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.
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
ℵ
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?
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:
A
≤
B
implies
A
⊆
B
.
A
⊆
B
⊆
C
and
A
≤
C
implies
A
≤
B
∅
≤
A
for all
A
∈
C
.
A
≤
B
implies
A
∩
C
≤
B
∩
C
for all
C
∈
C
.
If
f
:
A
→
A
′
is an isomorphism and
A
≤
B
, then
f
extends to an isomorphism
B
→
B
′
for some superset of
B
with
A
′
≤
B
′
.
An embedding
f
:
A
↪
D
is strong if
f
(
A
)
≤
D
.
We also want the pair (C,
≤
) to satisfy the amalgamation property: if
A
≤
B
1
,
A
≤
B
2
then there is a
D
∈
C
so that each
B
i
embeds strongly into
D
with the same image for
A
.
For infinite
D
, and
A
∈
C
, we say
A
≤
D
iff
A
≤
X
for
A
⊆
X
⊆
D
,
X
∈
C
. For any
A
⊆
D
, the closure of
A
(in
D
),
cl
D
(
A
)
is the smallest superset of
A
satisfying
cl
(
A
)
≤
D
.
Definition A countable structure
G
is a (C,
≤
)-generic if:
For
A
⊆
ω
G
,
A
∈
C
.
For
A
≤
G
, if
A
≤
B
then
B
there is a strong embedding of
B
into
G
over
A
G
has finite closures: for every
A
⊆
ω
G
,
cl
G
(
A
)
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.
