Set Of Primitive Words Research Articles

Generalized de Bruijn words for primitive words and powers

We show that for every n≥1 and over any finite alphabet, there is a word whose circular factors of length n have a one-to-one correspondence with the set of primitive words. In particular, we prove that such a word can be obtained by a greedy algorithm, or by concatenating all Lyndon words of length n in increasing lexicographic order. We also look into connections between de Bruijn graphs of primitive words and Lyndon graphs.Finally, we also show that the shortest word that contains every p-power of length pn over a k-letter alphabet has length between pkn and roughly (p+1k)kn, for all integers p≥1. An algorithm that generates a word which achieves the upper bound is provided.

Some properties of disjunctive languages on a free monoid

A language A on a free monoid X * generated by the finite alphabet X is called a disjunctive language if the principal congruence determined by A is the identity. In this paper we study some properties of disjunctive languages over X with | X | ⩾ 2. We show that there is no minimal (and hence no maximal) disjunctive language and that the cardinality of the family of disjunctive languages is the same as that of the reals. It is also shown that if a discrete disjunctive language is partitioned into two subsets then one of them must be disjunctive. We prove that every non-empty word which is not a power of a letter is a product of two primitive words and then show that Q n is regular for all n ⩾ 2, where Q is the set of primitive words. But in contrast with this, Q ( i ) Q ( j ) … Q ( l ) is shown to be disjunctive for all i + l ⩾ 3. Here Q ( i ) = če:italic>w i | w ε Q⩽ce:italic> .