We 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.

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ABSTRACT. LEMMA 1. If X is a cardinal with cf A > u>, then Dx implies that there is a A+-Aronszajn tree with an ui-ascent path, i.e. a sequence (xa : a < A+) with each xa = (i" : n < ui) a one-to-one sequence from Ta, such that for all a < ß < A+, z° precedes x" in the tree order for sufficiently large n.

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