Age (model theory)

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In model theory, a branch of mathematical logic, the age of a structure (or model) A is the class of all finitely generated structures which are embeddable in A (i.e. isomorphic to substructures of A). This concept is central in the construction of a Fraïssé limit.

The main point of Fraïssé's construction is to show how one can approximate a structure by its finitely generated substructures. Thus for example the age of any dense linear order without endpoints (DLO), \langle\mathbb{Q},<\rangle is precisely the set of all finite linear orderings, which are distinguished up to isomorphism only by their size. Thus the age of any DLO is countable. This shows in a way that a DLO is a kind of limit of finite linear orderings.

One can easily see that any class K which is an age of some structure satisfies the following two conditions:

Hereditary property
If A ∈ K and B is a finitely generated substructure of A, then B is isomorphic to a structure in K
Joint embedding property
If A and B are in K then there is C in K such that both A and B are embeddable in C.

Fraïssé proved that when K is any non-empty countable set of finitely generated σ-structures (with σ a signature) which has the above two properties, then it is an age of a countable structure.

Furthermore, suppose that K happens to satisfy the following additional property.

Amalgamation property
For any structures A, B and C in K such that A is embeddable in both B and C, there exists D in K to which B and C are both embeddable by embeddings which coincide on the image of A in both structures.

In that case there is a unique up to isomorphism structure which is countable, has the age K and is homogeneous. Homogeneous means here that any isomorphism between two finitely generated substructures can be extended to an automorphism. Again an example of this situation could be the ordered set of rational numbers \langle\mathbb{Q},<\rangle. It is the unique (up to isomorphism) homogenous countable structure whose age is the set of all finite linear orderings. Note that the ordered set of natural numbers \langle\mathbb{N},<\rangle has the same age as a DLO, but it is not homogenous since if we map {1, 3} to {5, 6}, it would not extend to any automorphism f since there should be an element between f(1) = 5 and f(3) = 6. The same applies to integers.

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