Identities of Hecke–Kiselman monoids

It is shown that the Hecke–Kiselman monoid \({\text {HK}}_{\Theta }\) associated to a finite oriented graph \(\Theta \) satisfies a semigroup identity if and only if \({\text {HK}}_{\Theta }\) does not have free noncommutative subsemigroups. It follows that this happens exactly when the semigroup algebra \(K[{\text {HK}}_{\Theta }]\) over a field K satisfies a polynomial identity. The latter is equivalent to a condition expressed in terms of the graph \(\Theta \). The proof allows to derive concrete identities satisfied by such monoids \({\text {HK}}_{\Theta }\).