ON KISELMAN QUOTIENTS OF 0-HECKE MONOIDS

Year 2011, Volume: 10 Issue: 10, 174 - 191, 01.12.2011

Olexandr Ganyushkin Volodymyr Mazorchuk

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 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.

Keywords

0-Hecke monoid, Kiselman type semigroup, action, isomorphism