nLab
abstract elementary class

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 $L_{{\omega }_1,\omega }$) as introduced by Saharon Shelah. Its theory is also more general than the homogeneous model theory.

An abstract elementary class is a nonempty class $K$ of structures for a given signature with language $L(K)$, that is closed under isomorphisms and equipped with a strong substructure relation ${\prec }_K$ (strong substructure relation means that if $M{\prec }_{K}N$ and $M_0\subset M$ is a substructure, then $M_0{\prec }_{K}N$) that is a partial order satisfying the axioms on union of chains (Tarski-Vaught), coherence and downward Loewenheim-Skolem properties. More precisely, ${\prec }_K$ is a partial order such that

(A0) if $M,N\in K$, $M{\prec }_{K}N$ then $M\subset N$

(A1) (closure under isomorphisms)

• (a) $M\in K$ and $N$ an $L(K)$ structure with $N\cong M$, then $N\in K$

• (b) if $N_1,N_2,M_1,M_2\in K$, $f_i:N_i\cong M_i$, $i=1,2$, $f_1\subset f_2$, with $M_1{\prec }_K{M}_2$ then $N_1{\prec }_K{N}_2$

(A2) for $M,N,P\in K$, if $M{\prec }_{K}P$, $N{\prec }_{K}P$, and $M\subset N$, then $M{\prec }_{K}P$

(A3) downward Loewenheim-Skolem. There exist a cardinal $\mathrm{LS}(K)=\mathrm{LS}(K,{\prec }_K)\ge \mid L(K)\mid +{\aleph }_0$ such that $\forall M\in K$, $\forall A\subset \mid M\mid$, $\exists N\in K$ with $A\subset \mid N\mid$, $N{\prec }_{K}M$, $\parallel N\parallel \le \mid A\mid +\mathrm{LS}(K)$.

(A4) (Tarski-Vaught chain condition) for every regular cardinal $\mu$

….

The usual elementary classes, i.e. the classes of the form $K=\mathrm{Mod}(T)$ for a first-order theory $T$, are abstract elementary with respect to the relation ${\prec }_K$ of being an elementary submodel, with $\mid \mathrm{LS}(K)\mid =\mid L(T)\mid +{\aleph }_0$ ($L(T)$ is the underlying language of the theory $T$).

• related $n$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, 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