On a Special Class of Primitive Words

When representing DNA molecules as words, it is necessary to take into account the fact that a word uencodes basically the same information as its Watson-Crick complement i¾?(u), where i¾?denotes the Watson-Crick complementarity function. Thus, an expression which involves only a word uand its complement can be still considered as a repeating sequence. In this context, we define and investigate the properties of a special class of primitive words, called i¾?-primitive, which cannot be expressed as such repeating sequences. For instance, we prove the existence of a unique i¾?-primitive root of a given word, and we give some constraints forcing two distinct words to share their i¾?-primitive root. Also, we present an extension of the well-known Fine and Wilf Theorem, for which we give an optimal bound.