Effective separation and reduction properties of Borel Wadge classes

In this paper, we use our descriptions of Borel Wadge classes from [A. R. Day, N. Greenberg, M. Harrison-Trainor and D. Turetsky, An effective classification of Borel Wadge classes] to characterize those Borel Wadge classes that have the separation property, and those that have the reduction property. Our analysis shows that both properties are equivalent to their effective versions. To do so, we give a characterization of containment between Borel Wadge classes based on their descriptions, and give a direct proof that all such classes admit admissible descriptions.