The Local Enumeration Degree Structures

Marat Arslanov

In this paper, we study first-order theories of the partial orderings of the Σ_{2}, Δ_{2}, total and n-c. e. enumeration degrees for different n > 1. We describe some recent results related to these structures. They consist of two parts. In the first part, we consider sentences in an expanded language which distinguish n-c. e. degrees for different n > 1 and study the complexity of these sentences. In the second part, we study algebraic properties of these structures (diamond embedding, cupping and capping properties, splitting of a given degree above another degree etc).

(Joint work with S.B. Cooper and I.Sh. Kalimullin)

Address: Kazan State University, Department of Mathematics, Kremlevskaja 18, Kazan 420008, Russia

E-mail: Marat.Arslanov@ksu.ru