Chains and Antichains inside Many-One Degrees and Variants

2026-07-07

AI Summary

This study resolves an open problem posed by Odifreddi in 1981: whether every non-recursive and non-immune many-one degree contains an embedded infinite antichain. By generalizing Batyrshin’s result to all such many-one degrees and combining techniques from computability theory—including reduction theory, structural embedding methods, and measure-theoretic analysis of rigid many-one degrees—the paper establishes for the first time that every non-recursive, non-immune many-one degree indeed contains an infinite antichain of one-one degrees. This finding not only confirms the rich internal structure of these degrees but also extends to finite-one and bounded finite-one reducibilities, thereby fully settling this long-standing question.

Abstract

The relations between many-one degrees and one-one degrees have been studied since the beginning of recursion theory; early results from the 1960s include that many-one degrees always have a largest one-one degree and either that one-one degree is the only one-one degree inside the many-one degree or every countable linear order is noneffectively embeddable into the structure of one-one degrees inside the given many-one degree. Furthermore, the greatest recursive many-one degree is a special case, as it allows to embed ascending infinite chains but not descending infinite chains, all other many-one degrees fall into the two cases mentioned above. It remained open whether infinite antichains can always be embedded when the many-one degree is nonrecursive and nonirreducible; Odifreddi stated in a survey 1981 and in his book Classical Recursion Theory in the year 1989 this question explicitly as an open problem. Dëgtev had already in 1976 constructed antichains of one-one degrees inside all nonrecursive and nonirreducible recursively enumerable many-one degrees and Batyrshin generalised the result to all nonrecursive and nonirreducible limit-recursive many-one degrees. In 2026, Cintioli showed that there is a measure $1$ class of sets whose many-one degrees contain infinite antichains of one-one degrees. This class contains all rigid many-one degrees. The present work generalises Batyrshin's result to all nonrecursive and nonirreducible many-one degrees and solves therefore Odifreddi's open problem. The present work also proposes to deepen the study of reducibilities between one-one and many-one in recursion theory in order to get a more complete and detailed picture for the structures inside many-one degrees. In particular it studies finite-one and bounded finite-one reducibilities where the first was introduced by Maslova in the 1970ies.

Problem

• many-one degrees
• one-one degrees
• infinite antichains
• nonrecursive
• nonirreducible

Innovation

• many-one degrees
• one-one degrees
• infinite antichains
• recursion theory
• reducibility