A strengthened form of Dëgtev's argument and $D$-maximal many-one degrees

Abstract

Dëgtev proved that if a noncomputable computably enumerable set \(U\) satisfies his property \((R)\) and the many-one degree of \(U\) contains no simple set, then the \(1\)-degree of \(U\) is minimal. We show that his proof yields the stronger conclusion that, for every noncomputable \(A\lem U\), \[ U\le_1 A. \] In particular, the injectivity of the reduction \(A\le_1 U\) with which Dëgtev begins his proof is not needed. Using the Cholak--Gerdes--Lange classification, we show that for a noncomputable \(D\)-maximal set \(D\), the hypothesis that its many-one degree contains no simple set is equivalent to \(D\) being of one of Types~3--10. Hence, for every such \(D\) and every noncomputable \(A\lem D\), one has \(D\le_1 A\). Together with Maslova's theorem for the simple-set cases, this recovers as a corollary the existence of a least finite-one degree in every noncomputable c.e.\ many-one degree containing a \(D\)-maximal set.

arXiv comment7 pages. Substantially revised and shortened. Following an observation of V. Yu. Shavrukov, Types 3--10 are recast via a strengthened reading of Dëgtev's 1976 argument. Companion to arXiv:2604.10879