On Kiselman quotients of 0-Hecke monoids

TL;DR

This paper introduces Kiselman quotients of 0-Hecke monoids linked to simply laced Dynkin diagrams, classifies them, explores their properties, and constructs various representations, revealing connections to Catalan numbers and Hecke-Kiselman monoids.

Contribution

It defines and classifies Kiselman quotients of 0-Hecke monoids, analyzes their algebraic structure, and constructs multiple representations, advancing understanding of these algebraic objects.

Findings

Kiselman quotients are $\,\mathcal{J}$-trivial monoids.

Type A monoids have maximal size given by Catalan numbers.

Kiselman quotients are isomorphic to monoids of order-preserving, order-decreasing transformations.

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.