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.