Irreducible representations of Hecke–Kiselman monoids

Let K[HKΘ] denote the Hecke–Kiselman algebra of a finite oriented graph Θ over an algebraically closed field K. All irreducible representations, and corresponding maximal ideals K[HKΘ], are characterized in case this satisfies polynomial identity. The latter condition corresponds to simple that can be expressed terms Θ. result shows surprising similarity classical results on representations semigroups; namely every representation either comes form idempotent monoid HKΘ (and hence it is 1-dimensional), or from certain semigroup matrix type. when cycle plays crucial role; prime spectrum completely case.