Tame abstract elementary class

In model theory, a discipline within the field of mathematical logic, a tame abstract elementary class is an abstract elementary class (AEC) which satisfies a locality property for types called tameness. Even though it appears implicitly in earlier work of Shelah, tameness as a property of AEC was first isolated by Grossberg and VanDieren, who observed that tame AECs were much easier to handle than general AECs.

Let K be an AEC with joint embedding, amalgamation, and no maximal models. Just like in first-order model theory, this implies K has a universal model-homogeneous monster model ${\displaystyle {\mathfrak {C}}}$. Working inside ${\displaystyle {\mathfrak {C}}}$, we can define a semantic notion of types by specifying that two elements a and b have the same type over some base model ${\displaystyle M}$ if there is an automorphism of the monster model sending a to b fixing ${\displaystyle M}$ pointwise (note that types can be defined in a similar manner without using a monster model). Such types are called Galois types.

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