Chinese monoid

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 {{harvtxt|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.{{Citation |last1=Cassaigne |first1=Julien |last2=Espie |first2=Marc |last3=Krob |first3=Daniel |last4=Novelli |first4=Jean-Christophe |last5=Hivert |first5=Florent | title=The Chinese monoid |url=http://www.worldscinet.com/ijac/11/1103/S0218196701000425.html | doi=10.1142/S0218196701000425 | mr=1847182 | zbl=1024.20046 | year=2001 | journal=International Journal of Algebra and Computation | issn=0218-1967 | volume=11 | issue=3 | pages=301–334}}

The Chinese monoid has a regular language cross-section

:$a^*\; \backslash \; (ba)^*b^*\; \backslash \; (ca)^*(cb)^*\; c^*\; \backslash cdots$

and hence polynomial growth of dimension $\backslash frac\{n(n+1)\}\{2\}$.{{citation | last1=Jaszuńska | first1=Joanna | last2=Okniński | first2=Jan | title=Structure of Chinese algebras. | zbl=1246.16022 | journal=J. Algebra | volume=346 | number=1 | pages=31–81 | issn=0021-8693 | arxiv=1009.5847| year=2011 | doi=10.1016/j.jalgebra.2011.08.020| s2cid=119280148 }}

The Chinese monoid equivalence class of a permutation is the preimage of an involution under the map $w\; \backslash mapsto\; w\; \backslash circ\; w^\{-1\}$ where $\backslash circ$ denotes the product in the Iwahori-Hecke algebra with $q\_s\; =\; 0$.{{Cite journal|last1=Hamaker|first1=Zachary|last2=Marberg|first2=Eric|last3=Pawlowski|first3=Brendan|date=2017-05-01|title=Involution words II: braid relations and atomic structures|url=https://doi.org/10.1007/s10801-016-0722-6|journal=Journal of Algebraic Combinatorics|language=en|volume=45|issue=3|pages=701–743|doi=10.1007/s10801-016-0722-6|issn=1572-9192|arxiv=1601.02269|s2cid=119330473}}