Cupping and jump classes in the computably enumerable degrees

Abstract

We show that there is a cuppable c.e. degree, all of whose cupping partners are high. In particular, not all cuppable degrees are -cuppable, or indeed cuppable for any n, refuting a conjecture by Li. On the other hand, we show that one cannot improve highness to superhighness. We also show that the -cuppable degrees coincide with the array computable-cuppable degrees, giving a full understanding of the latter class.