A hierarchy for cuppable degrees

Abstract

We say that a computably enumerable (c.e., for short) degree $\mathbf{a}$ is cuppable if there is a c.e. degree $\mathbf{b} \neq \mathbf{0}'$ such that $\mathbf{a} \vee \mathbf{b} \mathbf{0}'$. A c.e. degree $\mathbf{a}$ is called $low_{n}$-cuppable, $n \gt 0$, if there is a $\mathrm{low}_{n}$ c.e. degree $\mathbf{1}$ such that $\mathbf{a} \vee \mathbf{1} \mathbf{0}'$. Let $\mathbf{LC}_{n}$, be the set of all $\mathrm{low}_{n}$-cuppable c.e. degrees. In this paper, we show that $\mathbf{LC}_{1} \subset \mathbf{LC}_{2} \subseteq \mathbf{LC}_{3} \subseteq \cdots$, so giving a hierarchy for a class of cuppable degrees.