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• W2023215355 abstract "It is shown that there exist r.e. degrees other than <bold>0</bold> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold 0 prime> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn mathvariant=bold>0</mml:mn> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>mathbf {0}’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which have a greatest r.e. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree. This solves an old question of Rogers and Jockusch. We call such degrees <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula><italic>-topped</italic>. We show that there exist incomplete <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-topped degrees above any low r.e. degree, but also show that no nonzero low degree is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-topped. It then follows by known results that all incomplete <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-topped degrees are low<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=Subscript 2> <mml:semantics> <mml:msub> <mml:mi /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> but not low. We also construct cappable nonzero <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-topped r.e. degrees and examine the relationships between <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=application/x-tex>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-topped r.e. degrees and high r.e. degrees. Finally, we give an analysis of the “local” relationships of r.e. sets under various strong reducibilities. In particular, we analyze the structure of r.e. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=wtt hyphen> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>wtt-</mml:mtext> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {wtt-}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=tt> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>tt</mml:mtext> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {tt}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degrees within a single r.e. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>T</mml:mtext> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {T}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree. We show, for instance, that there is an r.e. degree which contains a greatest r.e. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=wtt hyphen> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>wtt-</mml:mtext> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {wtt-}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree and a least r.e. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=tt> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>tt</mml:mtext> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {tt}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree yet does not consist of a single r.e. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=wtt> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>wtt</mml:mtext> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {wtt}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree. This depends on a new construction of a nonzero r.e. <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>T</mml:mtext> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {T}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree with a least <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=tt> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>tt</mml:mtext> </mml:mrow> <mml:annotation encoding=application/x-tex>{text {tt}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-degree, which proves to have several further applications." @default.
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• W2023215355 date "1987-01-01" @default.
• W2023215355 modified "2023-10-18" @default.
• W2023215355 title "T-degrees, jump classes, and strong reducibilities" @default.
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