Hrushovski construction

In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure $\leq$ rather than $\subseteq$ . It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the generic. The specifics of $\leq$ determine various properties of the generic, with its geometric properties being of particular interest. It was initially used by Ehud Hrushovski to generate a stable structure with an "exotic" geometry, thereby refuting Zil'ber's Conjecture.

Three conjectures

The initial applications of the Hrushovski construction refuted two conjectures and answered a third question in the negative. Specifically, we have:

Lachlan's Conjecture Any stable $\aleph_0$ -categorical theory is totally transcendental.
Zil'ber's Conjecture Any uncountably categorical theory is either locally modular or interprets an algebraically closed field.
Cherlin's Question Is there a maximal (with respect to expansions) strongly minimal set?

The construction

Let L be a finite relational language. Fix C a class of finite L-structures which are closed under isomorphisms and substructures. We want to strengthen the notion of substructure; let $\leq$ be a relation on pairs from C satisfying:

$A \leq B$ implies $A \subseteq B$ .
$A \subseteq B \subseteq C$ and $A \leq C$ implies $A \leq B$
$\varnothing \leq A$ for all $A \in \C$ .
$A \leq B$ implies $A \cap C \leq B \cap C$ for all $C \in \C$ .
If $f: A \rightarrow A'$ is an isomorphism and $A \leq B$ , then $f$ extends to an isomorphism $B \rightarrow B'$ for some superset of $B$ with $A' \leq B'$ .

An embedding $f: A \hookrightarrow D$ is strong if $f(A) \leq D$ .

We also want the pair (C, $\leq$ ) to satisfy the amalgamation property: if $A \leq B_1, A \leq B_2$ then there is a $D \in \C$ so that each $B_i$ embeds strongly into $D$ with the same image for $A$ .

For infinite $D$ , and $A \in \C$ , we say $A \leq D$ iff $A \leq X$ for $A \subseteq X \subseteq D$ , $X \in \C$ . For any $A \subseteq D$ , the closure of $A$ (in $D$ ), $\operatorname{cl}_D(A)$ is the smallest superset of $A$ satisfying $\operatorname{cl}(A) \leq D$ .

Definition A countable structure $G$ is a (C, $\leq$ )-generic if:

For $A \subseteq_\omega G$ , $A \in \C$ .
For $A \leq G$ , if $A \leq B$ then $B$ there is a strong embedding of $B$ into $G$ over $A$
$G$ has finite closures: for every $A \subseteq_\omega G$ , $\operatorname{cl}_G(A)$ is finite.

Theorem If (C, $\leq$ ) has the amalgamation property, then there is a unique (C, $\leq$ )-generic.

The existence proof proceeds in imitation of the existence proof for Fraïssé limits. The uniqueness proof comes from an easy back and forth argument.

References

E. Hrushovski. A stable $\aleph_0$ -categorical pseudoplane. Preprint, 1988
E. Hrushovski. A new strongly minimal set. Annals of Pure and Applied Logic, 52:147–166, 1993.
Slides on Hrushovski construction from Frank Wagner

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