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<p>We prove that for every antichain <i>A</i> in the poset <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\langle [\omega ]^{&lt;\omega },\subseteq \rangle \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mi>ω</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo>&lt;</mo> <mi>ω</mi> </mrow> </msup> <mo>,</mo> <mo>⊆</mo> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> the set of maximal antichains which extend <i>A</i> is either finite or has the size of the continuum. As a consequence we prove a conjecture of de Jongh and Vargas-Sandoval about nepfi families of finite languages [<CitationRef CitationID="CR2">2</CitationRef>, <CitationRef CitationID="CR10">10</CitationRef>].</p>

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Extending antichains in the poset \(\langle [\omega ]^{<\omega },\subseteq \rangle \)

  • 期刊论文

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  • 发表日期：

    2025年5月21日

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  • Francisco Santiago Nieto-de la Rosa,
  • Ulises Ariet Ramos-García,
  • Ana Lucía Vargas-Sandoval,
  • Dick de Jongh

摘要

We prove that for every antichain A in the poset \(\langle [\omega ]^{<\omega },\subseteq \rangle \) ⟨ [ ω ] < ω , ⊆ ⟩ the set of maximal antichains which extend A is either finite or has the size of the continuum. As a consequence we prove a conjecture of de Jongh and Vargas-Sandoval about nepfi families of finite languages [2, 10].