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 ${\displaystyle \leq }$ rather than ${\displaystyle \subseteq }$. It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the generic or rich [1] model. The specifics of ${\displaystyle \leq }$ 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 ${\displaystyle \aleph _{0}}$-categorical theory is totally transcendental.[2]

• Zil'ber's Conjecture. Any uncountably categorical theory is either locally modular or interprets an algebraically closed field.[3]

• 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 ${\displaystyle \leq }$ be a relation on pairs from C satisfying:

• ${\displaystyle A\leq B}$ implies ${\displaystyle A\subseteq B.}$
• ${\displaystyle A\subseteq B\subseteq C}$ and ${\displaystyle A\leq C}$ implies ${\displaystyle A\leq B}$
• ${\displaystyle \varnothing \leq A}$ for all ${\displaystyle A\in \mathbf {C} .}$
• ${\displaystyle A\leq B}$ implies ${\displaystyle A\cap C\leq B\cap C}$ for all ${\displaystyle C\in \mathbf {C} .}$
• If ${\displaystyle f\colon A\to A'}$ is an isomorphism and ${\displaystyle A\leq B}$, then ${\displaystyle f}$ extends to an isomorphism ${\displaystyle B\to B'}$ for some superset of ${\displaystyle A'}$ with ${\displaystyle A'\leq B'.}$

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

Definition. The pair ${\displaystyle (\mathbf {C} ,\leq )}$ has the amalgamation property if ${\displaystyle A\leq B_{1},B_{2}}$ then there is a ${\displaystyle D\in \mathbf {C} }$ so that each ${\displaystyle B_{i}}$ embeds strongly into ${\displaystyle D}$ with the same image for ${\displaystyle A.}$

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

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

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

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

• For ${\displaystyle A\leq G,}$ if ${\displaystyle A\leq B}$ then there is a strong embedding of ${\displaystyle B}$ into ${\displaystyle G}$ over ${\displaystyle A.}$

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

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

References

1. Slides on Hrushovski construction from Frank Wagner
2. E. Hrushovski. A stable ${\displaystyle \aleph _{0}}$-categorical pseudoplane. Preprint, 1988
3. E. Hrushovski. A new strongly minimal set. Annals of Pure and Applied Logic, 52:147–166, 1993