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.

[1] The technique was developed in the 1950s by its namesake, French logician Roland Fraïssé.

[2] The main point of Fraïssé's construction is to show how one can approximate a (countable) structure by its finitely generated substructures.

satisfies certain properties (described below), then there exists a unique countable structure

, called the Fraïssé limit of

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.

-structure, we mean a logical structure having signature

, is the class of all finitely generated substructures of

that is the age of some structure satisfies the following two conditions: Hereditary property (HP) Joint embedding property (JEP) As above, we noted that for any

Fraïssé proved a sort-of-converse result: when

is any non-empty, countable set of finitely generated

-structures that has the above two properties, then it is the age of some countable structure.

Amalgamation property (AP) Essential countability (EC) In that case, we say that K is a Fraïssé class, and there is a unique (up to isomorphism), countable, homogeneous structure

[5] This structure is called the Fraïssé limit of

of all finite linear orderings, for which the Fraïssé limit is a dense linear order without endpoints (i.e. no smallest nor largest element).

, i.e. the rational numbers with the usual ordering.

[1] For any prime p, the Fraïssé limit of the class of finite fields of characteristic p is the algebraic closure

The Fraïssé limit of the class of finite abelian p-groups is

(the direct sum of countably many copies of the Prüfer group).

The Fraïssé limit of the class of all finite abelian groups is

The Fraïssé limit of the class of all finite groups is Hall's universal group.

The Fraïssé limit of the class of nontrivial finite Boolean algebras is the unique countable atomless Boolean algebra.

under consideration is called uniformly locally finite if for every

is uniformly locally finite, then the Fraïssé limit of

is finite, and consists only of relations and constants, then

is uniformly locally finite automatically.

For example, the class of finite dimensional vector spaces over a fixed field is always a Fraïssé class, but it is uniformly locally finite only if the field is finite.

The class of finite Boolean algebras is uniformly locally finite, whereas the classes of finite fields of a given characteristic, or finite groups or abelian groups, are not, as 1-generated structures in these classes may have arbitrarily large finite size.