Hrushovski construction

In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraisse limit by working with a notion of "strong substructure" $leq$ rather than $subseteq$. 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 $leq$ determine various properties of the generic, with it's 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 $aleph\_0$-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 andsubstructures. We want to strengthen the notion of substructure; let$leq$ be a relation on pairs from C satisfying:

* $A\; leq\; B$ implies $A\; subseteq\; B$.
* $A\; subseteq\; B\; subseteq\; C$ and $A\; leq\; C$ implies $A\; leq\; B$
* $varnothing\; leq\; A$ for all $A\; in\; C$.
* $A\; leq\; B$ implies $A\; cap\; C\; leq\; B\; cap\; C$ for all $C\; in\; C$.
* If $f:\; A\; ightarrow\; A\text{'}$ is an isomorphism and $A\; leq\; B$, then $f$ extends to an isomorphism $B\; ightarrow\; B\text{'}$ for some superset of $B$ with $A\text{'}\; leq\; B\text{'}$.

An embedding $f:\; A\; hookrightarrow\; D$ is "strong" if $f(A)\; leq\; D$.

We also want the pair (C, $leq$) to satisfy the "amalgamation property": if $A\; leq\; B\_1,\; A\; leq\; B\_2$ then there is a $D\; in\; C$so that each $B\_i$ embeds strongly into $D$ with the same image for$A$.

For infinite $D$, and $A\; in\; C$, we say $A\; leq\; D$ iff $A\; leq\; X$ for$A\; subseteq\; X\; subseteq\; D$, $X\; in\; C$. For any $A\; subseteq\; D$, the"closure" of $A$ (in $D$), $operatorname\{cl\}\_D(A)$ is the smallest superset of $A$satisfying $operatorname\{cl\}(A)\; leq\; D$.

Definition A countable structure $G$ is a (C, $leq$)-generic if:
* For $A\; subseteq\_omega\; G$, $A\; in\; C$.
* For $A\; leq\; G$, if $A\; leq\; B$ then $B$ there is a strong embedding of $B$ into $G$ over $A$
* $G$ has finite closures: for every $A\; subseteq\_omega\; G$, $operatorname\{cl\}\_G(A)$ is finite.

Theorem If (C, $leq$) has the amalgamation property, then there is aunique (C, $leq$)-generic.

The existence proof proceeds in imitation of the existence proof forFraisse limits. The uniquenss proof comes from an easy back and forthargument.

References

*E. Hrushovski. A stable $aleph\_0$-categorical pseudoplane. "Preprint, 1988"
*E. Hrushovski. A new strongly miminal set. "Annals of Pure and Applied Logic", 52:147–166, "1993".
* [http://math.univ-lyon1.fr/~wagner/nijmegen.pdf Slides on Hrushovski Construction from Frank Wagner]