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. 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.
- 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 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
- 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.
- Slides on Hrushovski construction from Frank Wagner