The intersection of flat subsets of a braid group

The braid groups are known to be automatic groups, and there is a quadratic-time algorithm for solving the word problem. But the normal forms corresponding to the automatic structure are not minimum-crossing braid words. Tatsuoko has proposed a polynomial time scheme for finding minimum crossing normal forms for braids, but the full details have never appeared, and his preliminary preprint contains errors. His method entails describing the braid group, and its Cayley graph, as a union of 'flat' subsets. The purpose of this paper is to state and prove a correct theorem about how these 'flat' subsets meet.