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 $\le$ 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 $\le$ 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 ${\mathrm{\aleph }}_0$-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 $\le$ be a relation on pairs from C satisfying:

• $A\le B$ implies $A\subseteq B.$
• $A\subseteq B\subseteq C$ and $A\le C$ implies $A\le B$
• $\varnothing \le A$ for all $A\in 𝐂.$
• $A\le B$ implies $A\cap C\le B\cap C$ for all $C\in 𝐂.$
• If $f:A\to A^{\prime }$ is an isomorphism and $A\le B$, then $f$ extends to an isomorphism $B\to B^{\prime }$ for some superset of $A^{\prime }$ with $A^{\prime }\le B^{\prime }.$

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

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

Definition. For infinite $D$ and $A\in 𝐂,$ we say $A\le D$ iff $A\le X$ for $A\subseteq X\subseteq D,X\in 𝐂.$

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

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

• For $A{\subseteq }_{\omega }G,A\in 𝐂.$

• For $A\le G,$ if $A\le B$ then there is a strong embedding of $B$ into $G$ over $A.$

• $G$ has finite closures: for every $A{\subseteq }_{\omega }G,{\mathrm{cl}}_G(A)$ is finite.

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