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<p>We consider effective versions of Stone duality for distributive <i>c</i>-posets and almost semispectral spaces with base. For any oracle <i>Z</i> ⊆ ω, the notions of <i>Z</i>-computably enumerable <i>c</i>-posets and <i>Z</i>-computably enumerable topological spaces with base are introduced. The dual equivalence of the categories <b>AS</b> and <b>DP</b> restricts to a dual equivalence of their full subcategories of <i>Z</i>-computably enumerable objects. Effective dualities are also obtained for distributive lattices, distributive meet semilattices, and distributive join semilattices. As an application, for any nontrivial countable structure <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation>, there exists a separable almost semispectral space with base whose degree spectrum coincides with that of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">S</mi> </math></EquationSource> </InlineEquation>.</p>

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A Computable Stone Duality

  • 期刊论文

  • 发表日期：

    2026年7月13日

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  • N. A. Bazhenov,
  • I. Sh. Kalimullin,
  • M. V. Schwidefsky

摘要

We consider effective versions of Stone duality for distributive c-posets and almost semispectral spaces with base. For any oracle Z ⊆ ω, the notions of Z-computably enumerable c-posets and Z-computably enumerable topological spaces with base are introduced. The dual equivalence of the categories AS and DP restricts to a dual equivalence of their full subcategories of Z-computably enumerable objects. Effective dualities are also obtained for distributive lattices, distributive meet semilattices, and distributive join semilattices. As an application, for any nontrivial countable structure \(\mathcal{S}\) S , there exists a separable almost semispectral space with base whose degree spectrum coincides with that of \(\mathcal{S}\) S .