Interaction Properties of Relational Periods

Similarity relations, i.e., compatibility relations on words induced by relations on letters were introduced as a generalization of partial words in [9](i). There and in [10, 11] the main focus was on the effect of these relations on coding properties and on the defect theorem of words. In the article [8] we started the study of the interaction properties of periods with respect to similarity relations. By the interaction property we mean that if a sufficiently long word has two periods then it also has another nontrivial period depending on the original periods. The theorem of Fine and Wilf is one of the cornerstones in combinatorics on words. In this theorem the derived period is the greatest common divisor of the original periods [7]. Actually, this topic was the starting point of the study of partial words in the seminal paper of J. Berstel and L. Boasson in 1999 [1]. They proved a variant of the theorem of Fine and Wilf for partial words with one hole. Since then several papers on period properties of partial words has been published [2, 3, 4, 5, 6, 12]. F. Blanchet-Sadri et al. studied the theorem of Fine and Wilf for partial words with local periods and with arbitrarily many holes. A.M. Shur and Yu.V. Gamzova investigated the case of global periods. We continue the study of this topic by introducing global, external and local relational periods as generalizations of periods of partial words. Using these periods we prove new variants of Fine and Wilf’s theorem. Especially, our aim is to compare the interaction properties of different types of periods.