Acta Informatica

, Volume 48, Issue 1, pp 1–18

# Some properties of the disjunctive languages contained in Q

Original Article

DOI: 10.1007/s00236-010-0127-2

Li, ZZ. & Tsai, Y.S. Acta Informatica (2011) 48: 1. doi:10.1007/s00236-010-0127-2

## Abstract

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 $${Q^{(i)} = \{f^i\,|\,f \in Q\}, i\geq 2}$$ are all disjunctive. Since then these disjunctive sets are often be quoted. Following Shyr and Thierrin showed that the half sets $${Q_{ev} = \{f \in Q\,|\,|f| = {\rm even}\}}$$ and Qod = Q \ Qev of Q are disjunctive, Chien proved that each of the set $${Q_{p,r}= \{u\in Q\,|\,|u|\equiv r\,(mod\,p) \},\,0\leq r < p}$$ is disjunctive, where p is a prime number. In this paper, we generalize this property to that all the languages $${Q_{n,r}= \{u\in Q\,|\,|u|\equiv r\,(mod\,n) \},\, 0\leq r < n}$$ are disjunctive languages, where n is any positive integer. We proved that for any n ≥ 1, k ≥ 2, (Qn,0)k are all regular languages. Some algebraic properties related to the family of languages {Qn,r | n ≥ 2, 0 ≤ r < n } are investigated.