Spectrum hierarchies and subdiagonal functions

The spectrum of a first-order sentence is the set of cardinalities of its finite models. Relatively little is known about the subclasses of spectra that are obtained by looking only at sentences with a specific signature. In this paper we study natural subclasses of spectra and their closure properties under simple subdiagonal functions. We show that many natural closure properties turn out to be equivalent to the collapse of potential spectrum hierarchies. We prove all of our results using explicit transformations on first-order structures.