Fraïssé Limits

Abstract

[en] One of the basic premises of model theory is the formalization of objects and the properties they satisfy as independent entities, called structures and theories, respectively. It goes without saying that this dichotomy leaves room for a more thorough description and handling of the classes of such objects, as well as their theories. Starting from a determinate set of properties from some field of mathematics, we may seek their translation into first-order logic and ask what kind of structures satisfy them. This will narrow down the possible characteristics of both our objects and their class as a whole. The question, as always, concerns what we can achieve from this approach. We can ask ourselves how to relate these structures beyond them fulfilling the formulas from a same theory. Under certain conditions of their class, Fraïssé [Fra54] proved the existence of a countable structure which can embed any of the structures and satisfies some additional homogeneity criteria. In other words, we obtain new information about the maps between the elements of the class. The constructions we will delve into nowadays constitute a paramount implement in some areas of model theory. Beyond this, their potential is manifested in their capability to attain various structures, which are pivotal in other fields of mathematics, solely through the application of a single method to different classes of lesser structures. Our goal throughout this work is to describe the properties which support Fraïssé’s framework, present the centrals results of his theory and provide other specific results for particular instances of structure classes. In this manner, we will be able to review and study some of the most celebrated examples of limit structures, while detailing a selection of their peculiarities. As any other widespread result, Fraïssé’s method is outlined in multiple reference texts on model theory, such as Hodges’ [Hod93], Evans’ [Eva94] or Tent and Ziegler’s [TZ12]. We will follow the formalization of the latter to provide a background for the central theorem and point out some of its results. Some other properties (including the central result or the amalgamation of finite Boolean algebras) were discussed over unpublished materials from Casanovas [Cas09][Cas14][Cas22] and during some meetings with him.