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Positive Definability Patterns

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}_π. It turns out that for sufficiently saturated models, the corresponding h-universal theories \mathcal{T} and \mathcal{T}_π 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}_π of \mathcal{T}_π, both of which embed into every type space. We discuss the properties of these canonical models, called cores, and give some concrete examples.