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, $L$ and $L_{\pi }$. It turns out that for sufficiently saturated models, the corresponding h-universal theories $T$ and $T_{\pi }$ are independent of the model. We show that there is a canonical model $J$ of $T$, and in many interesting cases there is an analogous canonical model $J_{\pi }$ of $T_{\pi }$, both of which embed into every type space. We discuss the properties of these canonical models, called cores, and give some concrete examples.