In mathematics, the Chinese monoid is a monoid generated by a totally ordered alphabet with the relations cba = cab = bca for every a ≤ b ≤ c. An algorithm similar to Schensted's algorithm yields characterisation of the equivalence classes and a cross-section theorem. It was discovered by Duchamp & Krob (1994) during their classification of monoids with growth similar to that of the plactic monoid, and studied in detail by Julien Cassaigne, Marc Espie, Daniel Krob, Jean-Christophe Novelli, and Florent Hivert in 2001.^[1]

The Chinese monoid has a regular language cross-section

${\displaystyle a^{*}\ (ba)^{*}b^{*}\ (ca)^{*}(cb)^{*}c^{*}\cdots }$

and hence polynomial growth of dimension ${\displaystyle {\frac {n(n+1)}{2}}}$.^[2]

The Chinese monoid equivalence class of a permutation is the preimage of an involution under the map ${\displaystyle w\mapsto w\circ w^{-1}}$ where ${\displaystyle \circ }$ denotes the product in the Iwahori-Hecke algebra with ${\displaystyle q_{s}=0}$.^[3]

See also

• Plactic monoid

References

• Duchamp, Gérard; Krob, Daniel (1994), "Plactic-growth-like monoids", Words, languages and combinatorics, II (Kyoto, 1992), World Sci. Publ., River Edge, NJ, pp. 124–142, MR 1351284, Zbl 0875.68720