On Kiselman Quotients of 0-hecke Monoids

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. This, in particular, implies a presentation for the latter monoid. Finally, we construct various representations of these monoids by matrices, total transformations and binary relations. 1. Definitions and description of the results Let Γ be a simply laced Dynkin diagram (or a disjoint union of simply laced Dynkin diagrams). Then the 0-Hecke monoid HΓ associated with Γ is the monoid generated by idempotents εi, where i runs through the set Γ0 of all vertexes of Γ, subject to the usual braid relation, namely, εiεj = εjεi in the case when i and j are not connected in Γ, and εiεjεi = εjεiεj in the case when i and j are connected in Γ (see e.g. [NT1]). The elements ofHΓ are in a natural bijection with the elements of the Weyl group WΓ of Γ. The latter follows e.g. from [Ma, Theorem 1.13] as the semigroup algebra of the monoid HΓ is canonically isomorphic to the specialization of the Hecke algebra Hq(WΓ) at q = 0, which also explains the name. This specialization was studied by several authors, see [No, Ca, McN, Fa, HNT, NT2] and references therein. The monoid HΓ appears for example in [FG, HST1, HST2]. One has to note that HΓ appears in articles where the emphasis is made on its semigroup algebra and not its structure as a monoid. Therefore semigroup properties of HΓ are not really spelled out in the above papers. However, with some efforts one can derive from the above literature that the monoid HΓ is J -trivial (we will show this in Subsection 2.1) and has 2 idempotents, where n is the number of vertexes in Γ (we will show this in Subsection 2.2). Another example of an idempotent generated J -trivial monoid with 2 idempotents (where n is the number of generators) is Kiselman’s semigroup Kn, defined as follows: it is generated by idempotents ei,