Some properties of the disjunctive languages contained in Q

• Published in: Acta informatica 2011-02, Vol.48 (1), p.1-18
• Main Authors: Li, Zheng-Zhu, Tsai, Y. S.
• Format: Article
• Language: English
• Subjects: Algebra; Alphabets; Applied sciences; Computer Science; Computer science; control theory; systems; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Exact sciences and technology; Information Systems and Communication Service; Language; Lectures; Logics and Meanings of Programs; Mathematical analysis; Mathematics; Miscellaneous; Number theory; Original Article; Prime numbers; Sciences and techniques of general use; Software Engineering/Programming and Operating Systems; Studies; Theoretical computing; Theory of Computation
• ISSN: 0001-5903; 1432-0525
• DOI: 10.1007/s00236-010-0127-2

The set of all primitive words Q over an alphabet X was first defined and studied by Shyr and Thierrin (Proceedings of the 1977 Inter. FCT-Conference, Poznan, Poland, Lecture Notes in Computer Science 56. pp. 171–176 ( 1977 )). It showed that for the case | X | ≥ 2, the set along with are all disjunctive. Since then these disjunctive sets are often be quoted. Following Shyr and Thierrin showed that the half sets and Q od = Q \ Q ev of Q are disjunctive, Chien proved that each of the set is disjunctive, where p is a prime number. In this paper, we generalize this property to that all the languages are disjunctive languages, where n is any positive integer. We proved that for any n  ≥ 1, k  ≥ 2, ( Q n ,0 ) k are all regular languages. Some algebraic properties related to the family of languages { Q n , r | n ≥ 2, 0 ≤ r < n } are investigated.