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

TL;DR

This paper extends the concept of canonical forms to Hecke-Kiselman monoids linked with simple oriented graphs, providing methods to obtain normal forms and a procedure for the lexicographically minimal form.

Contribution

It generalizes the canonical form concept to Hecke-Kiselman monoids and introduces a procedure to find a unique minimal normal form.

Findings

Normal forms can be obtained via elementary commutations.

Normal forms are not unique but related through elementary commutations.

A procedure for the lexicographically minimal normal form is described.

Abstract

We generalize Kudryavtseva and Mazorchuk's concept of canonical form of elements in Kiselman's semigroups to the setting of a Hecke-Kiselman monoid $\mathbf{HK}_\Gamma$ associated with a simple oriented graph $\Gamma$. We use confluence properties 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.