Fraïssé limit

In mathematical logic, specifically in the discipline of model theory, the Fraïssé limit (also called the Fraïssé construction or Fraïssé amalgamation) is a method used to construct (infinite) mathematical structures from their (finite) substructures. It is a special example of the more general concept of a direct limit in a category. The technique was developed in the 1950s by its namesake, French logician Roland Fraïssé.

The main point of Fraïssé's construction is to show how one can approximate a (countable) structure by its finitely generated substructures. Given a class ${\displaystyle \mathbf {K} }$ of finite relational structures, if ${\displaystyle \mathbf {K} }$ satisfies certain properties (described below), then there exists a unique countable structure ${\displaystyle \operatorname {Flim} (\mathbf {K} )}$, called the Fraïssé limit of ${\displaystyle \mathbf {K} }$, which contains all the elements of ${\displaystyle \mathbf {K} }$ as substructures.

The general study of Fraïssé limits and related notions is sometimes called Fraïssé theory. This field has seen wide applications to other parts of mathematics, including topological dynamics, functional analysis, and Ramsey theory.