On Kiselman quotients of 0-Hecke monoids

Abstract

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 $\mathcal{J}$-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.

arXiv comment14 pages; International Electronic Journal of Algebra, 2011

References in corpus (3)

• Hecke group algebras as quotients of affine Hecke algebras at level 0
• On Kiselman's semigroup
• Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras

Cited by in corpus (11)

• Quivers of monoids with basic algebras
• On the representation theory of finite J-trivial monoids
• Effective representations of Hecke-Kiselman monoids of type A
• Excursions into Algebra and Combinatorics at $q=0$
• Double Catalan monoids
• The Finite Basis Problem for Kiselman Monoids
• Algebraic and Affine Pattern Avoidance
• Categorification of the Catalan monoid
• Presentations for subsemigroups of $P{D}_n$
• Gröbner basis and the automaton property of Hecke--Kiselman algebras
• The word problem for Hecke--Kiselman monoids of type $A_n$ and ${\tilde{A}}_n$