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Mathematical logic and foundations


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Classification Theory for Abstract Elementary Classes

Saharon Shelah

An abstract elementary class is a class of structures of the same vocabulary (like a class of rings or a class of fields), with a partial order that generalises the relation "A is a substructure (or an elementary substructure) of B". The requirements are that the class is closed under isomorphism, and that isomorphic structures have isomorphic) generalised) substructures; we also require that our classes share some of the most basic properties of elementary classes, like closure under unions of increasing chains of substructures.

We would like to classify this general family; in the sense of proving dichotomies: either we can understand the structure of all models in our class or there are many to some extent. More specifically, we would like to generalise the theory about categoricity and superstability to this context.

24 April 2009

978-1-904987-71-0

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