The Infinity Project

We prove a preservation theorem for positive Horn definability inא0-categorical structures.In particular, we define and study a construction which we call the periodic power of a structure, anddefine a periomorphism of a structure to be a homomorphism from the periodic power of the structureto the structure itself. Our preservation theorem states that, over anא0-categorical structure, a relationis positive Horn definable if and only if it is preserved by all periomorphisms of the structure. We giveapplications of this theorem, including a new proof of the known complexity classification of quantifiedconstraint satisfaction on equality templates.