On n-power prefix languages

A word u is said to be an n-power prefix of another word w if w is an element of u(n) X*. Let P-i denote the family of words which have i-power prefixes but no (i +1)-power prefix. Words in P-1 are called p-primitive words. In this paper, some basic properties of words in sets P-1(i) and Pi are studied. It is shown that the sets P-i, P-i \ Q, P-i boolean AND Q, Q \ P-i, P-1(i) and P-i boolean OR Q for i >= 1 are disjunctive. That is, they are dense non-regular languages. A characterization of words in P-1((I)) for i >= 2 is derived. Several properties of words in P-n \ P-1(n) are considered too.