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 [math]\displaystyle{ \leq }[/math] rather than [math]\displaystyle{ \subseteq }[/math]. 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 [math]\displaystyle{ \leq }[/math] 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 [math]\displaystyle{ \aleph_0 }[/math]-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 [math]\displaystyle{ \leq }[/math] be a relation on pairs from C satisfying:

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

Definition. An embedding [math]\displaystyle{ f: A \hookrightarrow D }[/math] is strong if [math]\displaystyle{ f(A) \leq D. }[/math]

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

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

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

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

• For [math]\displaystyle{ A \subseteq_\omega G, A \in \mathbf{C}. }[/math]

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

• [math]\displaystyle{ G }[/math] has finite closures: for every [math]\displaystyle{ A \subseteq_\omega G, \operatorname{cl}_G(A) }[/math] is finite.

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