpuz, E.G.: Complete rewriting system for the Chinese monoid. Appl. Math. Sci. 4(22), 1081–1087 (2010)
Karpuz, E.G.: Finite derivation type property on the Chinese monoid. Appl. Math. Sci. 4, 1073–1080 (2010)
Knuth, D.E.: Permutations, matrices, and generalized young tableaux. Pacific J. Math. 34(3), 709–727 (1970)
Lascoux, A., Leclerc, B., Thibon, J.: The plactic monoid. In: Algebraic Combinatorics on Words . Cambridge University Press (2001)
Lascoux, A., Schützenberger, M.P.: Le monoïde plaxique. Non-Commutative Struct. Algebra Geom. Combin. 109, 129–156 (1981)
Lebed, V.: Plactic monoids: a braided approach. CoRR, abs/1612.05768 (2016)
Leclerc, B., Thibon, J.Y.: The robinson-schensted correspondence as the quantum straightening at q = 0. Electron. J. Combin. 3(2), R11 (1996)
Schensted, C.: Longest increasing and decreasing subsequences. Can. J. Math. 13, 179–191 (1961)
Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)
van Oostrom, V.: Confluence by decreasing diagrams. Theoret. Comput. Sci. 126(2), 259–280 (1994)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Endrullis, J., Klop, J.W. (2019). Confluence of the Chinese Monoid. In: Alvim, M., Chatzikokolakis, K., Olarte, C., Valencia, F. (eds) The Art of Modelling Computational Systems: A Journey from Logic and Concurrency to Security and Privacy. Lecture Notes in Computer Science(), vol 11760. Springer, Cham. https://doi.org/10.1007/978-3-030-31175-9_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-31175-9_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-31174-2
Online ISBN: 978-3-030-31175-9
eBook Packages: Computer ScienceComputer Science (R0)