Classifications of Dense Languages

Abstract

Let X be a finite alphabet containing more than one letter. A dense language over X is a language containing a disjunctive language. A language L is an n-dense language if for any distinct n words $w_1, w_2, \ldots,w_n \in X^+,$ there exist two words$u, v \in X^*$ such that$uw_1v, uw_2v, \ldots uw_nv \in L.$ In this paper we classify dense languages into strict n-dense languages and study some of their algebraic properties. We show that for each n  ≥  0, the n-dense language exists. For an n-dense language L, n  ≠  1, the language L ∩ Q is a dense language, where Q is the set of all primitive words over X. Moreover, for a given n  ≥  1, the language L is such that $L \cap Q\in D_n(X)$, then $L\in D_m(X)$ for some m, n  ≤  m  ≤  2n + 1. Characterizations on 0-dense languages and n-dense languages are obtained. It is true that for any dense language L, there exist $w_1\neq w_2\in X^+$ such that$uw_1v,uw_2v\in L$ for some$u,v\in X^\ast$. We show that everyn-dense language, n ≥  0, can be split into disjoint union of infinitely many n-dense languages.