Hecke and Artin monoids and their homomorphisms

Abstract

The aim of the present work is to systematically study homomorphisms of Hecke and Artin monoids and thus to develop their comprehensive theory. Our original motivation was the striking observation that parabolic projections of Hecke monoids respect all parabolic elements. We found other classes of homomorphisms of Hecke monoids with the same property and discovered that many of them lift to homomorphisms of covering Artin monoids with a similar property. It turned out that they belong to a much larger class (in fact, a category) of homomorphisms of Artin monoids, most of which appear to be new.