Some Results on Primitive Words and Disjunctive Languages

Abstract

Let X be a finite alphabet and Q be the set of all primitive words over X. In 1978, Chu and Town proved that if a ² = c ^k x where k ≥ 2, x is a prefix word of c and a, c ∈ Q, then a = c. In this paper, we improve it into that if a ^m = c ^k x where m, k ≥ 2, x is a prefix word of c and a, c ∈ Q, then a = c. On the other hand, we give some disjunctive languages and a regular language which are related to primitive words. Languages D ₁ = Da ∪ X* wb and D ₂ = aD ∪ bw X* are all disjunctive languages for an arbitrary disjunctive language D, a, b ∈ X, a ≠ b and w ∈ X*, which were proved by Reis and Shyr in 1978. But there are some flaws in their proof. Also in the paper, we provide the other proof for them.

This work is supported by a grant from the Applied Basic Research Programs of Science and Technology Department Foundation of Yunnan Province of Chian # 2010CD21.