Identities of Hecke–Kiselman monoids

Abstract

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