$m$-Rigidity and Finite-One Degrees Inside Typical Many-One Degrees
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911492226416640 |
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| author | Cintioli, Patrizio |
| author_facet | Cintioli, Patrizio |
| contents | In recent work, the notion of $m$-rigidity was introduced as a sufficient condition for the existence of infinite antichains of $1$-degrees inside many-one degrees. Motivated by a recent preprint of Richter, Stephan, and Zhang on finite-one degrees inside many-one degrees, we study the finite-one structure of the many-one degree of an $m$-rigid set.
First, combining bi-immunity of $m$-rigid sets with a theorem of Richter, Stephan, and Zhang, we show that for Lebesgue-almost every set $A$, and for a comeager class of sets $A$, the many-one degree $°_m(A)$ contains a least finite-one degree. Second, we prove that if $A$ is $m$-rigid, then $°_m(A)$ contains infinitely many pairwise incomparable finite-one degrees. More precisely, we construct representatives $B_S \equiv_m A$, indexed by computable sets $S$, such that $T \setminus S$ infinite implies $B_T \not\leq_{fin} B_S$. Third, inside a single finite-one degree we build a strict ascending chain \[ A_{(1)} <_1 A_{(2)} <_1 \cdots \] of $1$-degrees.
These results yield almost-sure and comeager partial answers to the first two open problems posed by Richter, Stephan, and Zhang. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02600 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $m$-Rigidity and Finite-One Degrees Inside Typical Many-One Degrees Cintioli, Patrizio Logic 03D30, 03D32 In recent work, the notion of $m$-rigidity was introduced as a sufficient condition for the existence of infinite antichains of $1$-degrees inside many-one degrees. Motivated by a recent preprint of Richter, Stephan, and Zhang on finite-one degrees inside many-one degrees, we study the finite-one structure of the many-one degree of an $m$-rigid set. First, combining bi-immunity of $m$-rigid sets with a theorem of Richter, Stephan, and Zhang, we show that for Lebesgue-almost every set $A$, and for a comeager class of sets $A$, the many-one degree $°_m(A)$ contains a least finite-one degree. Second, we prove that if $A$ is $m$-rigid, then $°_m(A)$ contains infinitely many pairwise incomparable finite-one degrees. More precisely, we construct representatives $B_S \equiv_m A$, indexed by computable sets $S$, such that $T \setminus S$ infinite implies $B_T \not\leq_{fin} B_S$. Third, inside a single finite-one degree we build a strict ascending chain \[ A_{(1)} <_1 A_{(2)} <_1 \cdots \] of $1$-degrees. These results yield almost-sure and comeager partial answers to the first two open problems posed by Richter, Stephan, and Zhang. |
| title | $m$-Rigidity and Finite-One Degrees Inside Typical Many-One Degrees |
| topic | Logic 03D30, 03D32 |
| url | https://arxiv.org/abs/2603.02600 |