DOL Schemes and the Periodicity of String Embeddings

Abstract For a DOL scheme ( A , h ), integers i and k with i k are specified for which for every string w in A ∗ , h i ( w ) is a subsequence of h k ( w ). For a finite non-empty set A integers I and K with I K are specified for which for every DOL system ( A , h , w ), h I ( w ) is a subsequence of h K ( w ). These results allow clarification and simplification of earlier results in the literature of DOL languages. Conclusions are drawn concerning the family of finite DOL languages sharing a common scheme and the family of finite DOL languages sharing a common alphabet. The stationary sets of DOL schemes are shown to be finitely generated free monoids.