On the longest common prefix of suffixes in an inverse Lyndon factorization and other properties

The Lyndon factorization of a word has been largely studied and recently variants of it have been introduced and investigated with different motivations. In particular, the canonical inverse Lyndon factorization ICFL(w) of a word w, previously introduced, maintains the main properties of the Lyndon factorization since it can be computed in linear time and it is uniquely determined. In this paper we investigate new properties of this factorization with the aim of exploring their use in some classical queries on w. The main property we prove is related to a classical query on words. We prove that there are relations between the length of the longest common prefix (or longest common extension) lcp(x,y) of two different suffixes x,y of a word w and the maximum length M of two consecutive factors of ICFL(w). More precisely, M is an upper bound on the length of lcp(x,y). A main tool used in the proof of the above result is a property that we state for factors m_i with nonempty borders in ICFL(w): a nonempty border of m_i cannot be a prefix of the next factor m_{i+1}. Another interesting result relates sorting of global suffixes, i.e., suffixes of a word w, and sorting of local suffixes, i.e., suffixes of products of factors in ICFL(w). This is the counterpart for ICFL(w) of the compatibility property, proved for the Lyndon factorization by other authors. Roughly, the compatibility property allows us to extend the mutual order between suffixes of products of the (inverse) Lyndon factors to the suffixes of the whole word. The last property we prove focuses on the Lyndon factorizations of a word and its factors. It suggests that the Lyndon factorizations of two words sharing a common overlap could be used to capture the common overlap of these two words.