The word problem for Hecke--Kiselman monoids of type $A_n$ and $\widetilde{A}_n$

TL;DR

This paper provides explicit bijections between Hecke--Kiselman monoids of types A_n and Ã_n, braid diagrams, and integer sequences, enabling a fast solution to the word problem and normal forms.

Contribution

It introduces a novel approach using bijections and Yang--Baxter actions to efficiently solve the word problem for these monoids.

Findings

Explicit bijections between monoids, diagrams, and sequences

Fast algorithms for the word problem

Efficient normal forms for HK monoids

Abstract

We exhibit explicit and easily realisable bijections between Hecke--Kiselman monoids of type $A_n$/$\widetilde{A}_n$; certain braid diagrams on the plane/cylinder; and couples of integer sequences of particular types. This yields a fast solution of the word problem and an efficient normal form for these HK monoids. Yang--Baxter type actions play an important role in our constructions.