Confluence of the Chinese Monoid

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: \(\forall u, v. \;\exists x, y. \;ux = vy\).