A Computably Enumerable $tt$-Degree Without Computably Enumerable Irreducible $m$-Degrees

TL;DR

This paper constructs a computably enumerable $tt$-degree that contains no c.e. irreducible $m$-degrees, answering a long-standing open problem negatively and showing the limits of classical theorems.

Contribution

It provides the first example of a c.e. $tt$-degree lacking any c.e. irreducible $m$-degree, disproving a previous conjecture and demonstrating the optimality of Jockusch's 1969 theorem.

Findings

Existence of a c.e. $tt$-degree without c.e. irreducible $m$-degrees

The unique c.e. $m$-degree within such a $tt$-degree consists of simple sets

Classical 1969 theorem by Jockusch Jr. is shown to be strictly optimal

Abstract

In this paper, we provide a negative solution to Problem 3 formulated by P.~Odifreddi in his survey articles \textit{``Strong Reducibilities''} (1981) and \textit{``Reducibilities''} (1999). The problem asks whether every computably enumerable (c.e.) $tt$-degree contains a c.e.\ \textit{irreducible} $m$-degree (i.e., an $m$-degree consisting of only one $1$-degree). We answer this question in the negative by proving the existence of a c.e.\ $tt$-degree that does not contain any c.e.\ irreducible $m$-degree. Our proof relies on the structural properties of c.e.\ semirecursive sets with a rigid complement, originally constructed by A.~N.~Degtev. We show that the unique c.e.\ $m$-degree contained within the $tt$-degree of such a set consists of simple sets, which cannot be cylinders, and therefore necessarily splits into multiple $1$-degrees. Furthermore, our result demonstrates that a classical 1969 theorem by C.~G.~Jockusch Jr. -- which guarantees the existence of an irreducible $m$-degree within every c.e.\ $tt$-degree -- is strictly optimal and cannot be generally strengthened to require such an $m$-degree to be computably enumerable.