Definable relations in structures of Turing degrees

In this paper we investigate questions about the definability of classes of $n$-computably enumerable (c.e.) sets and degrees in the Ershov difference hierarchy. It is proved that the class of all c.e. sets it is definable under the set inclusion $\subseteq$ in all finite levels of the difference hierarchy. It is also proved the definability of all $m$-c.e. degrees in each higher level of the hierarchy. Besides, for each level $n$, $n\ge2$, of the hierarchy a definable non-trivial subset of $n$-c.e. degrees is established.