Merges of Smooth Classes and their Properties

Abstract

We work with classes of finite structures called smooth classes, a notion which originates from Hrushovski’s constructions in [Hru93]. Under sufficient conditions, these classes have generic limits and can be thought of as generalizations of Fraisse classes and limits. Given two smooth classes, we ask whether we can merge the two classes into one class with a generic limit in a new language. This is an operation one can easily do when dealing with two traditional Fraisse classes, but smooth classes add complexity. We define these merges of classes and study the model theoretic properties of these merges and their generics. Namely, we give a characterization of the generic of the merged class relative to the generics of the original classes, strengthening a result in [EHN19]. We also discuss the properties of the generic of the merged class being atomic, saturated, and ω-categorical relative to the original generics. Merges of smooth classes have a historical place within the realm of structural Ramsey theory and the study of the Hrushovski property (EPPA). We extend merges of smooth classes tothis context, and, generalizing work done in ([EHN19], [EHN21], [HKN22]), we give examples of merges of smooth classes with the Ramsey and EPPA properties.