Are there any good references that go into some detail of known 'translations' between properties of the type space of a model and the model theoretic properties of the model? All I seem to find are the standard examples in terms of compactness and omitting types (= Baire category theorem).
I would be happy with any list of other known examples with references. For example, any elementary extension $f: N \rightarrow M$ induces a homomorphism of Lindenbaum rings/a continuous map of type spaces (with respect to some subset of parameters). Are there any known results on when such maps necessarily are induced by an elementary extension? Any partial result in this direction would be nice.