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 ).