Lyndon-like and V-order factorizations of strings

We say a family W of strings is an UMFF if every string has a unique maximal factorization over W. Then W is an UMFF iff xy, yz ∈ W and y non-empty imply xyz ∈ W. Let L-order denote lexicographic order. Danh and Daykin discovered V -order, B-order and T -order. Let R be L, V , B or T . Then we call r an R-word if it is strictly first in R-order among the cyclic permutations of r. The set of R-words form an UMFF. We show a large class of B-like UMFF. The well-known Lyndon factorization of Chen, Fox and Lyndon is the L case, and it motivated our work.