Homomorphisms of Hecke-Kiselman Monoids Associated to Simple Oriented Graphs

Abstract

Let $\Gamma$ and $\Theta$ be finite simple oriented graphs, and let ${\mathbf{HK}}_{\Gamma }$ and ${\mathbf{HK}}_{\Theta }$ denote the corresponding Hecke-Kiselman monoids. We present a set of Boolean matrices that is in bijection with the set of monoid homomorphisms from ${\mathbf{HK}}_{\Gamma }$ to ${\mathbf{HK}}_{\Theta }$. In particular, we obtain a Boolean matrix monoid isomorphic to the endomorphism monoid of ${\mathbf{HK}}_{\Theta }$.

16 pages