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 (up to isomorphism) finitely generated structures, which are embeddable in A. This concept is central in the so called Fraïssé construction. The main point of this construction is to show how one can approximate a structure by its finitely generated substructures. Thus for example the age of the dense linear ordering without enpoints (DLO), \langle\mathbb{Q},<\rangle is precisely the set of all finite linear orderings, which are distiguished up to isomorphism only by their size. Thus the age of DLO is countable. This shows in a way that DLO is a kind of a limit of finite linear orderings.

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

Hereditary property: if A\in\mathbb{K} and B is a finitely generated substructure of A, then B is isomorphic to a structure in \mathbb{K}

Joint embedding property: if A and B are in \mathbb{K} then there is C in \mathbb{K} such that both A and B are embeddable in C.

Fraïssé proved in his construction that when \mathbb{K} is any non-empty countable set of finitely generated L-structures (with L a first-order language) which has the above two properties, it is an age of a countable structure.

Furthermore suppose that \mathbb{K} happens to satisfy an additional property, called amalgamation property, that is for any structures A, B and C in \mathbb{K} such that A is embeddable in both B and C, there exists D in \mathbb{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 \mathbb{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 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. Same applies to integers.

    [edit] References

    1. Wilfrid Hodges, A shorter model theory (1997) Cambridge University Press ISBN 0-521-58713-1
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