Positive Definability Patterns

Abstract

We reformulate Hrushovski's definability patterns from the setting of first order logic to the setting of positive logic. Given an h-universal theory T we put two structures on the type spaces of models of T in two languages, \mathcal{L} and \mathcal{L}_{\pi}. It turns out that for sufficiently saturated models, the corresponding h-universal theories \mathcal{T} and \mathcal{T}_{\pi} are independent of the model. We show that there is a canonical model \mathcal{J} of \mathcal{T}, and in many interesting cases there is an analogous canonical model \mathcal{J}_{\pi} of \mathcal{T}_{\pi}, both of which embed into every type space. We discuss the properties of these canonical models, called cores, and give some concrete examples.