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Combining the definition of 0-Hecke monoids with that of Kiselman semigroups,we define what we call Kiselman quotients of 0-Hecke monoids associated withsimply laced Dynkin diagrams. We classify these monoids up to isomorphism,determine their idempotents and show that they are $\mathcal{J}$-trivial. Fortype $A$ we show that Catalan numbers appear as the maximal cardinality of ourmonoids, in which case the corresponding monoid is isomorphic to the monoid ofall order-preserving and order-decreasing total transformations on a finitechain. We construct various representations of these monoids by matrices, totaltransformations and binary relations. Motivated by these results, with a mixedgraph we associate a monoid, which we call a Hecke-Kiselman monoid, andclassify such monoids up to isomorphism. Both Kiselman semigroups and Kiselmanquotients of 0-Hecke monoids are natural examples of Hecke-Kiselman monoids.

On Kiselman quotients of 0-Hecke monoids