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• W2085333977 abstract "In this paper, we study effective monotonic approximations of sets and sequences of sets. We show that there is a sequence of sets which has no uniform computable monotonic approximation but has an <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold x> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-computable monotonic approximation for every hyperimmune degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold x> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also construct a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma 2 Superscript 0> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>Sigma ^0_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> set which is not limitwise monotonic but is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold x> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-limitwise monotonic relative to every non-zero <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Delta 2 Superscript 0> <mml:semantics> <mml:msubsup> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mn>2</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding=application/x-tex>Delta ^0_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold x> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that if a sequence of sets is uniformly limitwise monotonic in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold x> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all except countably many degrees <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold x> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {x}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then it has to be uniformly limitwise monotonic. Finally, we apply these results to investigate degree spectra of abelian groups, equivalence relations, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal alef 1> <mml:semantics> <mml:msub> <mml:mi mathvariant=normal>ℵ<!-- ℵ --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>aleph _1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categorical structures." @default.
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• W2085333977 date "2013-05-31" @default.
• W2085333977 modified "2023-09-27" @default.
• W2085333977 title "Limitwise monotonic sequences and degree spectra of structures" @default.
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