Normal form in Hecke-Kiselman monoids associated with simple oriented graphs

We generalize Kudryavtseva and Mazorchuk’s concept of canonical form of elements in Kiselman’s semigroups to the setting of a Hecke-Kiselman monoid ${\mathbf{H}\mathbf{K}}_{\mathrm{\Gamma }}$ associated with a simple oriented graph $\mathrm{\Gamma }$. We use confluence properties to associate with each element in ${\mathbf{H}\mathbf{K}}_{\mathrm{\Gamma }}$ a normal form; normal forms are not unique, and we show that they can be obtained from each other by a sequence of elementary commutations. We finally describe a general procedure to recover a (unique) lexicographically minimal normal form.