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\section*{abstract elementary class}

In [[model theory]] abstract elementary classes are a vast generalizations of [[elementary classes]] of [[structure in model theory|structures]] beyond [[first-order theory|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 \textbf{abstract elementary class} is a \textbf{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)

\begin{itemize}%
\item (a) $M\in K$ and $N$ an $L(K)$ structure with $N\cong M$, then $N\in K$


\item (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 $N_1\prec_K N_2$



\end{itemize}
(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 $\mu$

\ldots{}.

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 $T$).

\begin{itemize}%
\item related $n$Lab entries: [[amalgamation]]
\item wikipedia \href{http://en.wikipedia.org/wiki/Abstract_elementary_class}{abstract elementary class}
\item [[Saharon Shelah]], \emph{Classification theory for elementary abstract classes I, II}, Studies in Logic (London), \textbf{18}, \textbf{20}, College Publications, London 2009
\item John Baldwin, \emph{Categoricity}, Amer. Math. Soc. 2011, \href{http://www.math.uic.edu/~jbaldwin/pub/AEClec.pdf}{pdf}
\item D. W. Kueker, \emph{Abstract elementary classes and infinitary logic}, Ann. Pure Appl. Logic \textbf{156} (2008), 274-286.

\end{itemize}
AECs can also be essentially identified with [[accessible categories]] in which all morphisms are [[monomorphisms]]. Some recent papers which study them from this viewpoint include:

\begin{itemize}%
\item [[Tibor Beke]], [[Jiří Rosický]] \emph{Abstract elementary classes and accessible categories}, Annals of Pure and Applied Logic \textbf{163} (2012) 2008-2017, \href{http://arxiv.org/abs/1005.2910}{arxiv/1005.2910}


\item Michael Lieberman, Jiří Rosický, Sebastien Vasey, \emph{Internal sizes in μ-abstract elementary classes}, \href{https://arxiv.org/abs/1708.06782}{arxiv}


\item Michael Lieberman, Jiří Rosický, Sebastien Vasey, \emph{Set-theoretic aspects of accessible categories}, \href{https://arxiv.org/abs/1902.06777}{arxiv}


\item M. J. Lieberman, \emph{Topological and category-theoretic aspects of abstract elementary classes}, Thesis, The University of Michigan 2009, \href{http://deepblue.lib.umich.edu/bitstream/2027.42/63854/1/liebermm_1.pdf}{pdf}; defense slides \href{http://www.math.upenn.edu/~mlieb/defense.pdf}{pdf}; \emph{Category theoretic aspects of abstract elementary classes}, Annals Pure Appl. Logic \textbf{162} (2011), 903-915; \emph{A topology for Galois types in AECs}, \href{http://arxiv.org/abs/0906.3573}{arxiv/0906.3573}



\end{itemize}
[[!redirects abstract elementary classes]]



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