Enumeration Degrees of the Bounded Sets

Abstract

The term ``total enumeration degree'' is related to the fact that the e-degree is total if and only if it contains a graph of some total function. In a number of works by the author and a group of mathematicians from the University of Wisconsin-Madison, the so-called ``graph-cototal enumeration degrees'' were considered, i.e. e-degrees containing the complement of the graph of some total function $f(x)$. In this article, the next step is taken -- the enumeration degrees of sets bounded from above or below by a graph of a total function are considered. More precisely, the set A is bounded from above if $A=\\{\\langle x,y\\rangle:y < f(x)\\}$ for some total function $f(x)$ and the set A is bounded from below if $A=\\{\\langle x,y\\rangle:y > f(x)\\}$ for some total function $f(x)$. The article presents a number of results showing the existence of nontotal enumeration degrees containing bounded sets, and the constructed e-degrees are quasi-minimal. An important result is the one stating that bounded sets have the Friedberg property related to the jump inversion.