Fast algorithms for identification and comparison of braids

A method for constructing various algorithms for mapping-class groups (e.g., for the braid group) is presented, and a family of one-side invariant orders on the mapping-class group of a surface with boundary is described. A method for constructing comparison algorithms for all finite orders on the mapping-class group of any surface with boundary is described, a fast and simple comparison algorithm for the Dehornois order on the braid group is presented, examples of normal forms for braid groups are given, and algorithms for finding the forms are indicated.