Geodesic
Coxeter groups, Coxeter monoids and the Bruhat order
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 719-731.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Associated with any Coxeter group is a Coxeter monoid, which has the same elements, and the same identity, but a different multiplication. (Some authors call these Coxeter monoids 0-Hecke monoids, because of their relation to the 0-Hecke algebras -- the $q=0$ case of the Hecke algebra of a Coxeter group.) A Coxeter group is defined as a group having a particular presentation, but a pair of isomorphic groups could be obtained via non-isomorphic presentations of this form. We show that when we have both the group and the monoid structure, we can reconstruct the presentation uniquely up to isomorphism and present a characterisation of those finite group and monoid structures that occur as a Coxeter group and its corresponding Coxeter monoid. The Coxeter monoid structure is related to this Bruhat order. More precisely, multiplication in the Coxeter monoid corresponds to element-wise multiplication of principal downsets in the Bruhat order. Using this property and our characterisation of Coxeter groups among structures with a group and monoid operation, we derive a classification of Coxeter groups among all groups admitting a partial order.
Classification : 05E15, 20F55
Keywords: Coxeter group, Coxeter monoid, 0-Hecke monoid, Bruhat order 
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Kenney, Toby. Coxeter groups, Coxeter monoids and the Bruhat order. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 719-731. http://geodesic.mathdoc.fr/item/JAC_2014__39_3_a1/