A Borel reductibility theory for classes of countable structures

Abstract

AbstractWe introduce a reducibility preordering between classes of countable structures, each class containing only structures of a given similarity type (which is allowed to vary from class to class). Though we sometimes work in a slightly larger context, we are principally concerned with the case where each class is an invariant Borel class (i.e. the class of all models, with underlying set = ω, of an Lω1ω sentence; from this point of view, the reducibility can be thought of as a (rather weak) sort of Lω1ω-interpretability notion). We prove a number of general results about this notion, but our main thrust is to situate various mathematically natural classes with respect to the preordering, most notably classes of algebraic structures such as groups and fields.