The authors investigate the elementary theories of these degrees and examine their structural properties. They discuss various aspects related to the complexity of computations and the classification of degrees based on computability theory.**What are d-c.e. and n-c.e. degrees?**

The d-c.e. (difference computably enumerable) degrees and n-c.e. (nondifference computably enumerable) degrees are two classes of degrees in computability theory that measure the complexity of computations. They provide a way to classify sets based on their computability properties.**What are the elementary theories of d-c.e. and n-c.e. degrees?**

The authors investigate the elementary theories of d-c.e. and n-c.e. degrees, which refer to the logical theories that capture the fundamental properties and relationships between these degrees. These theories include theorems, axioms, and logical principles specific to the study of these degrees.**What are the structural properties of d-c.e. and n-c.e. degrees?**

The authors explore the structural properties of d-c.e. and n-c.e. degrees, which involve examining the characteristics and relationships between different degrees within these classes. This include studying properties such as degree spectra, degrees of unsolvability, and the structure of the degrees under various operations.**How do d-c.e. and n-c.e. degrees contribute to our understanding of computability theory?**

The authors discuss the significance of d-c.e. and n-c.e. degrees in the broader context of computability theory. These classes of degrees provide insights into the complexity of computations and the classification of sets based on their computability properties, thus contributing to our understanding of what can and cannot be effectively computed.