Hecke-Kiselman Monoids of Small Cardinality

In this paper, we give a characterization of digraphs $Q,\mathrm{\mid }Q\mathrm{\mid }\le 4$ such that the associated Hecke-Kiselman monoid $H_Q$ is finite. In general, a necessary condition for $H_Q$ to be a finite monoid is that $Q$ is acyclic and its Coxeter components are Dynkin diagram. We show, by constructing examples, that such conditions are not sufficient.