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 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 ℵ 0 {\displaystyle \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 ≤ {\displaystyle \leq } be a relation on pairs from C satisfying:
A ≤ B {\displaystyle A\leq B} implies A ⊆ B . {\displaystyle A\subseteq B.}
A ⊆ B ⊆ C {\displaystyle A\subseteq B\subseteq C} and A ≤ C {\displaystyle A\leq C} implies A ≤ B {\displaystyle A\leq B}
∅ ≤ A {\displaystyle \varnothing \leq A} for all A ∈ C . {\displaystyle A\in \mathbf {C} .}
A ≤ B {\displaystyle A\leq B} implies A ∩ C ≤ B ∩ C {\displaystyle A\cap C\leq B\cap C} for all C ∈ C . {\displaystyle C\in \mathbf {C} .}
If f : A → A ′ {\displaystyle f\colon A\to A'} is an isomorphism and A ≤ B {\displaystyle A\leq B} , then f {\displaystyle f} extends to an isomorphism B → B ′ {\displaystyle B\to B'} for some superset of A ′ {\displaystyle A'} with A ′ ≤ B ′ . {\displaystyle A'\leq B'.}
Definition. An embedding f : A ↪ D {\displaystyle f:A\hookrightarrow D} is strong if f ( A ) ≤ D . {\displaystyle f(A)\leq D.}
Definition. The pair ( C , ≤ ) {\displaystyle (\mathbf {C} ,\leq )} has the amalgamation property if A ≤ B 1 , B 2 {\displaystyle A\leq B_{1},B_{2}} then there is a D ∈ C {\displaystyle D\in \mathbf {C} } so that each B i {\displaystyle B_{i}} embeds strongly into D {\displaystyle D} with the same image for A . {\displaystyle A.}
Definition. For infinite D {\displaystyle D} and A ∈ C , {\displaystyle A\in \mathbf {C} ,} we say A ≤ D {\displaystyle A\leq D} iff A ≤ X {\displaystyle A\leq X} for A ⊆ X ⊆ D , X ∈ C . {\displaystyle A\subseteq X\subseteq D,X\in \mathbf {C} .} Definition. For any A ⊆ D , {\displaystyle A\subseteq D,} the closure of A {\displaystyle A} in D , {\displaystyle D,} denoted by cl D ( A ) , {\displaystyle \operatorname {cl} _{D}(A),} is the smallest superset of A {\displaystyle A} satisfying cl ( A ) ≤ D . {\displaystyle \operatorname {cl} (A)\leq D.}
Definition. A countable structure G {\displaystyle G} is ( C , ≤ ) {\displaystyle (\mathbf {C} ,\leq )} -generic if:
For A ⊆ ω G , A ∈ C . {\displaystyle A\subseteq _{\omega }G,A\in \mathbf {C} .}
For A ≤ G , {\displaystyle A\leq G,} if A ≤ B {\displaystyle A\leq B} then there is a strong embedding of B {\displaystyle B} into G {\displaystyle G} over A . {\displaystyle A.}
G {\displaystyle G} has finite closures: for every A ⊆ ω G , cl G ( A ) {\displaystyle A\subseteq _{\omega }G,\operatorname {cl} _{G}(A)} is finite. Theorem. If ( C , ≤ ) {\displaystyle (\mathbf {C} ,\leq )} has the amalgamation property, then there is a unique ( C , ≤ ) {\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.
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