ON KISELMAN QUOTIENTS OF 0-HECKE MONOIDS

Combining the definition of 0-Hecke monoids with that of Kisel-man 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 semi groups and Kiselman quotients of 0-Hecke monoids are natural examples of Hecke-Kiselman monoids.