# Abstract elementary class

In model theory, a discipline within mathematical logic, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of an elementary substructure of an elementary class in first-order model theory. They were introduced by Saharon Shelah.[1]

## Definition

${\displaystyle \langle K,\prec _{K}\rangle }$, for ${\displaystyle K}$ a class of structures in some language ${\displaystyle L=L(K)}$, is an AEC if it has the following properties:

Note that we usually do not care about the models of size less than the Löwenheim–Skolem number and often assume that there are none (we will adopt this convention in this article). This is justified since we can always remove all such models from an AEC without influencing its structure above the Löwenheim–Skolem number.

## Examples

The following are examples of abstract elementary classes:[2]

## Common assumptions

AECs are very general objects and one usually make some of the assumptions below when studying them:

Note that in elementary classes, joint embedding holds whenever the theory is complete, while amalgamation and no maximal models are well-known consequences of the compactness theorem. These three assumptions allow us to build a universal model-homogeneous monster model ${\displaystyle {\mathfrak {C}}}$, exactly as in the elementary case.

Another assumption that one can make is tameness.

## Shelah's categoricity conjecture

Shelah introduced AECs to provide a uniform framework in which to generalize first-order classification theory. Classification theory started with Morley's categoricity theorem, so it is natural to ask whether a similar result holds in AECs. This is Shelah's categoricity conjecture. It states that there should be a Hanf number for categoricity:

For every AEC K there should be a cardinal ${\displaystyle \mu }$ depending only on ${\displaystyle \operatorname {LS} (K)}$ such that if K is categorical in some ${\displaystyle \lambda \geq \mu }$ (i.e. K has exactly one (up to isomorphism) model of size ${\displaystyle \lambda }$), then K is categorical in ${\displaystyle \theta }$ for all ${\displaystyle \theta \geq \mu }$.

Several approximations have been published (see for example the results section below), assuming set-theoretic assumptions (such as the existence of large cardinals or variations of the generalized continuum hypothesis), or model-theoretic assumptions (such as amalgamation or tameness). As of 2014, the original conjecture remains open.

## Results

The following are some important results about AECs. Except for the last, all results are due to Shelah.

## Notes

1. Template:Harvnb, Section 1.
2. Template:Harvnb, Theorem 3.4.
3. Template:Harvnb, Corollary 3.5. Note that there is a typo there and that ${\displaystyle 2^{2^{\operatorname {LS} (K)}}}$ should be replaced by ${\displaystyle 2^{\operatorname {LS} (K)}}$.
4. Template:Harvnb, Theorem 4.3.
5. Template:Harvnb, Theorem 5.1.
6. This is due to Will Boney, but combines results of many people, including Grossberg, Makkai, Shelah, and VanDieren. A proof appears in Template:Harvnb, Theorem 7.5.

## References

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