V-Words, Lyndon Words and Galois Words

We say that a family $\mathcal{W}$ of strings over ${\mathrm{\Sigma }}^{+}$ forms a Unique Maximal Factorization Family (UMFF) if and only if every $w\in \mathcal{W}$ has a unique maximal factorization. Further, an UMFF $\mathcal{W}$ is called a circ-UMFF whenever it contains exactly one rotation of every primitive string $x\in {\mathrm{\Sigma }}^{+}$. $V$-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.