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 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 [1] 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.[2]
- Zil'ber's Conjecture Any uncountably categorical theory is either locally modular or interprets an algebraically closed field.[3]
- 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 .
An embedding is strong if .
We also want the pair (C, ) to satisfy the amalgamation property: if then there is a so that each embeds strongly into with the same image for .
For infinite , and , we say iff for , . For any , the closure of (in ), is the smallest superset of satisfying .
Definition A countable structure is a (C, )-generic if:
- For , .
- For , if then there is a strong embedding of into over
- has finite closures: for every , 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.
References
- ↑ Slides on Hrushovski construction from Frank Wagner
- ↑ E. Hrushovski. A stable -categorical pseudoplane. Preprint, 1988
- ↑ E. Hrushovski. A new strongly minimal set. Annals of Pure and Applied Logic, 52:147–166, 1993