Effective representations of Hecke-Kiselman monoids of type A

Abstract

We prove effectiveness of certain representations of Hecke-Kiselman monoids of type A constructed by Ganyushkin and Mazorchuk and also construct further classes of effective representations for these monoids. As a consequence the effective dimension of monoids of type A is determined. We also show that odd Fibonacci numbers appear as the cardinality of certain bipartite HK-monoids and count the number of multiplicity free elements in any HK-monoid of type A.