Borderedness-preserving homomorphisms

This paper aims at investigating properties of homomorphisms which preserve the bordered words. The bordered words are classified into d1d1-words and d2d2-words, where the length of the proper border of d2d2-word is greater than half of it. Some characterizations of d1d1-word-preserving homomorphism are studied. Some relationships among d-primitivity-preserving homomorphisms, d1d1-word-preserving homomorphisms, and d2d2-word-preserving homomorphisms are investigated. We show that every d-primitivity-preserving homomorphism is d1d1-word-preserving and every d1d1-word-preserving homomorphism is d2d2-word-preserving.