2010/06/30 by Olexandr Ganyushkin, Volodymyr Mazorchuk · 1 citation
Mathematics · #math.GR #math.CO #msc:20M10 #msc:20M20 #msc:20M30
published as Int. Electron. J. Algebra 10 (2011) 174-191 · 14 pages; International Electronic Journal of Algebra, 2011
arxiv created 2011/03/27 · arxiv updated 2011/07/26
Combining the definition of 0-Hecke monoids with that of Kiselman semigroups, we define what we call Kiselman quotients of 0-Hecke monoids associated with simply laced Dynkin diagrams. We classify these monoids up to isomorphism, determine their idempotents and show that they are J-trivial. For type A we show that Catalan numbers appear as the maximal cardinality of our monoids, in which case the corresponding monoid is isomorphic to the monoid of all order-preserving and order-decreasing total transformations on a finite chain. We construct various representations of these monoids by matrices, total transformations and binary relations. Motivated by these results, with a mixed graph we associate a monoid, which we call a Hecke-Kiselman monoid, and classify such monoids up to isomorphism. Both Kiselman semigroups and Kiselman quotients of 0-Hecke monoids are natural examples of Hecke-Kiselman monoids.