nLab Galois type

Idea

Galois types are a generalization of types for abstract elementary classes with amalgamation property.

Consider the type (in model theory) $tp(a/M)$ for a first order theory in larger uncountable saturated model $M'$ (of cardinality strictly larger than of the underlying language). By a standard result it can be identified with the orbit of $a$ under the automorphism group $Aut_M(M')$ of $M'$ fixing $M$ pointwise. This is taken as a definition for some infinitary theories in the case we have the monster model. A slightly weaker version can be rephrased in terms of amalgamation.

Definition

Let $(K,\lt)$ be an abstract elementary class of structures.

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Related entries

• amalgamation property

• type in model theory

References

• Rami Grossberg, Classification theory for abstract elementary classes (section 6), Logic and Algebra, ed. Yi Zhang, Contemporary Mathematics 302, AMS, (2002) 165–204, pdf
• Rami Grossberg, Monica VanDieren, Galois-stability for tame abstract elementary classes, Journal of Mathematical Logic 6, No. 1 (2006) 25–49, pdf