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.
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Li, ZZ., Shyr, H.J. & Tsai, Y.S. Classifications of Dense Languages. Acta Informatica 43, 173–194 (2006). https://doi.org/10.1007/s00236-006-0015-y
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DOI: https://doi.org/10.1007/s00236-006-0015-y