Positive Definability Patterns

TL;DR

This paper extends Hrushovski's definability patterns to positive logic, constructing canonical models called cores from type spaces of models of an h-universal theory, which are independent of specific models.

Contribution

It introduces a reformulation of definability patterns in positive logic and constructs canonical core models that embed into all type spaces, demonstrating model independence.

Findings

Canonical models (cores) exist for h-universal theories.

These cores embed into every type space of the theory.

The construction is independent of the choice of sufficiently saturated models.

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.