Abstract Elementary Classes

Definition + Intuition

$⟨\mathcal{K},{\prec }_{\mathcal{K}}⟩$, for $\mathcal{K}$ is a class of structures in some language in some language $L=L(\mathcal{K})$ is an AEC (Abstract Elementary Class) if it has the following properties:

• ${\prec }_{\mathcal{K}}$ is a partial-order on $\mathcal{K}$
• if $\mathcal{M}{\prec }_{\mathcal{K}}\mathcal{N}$ if $\mathcal{M}$ is a substructure of $\mathcal{N}$
• $\mathcal{K}$ is closed under isomorphisms, that is if $\mathcal{M}\in \mathcal{K}$ and $\mathcal{M}\cong \mathcal{N}$, then $\mathcal{N}\in \mathcal{K}$. Also, if $\mathcal{M},\mathcal{N},{\mathcal{M}}^{\prime },{\mathcal{N}}^{\prime }\in \mathcal{K},f:\mathcal{M}\cong {\mathcal{M}}^{\prime },g:\mathcal{N}\cong {\mathcal{N}}^{\prime }.g\subseteq f$ and $\mathcal{M}{\prec }_{\mathcal{K}}\mathcal{N}$, then ${\mathcal{M}}^{\prime }\prec {\mathcal{N}}^{\prime }$.
• If ${\mathcal{M}}_{1{\prec }_{\mathcal{K}}}{\mathcal{M}}_3,{\mathcal{M}}_2\prec {\mathcal{M}}_3,M_1\subseteq M_2$, then ${\mathcal{M}}_1{\prec }_{\mathcal{K}}{\mathcal{M}}_2$
• Tarski-Vaught chain axioms: If $\gamma$ is an ordinal and {$M_a$ | $\alpha <\gamma$} $\subseteq \mathcal{K}$ is a chain (i.e., $\alpha <\beta <\gamma ⟹{\mathcal{M}}_{\alpha }{\prec }_{\mathcal{K}}{\mathcal{M}}_{\beta }$), then
  • $\bigcup _{\alpha <\gamma }{\mathcal{M}}_{\alpha }\in \mathcal{K}$
  • If ${\mathcal{M}}_{\alpha }{\prec }_{\mathcal{K}}\mathcal{N},$ for all $\alpha <\gamma ,$ then $\bigcup _{\alpha <\gamma }{\mathcal{M}}_{\alpha {\prec }_{\mathcal{K}}}\mathcal{N}$
• Löwenheim-Skolem axiom: There exists a cardinal $\mu \ge |L(K)|+{\aleph }_{0,}$ such that if $A\subseteq Dom(M),$ then there is a $\mathcal{N}\in \mathcal{K}$ whose universe contains A such that $||N||\le |A|+\mu$ and $\mathcal{N}{\prec }_K\mathcal{M}.$ Let $LS(\mathcal{K})$ denote the least such $\mu$ and call it the Löwenheim-Skolem number of $\mathcal{K}$

Abstract elementary class allows us to study a class of objects w/out going into the specifics of logics (i.e., syntax). AEC’s provide a diverse framework to work w/in to study semantics and model-theoretic properties of structures.

The relation ${\prec }_{\mathcal{K}}$ is a ‘strong’ substructure relation, which is analogous to a generalization of the elementary substructure relation (in the context of AEC’s).

A simple example of an AEC is an elementary class. If T is a first-order theory, then Mod$(|T|)$ (the class of all models of $|T|$) together with elementary substructure forms an AEC w/ Löwenheim–Skolem number $|T|$.

Useful Properties for AEC’s

• It is often assumed that $AE{C}^{\prime }s$ do not contain maximal models (w/ respect to cardinality and the '${⪯}_{\mathcal{K}}$' relation), that is, it is assumed there does not exists a structure $\mathcal{N}\in \mathcal{K}$ such that for all $\mathcal{M}\in \mathcal{K},\mathcal{M}{⪯}_k\mathcal{N}$.

The following properties are also often assumed for AEC’s:

Tameness: Fragment of compactness for AEC’s

An AEC $\mathcal{K}$ is $(\chi ,\mu )$-weakly tame if for any saturated model $\mathcal{N}\in \mathcal{K}$ such that $|N|=\mu ,$ if $p,q\in S(N)$ with $q\ne p$ then for some $\mathcal{M}{⪯}_{\kappa }\mathcal{N}$ w/ $|M|\le \chi ,p\upharpoonright \mathcal{M}\ne q\upharpoonright \mathcal{M}$.

Notation: $p\upharpoonright \mathcal{M}$ means the maximally consistent set of formulas $\phi (a_1,\dots ,a_n)$ that only has parameters $a_1,\dots ,a_n\in M$

i.e., Any two Galois types that differ have a small sub model $\mathcal{M}$

An AEC $\mathcal{K}$ is $(\chi ,\mu )$-tame if the above holds for all $\mathcal{N}$ with cardinality $\mu$

Joint Elementary Property: An AEC $\mathcal{K}$ has the joint elementary property (JEP) if for any two structures $N_1,N_2\in \mathcal{K}$, then there exists a structure $M\in \mathcal{K}$ such that there are strong embeddings $f:N_1\to M$ and $g:N_2\to M$.

Amalgamation Property: If you have a structure $M\in \mathcal{K}$ and two extensions of $M,N_1,N_2\in \mathcal{K}$, then there exists another structure $N_3\in \mathcal{K}$ such that there is $f:N_1\to N_3$ and $g:N_2\to N_3$ over $M$.

Note: For elementary classes, the joint elementary property holds whenever the theory is complete. The amalgamation property and the non-existence of maximal models in $\mathcal{K}$ are consequences of the compactness property.

Monster Model: $\mathbb{M}\in \mathcal{K}$ is a monster model if it $\kappa$-saturated for a sufficiently large $\kappa$ ($⟹\kappa$-homogeneity), where all Galois types over substructures of size $<\kappa$ and by implication, any partial elementary map between ‘small’ ($<\kappa$) substructures extends to an automorphism of $\mathbb{M}$.

Note: ‘Large enough’ meaning larger than all models you intend to work over, and so every relevant structure $\mathcal{M}$ embeds into $\mathbb{M}$. Choosing $|\mathbb{M}|=2^{\kappa }$ for $\mathbb{M}$ is $\kappa$-saturated, or a strong inaccessible cardinal.

Embedding reducts.

Abstract elementary class - Wikipedia] AEClec.pdf The Löwenheim Number Stability Theory Logics, Structures & Models