In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure. It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the generic or rich 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.-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 be a relation on pairs from C satisfying:

Definition. is strong

Definition. has the amalgamation property

Definition.

Definition. the closure

Definition. is -generic if:

• is finite.

Theorem. has the amalgamation property-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