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Cone Avoidance and the No-Least-Join Theorem

  • Patrizio Cintioli
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Abstract

We give a column-avoidance modification of the no-least-join construction of Downey, Greenberg, Lewis, and Montalbán. We show that if C is computably enumerable, A,B<T C, and Anot≤T B, then for every noncomputable set E there is a set X≤T C such that B≤T A⊕ X and Enot≤T X. Consequently, for the corresponding Turing degrees a,b,c, the degrees x≤ c satisfying b≤ a∨ x have no nonzero common lower bound; equivalently, no nonzero Turing cone contains all of them. This gives a negative answer to their Question 1.19.

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Cite

BibTeXRIS
BibTeX
@misc{cintioli2026cone,
  title = {{Cone Avoidance and the No-Least-Join Theorem}},
  author = {Patrizio Cintioli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02027},
  eprint = {2610.02027},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02027},
}
RIS
TY  - UNPB
TI  - Cone Avoidance and the No-Least-Join Theorem
AU  - Patrizio Cintioli
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.02027
UR  - https://arxiv.org/abs/2610.02027
AB  - We give a column-avoidance modification of the no-least-join construction of Downey, Greenberg, Lewis, and Montalbán. We show that if C is computably enumerable, A,B<_T C, and Anot≤_T B, then for every noncomputable set E there is a set X≤_T C such that B≤_T A⊕ X and Enot≤_T X. Consequently, for the corresponding Turing degrees a,b,c, the degrees x≤ c satisfying b≤ a∨ x have no nonzero common lower bound; equivalently, no nonzero Turing cone contains all of them. This gives a negative answer to their Question 1.19.
ER  -