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.
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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].
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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
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