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 $M\prec_K N$
(A3) downward Loewenheim-Skolem. There exist a cardinal $LS(K) = LS(K,\prec_K)\geq |L(K)|+\aleph_0$ such that $\forall M\in K$, $\forall A\subset |M|$, $\exists N\in K$ with $A\subset |N|$, $N\prec_K M$, $\|N\|\leq |A|+LS(K)$.
(A4) (Tarski-Vaught chain condition) for every regular cardinal $\mu$
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The usual elementary classes, i.e. the classes of the form $K = Mod(T)$ for a first-order theory $T$, are abstract elementary with respect to the relation $\prec_K$ of being an elementary submodel, with ${|LS(K)|} = {|L(T)|}+\aleph_0$ ($L(T)$ is the underlying language of the theory $T$).
Last revised on August 3, 2016 at 00:13:21. See the history of this page for a list of all contributions to it.