Английская Википедия: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 <math>\leq</math> rather than <math>\subseteq</math>. 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 <math>\leq</math> 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 <math>\aleph_0</math>-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 <math>\leq</math> be a relation on pairs from C satisfying:
- <math>A \leq B</math> implies <math>A \subseteq B.</math>
- <math>A \subseteq B \subseteq C</math> and <math>A \leq C</math> implies <math>A \leq B</math>
- <math>\varnothing \leq A</math> for all <math>A \in \mathbf{C}.</math>
- <math>A \leq B</math> implies <math>A \cap C \leq B \cap C</math> for all <math>C \in \mathbf{C}.</math>
- If <math>f\colon A \to A'</math> is an isomorphism and <math>A \leq B</math>, then <math>f</math> extends to an isomorphism <math>B \to B'</math> for some superset of <math>B</math> with <math>A' \leq B'.</math>
Definition. An embedding <math>f: A \hookrightarrow D</math> is strong if <math>f(A) \leq D.</math>
Definition. The pair <math>(\mathbf{C}, \leq)</math> has the amalgamation property if <math>A \leq B_1, B_2</math> then there is a <math>D \in \mathbf{C}</math> so that each <math>B_i</math> embeds strongly into <math>D</math> with the same image for <math>A.</math>
Definition. For infinite <math>D</math> and <math>A \in \mathbf{C},</math> we say <math>A \leq D</math> iff <math>A \leq X</math> for <math> A \subseteq X \subseteq D, X \in \mathbf{C}.</math>
Definition. For any <math>A \subseteq D,</math> the closure of <math>A</math> in <math>D,</math> denoted by <math>\operatorname{cl}_D(A),</math> is the smallest superset of <math>A</math> satisfying <math>\operatorname{cl}(A) \leq D.</math>
Definition. A countable structure <math>G</math> is <math>(\mathbf{C}, \leq)</math>-generic if:
- For <math>A \subseteq_\omega G, A \in \mathbf{C}.</math>
- For <math>A \leq G,</math> if <math>A \leq B</math> then there is a strong embedding of <math>B</math> into <math>G</math> over <math>A.</math>
- <math>G</math> has finite closures: for every <math>A \subseteq_\omega G, \operatorname{cl}_G(A)</math> is finite.
Theorem. If <math>(\mathbf{C},\leq)</math> has the amalgamation property, then there is a unique <math>(\mathbf{C},\leq)</math>-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 <math>\aleph_0</math>-categorical pseudoplane. Preprint, 1988
- ↑ E. Hrushovski. A new strongly minimal set. Annals of Pure and Applied Logic, 52:147–166, 1993
