In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure ≤ rather than ⊆ . It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the generic. The specifics of ≤ 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.
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 -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?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 ≤ be a relation on pairs from C satisfying:
A ≤ B implies A ⊆ B . A ⊆ B ⊆ C and A ≤ C implies A ≤ B ∅ ≤ A for all A ∈ C . A ≤ B implies A ∩ C ≤ B ∩ C for all C ∈ C .If f : A → A ′ is an isomorphism and A ≤ B , then f extends to an isomorphism B → B ′ for some superset of B with A ′ ≤ B ′ .An embedding f : A ↪ D is strong if f ( A ) ≤ D .
We also want the pair (C, ≤ ) to satisfy the amalgamation property: if A ≤ B 1 , A ≤ B 2 then there is a D ∈ C so that each B i embeds strongly into D with the same image for A .
For infinite D , and A ∈ C , we say A ≤ D iff A ≤ X for A ⊆ X ⊆ D , X ∈ C . For any A ⊆ D , the closure of A (in D ), cl D ( A ) is the smallest superset of A satisfying cl ( A ) ≤ D .
Definition A countable structure G is a (C, ≤ )-generic if:
For A ⊆ ω G , A ∈ C .For A ≤ G , if A ≤ B then B there is a strong embedding of B into G over A G has finite closures: for every A ⊆ ω G , cl G ( A ) is finite.Theorem If (C, ≤ ) has the amalgamation property, then there is a unique (C, ≤ )-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.
