V-Words, Lyndon Words and Substring circ-UMFFs

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