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2009 Immunity and Non-Cupping for Closed Sets
Douglas Cenzer, Takayuki Kihara, Rebecca Weber, Guohua Wu
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Tbilisi Math. J. 2: 77-94 (2009). DOI: 10.32513/tbilisi/1528768843

Abstract

We extend the notion of immunity to closed sets and to $\Pi^0_1$ classes in particular in two ways: immunity meaning the corresponding tree has no infinite computable subset, and tree-immunity meaning it has no infinite computable subtree. We separate these notions from each other and that of being special, and show separating classes for computably inseparable c.e. sets are immune and perfect thin classes are tree-immune. We define the notion of prompt immunity and construct a positive-measure promptly immune $\Pi^0_1$ class. We show that no immune-free $\Pi^0_1$ class $P$ cups to the Medvedev complete class $\mathrm{DNC}$ of diagonally noncomputable sets, where $P$ cups to $Q$ in the Medvedev degrees of $\Pi^0_1$ classes if there is a class $R$ such that the product $P \otimes R \equiv_\textrm{M} Q$. We characterize the interaction between (tree-)immunity and Medvedev meet and join, showing the (tree-)immune degrees form prime ideals in the Medvedev lattice. We show that every random closed set is immune and not small, and every small special class is immune.

Funding Statement

This research was partially supported by NSF grants DMS-0554841 and DMS-0652372. Weber was partially supported by NSF grant DMS-0652326 and Wu was partially supported by NTU grant RG58/06 MS21100023.

Citation

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Douglas Cenzer. Takayuki Kihara. Rebecca Weber. Guohua Wu. "Immunity and Non-Cupping for Closed Sets." Tbilisi Math. J. 2 77 - 94, 2009. https://doi.org/10.32513/tbilisi/1528768843

Information

Received: 28 September 2009; Revised: 25 November 2009; Accepted: 24 December 2009; Published: 2009
First available in Project Euclid: 12 June 2018

zbMATH: 1208.03047
MathSciNet: MR2610163
Digital Object Identifier: 10.32513/tbilisi/1528768843

Subjects:
Primary: 03D45
Secondary: 03D25, 03D30, 03D32

Rights: Copyright © 2009 Tbilisi Centre for Mathematical Sciences

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