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