V-Words, Lyndon Words and Substring circ-UMFFs

We say that a family W$$\mathcal {W}$$ of strings over Σ+$$\varSigma ^+$$ forms a Unique Maximal Factorization Family if and only if for every w∈W$${\boldsymbol{w}} \in \mathcal {W}$$, w$${\boldsymbol{w}}$$ has a unique maximal factorization. Then an UMFF W$$\mathcal {W}$$ is a circ-UMFF whenever it contains exactly one rotation of every primitive string x∈Σ+$${\boldsymbol{x}} \in \varSigma ^+$$. 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 the combinatorial research on V-order by investigating connections to Lyndon words. Then we extend concepts to considering any total order. Applications of this research arise in efficient text indexing, compression, and search tasks.