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 or rich Slides on Hrushovski construction from Frank Wagner model. 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.
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 -categorical theory is totally transcendental.E. Hrushovski. A stable -categorical pseudoplane. Preprint, 1988
Zil'ber's Conjecture. Any uncountably categorical theory is either locally modular or interprets an algebraically closed field.E. Hrushovski. A new strongly minimal set. Annals of Pure and Applied Logic, 52:147–166, 1993
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 be a relation on pairs from C satisfying:
implies and implies for all implies for all If is an isomorphism and , then extends to an isomorphism for some superset of with
Definition. An embedding is strong if
Definition. The pair has the amalgamation property if then there is a so that each embeds strongly into with the same image for
Definition. For infinite and we say iff for
Definition. For any the closure of in denoted by is the smallest superset of satisfying
Definition. A countable structure is -generic if:
For
For if then there is a strong embedding of into over
has finite closures: for every is finite.
Theorem. If has the amalgamation property, then there is a unique -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