Some Properties of the Overlapping Words

This paper aims to investigate some properties of the overlapping words. A word w is called an overlapping word if there exists a nonempty word v such that v <;_d w, where the notation <;_d is the partial order relation. The overlapping words consist of d₁-words and d₂-words. We called that an overlapping word u is d₂-word if there exists a nonempty word v such that v <;d u and lg(v) >; (1/2) lg(u). In the meanwhile, an overlapping word is a d₁-word if it is not a d₂-word. The characterizations of d₁-words and d₂-words are studied in this paper. The procedure of constructing the d₂-words is provided. We also study the characteristics of d-primitive word preserving homomorphism, d₁-word preserving homomorphism, and d₂-word preserving homomorphism. Furthermore, the relationships between d-primitive word preserving homomorphism, d₁-word preserving homomorphism, and d₂-word preserving homomorphism are considered. We have the following results. The d-primitive word preserving homomorphism is d₁-word preserving and the d₁-word preserving homomorphism is d₂-word preserving.