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<p>For computable families of computably enumerable sets we find sufficient conditions under which, for any n &gt; 1, the Rogers semilattices of these families have ideals containing exactly n minimal elements, each specified by positive undecidable numberings. It is shown that the Rogers semilattice of every <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\Sigma }_{\mathrm{m}}^{0}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow> <mi mathvariant="normal">m</mi> </mrow> <mn>0</mn> </msubsup> </math></EquationSource> </InlineEquation>-computable family, m &gt; 1, possessing a Friedberg <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({\Sigma }_{\mathrm{m}}^{0}\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow> <mi mathvariant="normal">m</mi> </mrow> <mn>0</mn> </msubsup> </math></EquationSource> </InlineEquation>-computable numbering, for any n &gt; 1 possesses an ideal that also contains exactly n minimal elements specified by positive undecidable numberings.</p>

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On the Number of Minimal Elements of Ideals of Rogers Semilattices

  • 期刊论文

  • 发表日期：

    2025年2月1日

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  • M. Kh. Faizrahmanov

摘要

For computable families of computably enumerable sets we find sufficient conditions under which, for any n > 1, the Rogers semilattices of these families have ideals containing exactly n minimal elements, each specified by positive undecidable numberings. It is shown that the Rogers semilattice of every \({\Sigma }_{\mathrm{m}}^{0}\) Σ m 0 -computable family, m > 1, possessing a Friedberg \({\Sigma }_{\mathrm{m}}^{0}\) Σ m 0 -computable numbering, for any n > 1 possesses an ideal that also contains exactly n minimal elements specified by positive undecidable numberings.