We say that a family $\mathcal{W}$ of strings over $\Sigma^+$ 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
$x \in \Sigma^+$. $V$-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.