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.