V-Words, Lyndon Words and Galois Words

We say that a family W of strings over Σ⁺ forms a Unique Maximal Factorization Family (UMFF) if and only if every w ∈ W has a unique maximal factorization. Further, an UMFF W is called a circ-UMFF whenever it contains exactly one rotation of every primitive string x ∈ Σ⁺. V-order is a non-lexicographical total ordering on strings that determines a circ-UMFF. In this paper we propose a generalization of circ-UMFF called the substring circ-UMFF and extend combinatorial research on V-order by investigating connections to Lyndon words. Then we extend these concepts to any total order. Applications of this research arise in efficient text indexing, compression, and search problems.