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 or rich model. 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 $$A \in \mathbf{C}.$$
 * $$A \leq B$$ implies $$A \cap C \leq B \cap C$$ for all $$C \in \mathbf{C}.$$
 * If $$f\colon A \to A'$$ is an isomorphism and $$A \leq B$$, then $$f$$ extends to an isomorphism $$B \to B'$$ for some superset of $$B$$ with $$A' \leq B'.$$

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

Definition. The pair $$(\mathbf{C}, \leq)$$ has the amalgamation property if $$A \leq B_1, B_2$$ then there is a $$D \in \mathbf{C}$$ so that each $$B_i$$ embeds strongly into $$D$$ with the same image for $$A.$$

Definition. For infinite $$D$$ and $$A \in \mathbf{C},$$ we say $$A \leq D$$ iff $$A \leq X$$ for $$ A \subseteq X \subseteq D, X \in \mathbf{C}.$$

Definition. For any $$A \subseteq D,$$ the closure of $$A$$ in $$D,$$ denoted by $$\operatorname{cl}_D(A),$$ is the smallest superset of $$A$$ satisfying $$\operatorname{cl}(A) \leq D.$$

Definition. A countable structure $$G$$ is $$(\mathbf{C}, \leq)$$-generic if:


 * For $$A \subseteq_\omega G, A \in \mathbf{C}.$$


 * For $$A \leq G,$$ if $$A \leq B$$ then 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 $$(\mathbf{C},\leq)$$ has the amalgamation property, then there is a unique $$(\mathbf{C},\leq)$$-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.