In model theory abstract elementary classes are a vast generalizations of elementary classes of structures beyond first-order theories (e.g. for the infinitary logic ) as introduced by Saharon Shelah. Its theory is also more general than the homogeneous model theory.
An abstract elementary class is a nonempty class of structures for a given signature with language , that is closed under isomorphisms and equipped with a strong substructure relation (strong substructure relation means that if and is a substructure, then ) that is a partial order satisfying the axioms on union of chains (Tarski-Vaught), coherence and downward Loewenheim-Skolem properties. More precisely, is a partial order such that
(A0) if , then
(A1) (closure under isomorphisms)
(a) and an structure with , then
(b) if , , , , with then
(A2) for , if , , and , then
(A3) downward Loewenheim-Skolem. There exist a cardinal such that , , with , , .
(A4) (Tarski-Vaught chain condition) for every regular cardinal
The usual elementary classes, i.e. the classes of the form for a first-order theory , are abstract elementary with respect to the relation of being an elementary submodel, with ( is the underlying language of the theory ).
- related Lab entries: amalgamation
- wikipedia abstract elementary class
- Saharon Shelah, Classification theory for elementary abstract classes I, II, Studies in Logic (London), 18, 20, College Publications, London 2009
- Tibor Beke, Jiri Rosicky, Abstract elementary classes and accessible categories, Annals of Pure and Applied Logic 163 (2012) 2008-2017, arxiv/1005.2910
- John Baldwin, Categoricity, Amer. Math. Soc. 2011, pdf
- D. W. Kueker, Abstract elementary classes and infinitary logic, Ann. Pure Appl. Logic 156 (2008), 274-286.
- M. J. Lieberman, Topological and category-theoretic aspects of abstract elementary classes, Thesis, The University of Michigan 2009, pdf; defense slides pdf; Category theoretic aspects of abstract elementary classes, Annals Pure Appl. Logic 162 (2011), 903-915; A topology for Galois types in AECs, arxiv/0906.3573