nLab
Galois type

Idea

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

Consider the type (in model theory) $\mathrm{tp}(a/M)$ for a first order theory in larger uncountable saturated model $M\prime$ (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 ${\mathrm{Aut}}_M(M\prime )$ of $M\prime$ 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 $(K,<)$ be an abstract elementary class of structures.

…

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