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

Abstract

We generalize Kudryavtseva and Mazorchuk's concept of a canonical form of elements [9] in Kiselman's semigroups to the setting of a Hecke-Kiselman monoid \(\mathbf{HK}_\Gamma\) associated with a simple oriented graph \(\Gamma\). We use confluence properties from [7] to associate with each element in \(\mathbf{HK}_\Gamma\) 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.