On definability of c.e. degrees in the 2-c.e. degree structures

In our talk we will consider the degree structures induced by various algorithmic reducibilities within the finite levels of the Ershov hierarchy. The most attention will be paid for the first two levels known as c.e. and 2-c.e. degrees. The problem of definability of c.e. Turing degrees in the partial order of 2-c.e. Turing degrees was posed by Cooper and Li in 2002. Nowadays it is one the most intriguing questions about model-theoretic properties of 2-c.e. Turing degrees. In our talk we will consider these and related questions for different settings. Thus, we will show how the problem can be solved for the finite levels of the Ershov hierarchy for $m$-degrees. In the setting of Turing degrees we will show how weaker versions of these questions can be solved. Additionally we review and outline possible approached for definability of c.e. degrees in the structures of 2-c.e. $wtt$-degrees and 2-c.e. Turing degrees, respectively.