Endomorphisms of Hecke-Kiselman Monoids Associated to Simple Oriented Graphs

Let ${\mathrm{H}\mathrm{K}}_{\mathrm{\Theta }}$ denote the Hecke-Kiselman monoid associated to a finite simple oriented graph $\mathrm{\Theta }$. We present a Boolean matrix monoid that is isomorphic to the endomorphism monoid $\mathrm{E}\mathrm{n}\mathrm{d}({\mathrm{H}\mathrm{K}}_{\mathrm{\Theta }})$.