In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure
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.
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: 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: Definition.
An embedding
The pair
has the amalgamation property if
embeds strongly into
with the same image for
the closure of
is the smallest superset of
satisfying
A countable structure
-generic if: Theorem.
has the amalgamation property, then there is a unique
The existence proof proceeds in imitation of the existence proof for Fraïssé limits.
The uniqueness proof comes from an easy back and forth argument.