Hecke and Artin monoids and their homomorphisms

The aim of this work is to build a theory of homomorphisms of Hecke and Artin monoids with possible applications to representation 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.
Comment: AMSLaTeX, 112 pages