A Normal Form for a Class of Monoids Including the Singular Braid Monoids

The first part of this paper investigates a class of homogeneously presented monoids. Constructions which enable division and multiplication to be computed are described. The word problem and the division problem are solved, and a unique normal form is given for monoids in this class. The second part deals with the motivating examples of this class—the positive singular Artin monoids. Finally, the singular Artin monoids are introduced. The word and division problems are solved for all singular Artin monoids of finite type, including, for example, the singular braid monoids introduced by Baez and Birman.