Hrushovski construction

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 ${\displaystyle \aleph _{0}}$-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?

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 ${\displaystyle \leq }$ be a relation on pairs from C satisfying:

• ${\displaystyle A\leq B}$ implies ${\displaystyle A\subseteq B.}$
• ${\displaystyle A\subseteq B\subseteq C}$ and ${\displaystyle A\leq C}$ implies ${\displaystyle A\leq B}$
• ${\displaystyle \varnothing \leq A}$ for all ${\displaystyle A\in \mathbf {C} .}$
• ${\displaystyle A\leq B}$ implies ${\displaystyle A\cap C\leq B\cap C}$ for all ${\displaystyle C\in \mathbf {C} .}$
• If ${\displaystyle f\colon A\to A'}$ is an isomorphism and ${\displaystyle A\leq B}$, then ${\displaystyle f}$ extends to an isomorphism ${\displaystyle B\to B'}$ for some superset of ${\displaystyle A'}$ with ${\displaystyle A'\leq B'.}$

Definition. An embedding ${\displaystyle f:A\hookrightarrow D}$ is strong if ${\displaystyle f(A)\leq D.}$

Definition. The pair ${\displaystyle (\mathbf {C} ,\leq )}$ has the amalgamation property if ${\displaystyle A\leq B_{1},B_{2}}$ then there is a ${\displaystyle D\in \mathbf {C} }$ so that each ${\displaystyle B_{i}}$ embeds strongly into ${\displaystyle D}$ with the same image for ${\displaystyle A.}$

Definition. For infinite ${\displaystyle D}$ and ${\displaystyle A\in \mathbf {C} ,}$ we say ${\displaystyle A\leq D}$ iff ${\displaystyle A\leq X}$ for ${\displaystyle A\subseteq X\subseteq D,X\in \mathbf {C} .}$

Definition. For any ${\displaystyle A\subseteq D,}$ the closure of ${\displaystyle A}$ in ${\displaystyle D,}$ denoted by ${\displaystyle \operatorname {cl} _{D}(A),}$ is the smallest superset of ${\displaystyle A}$ satisfying ${\displaystyle \operatorname {cl} (A)\leq D.}$

Definition. A countable structure ${\displaystyle G}$ is ${\displaystyle (\mathbf {C} ,\leq )}$-generic if:

• For ${\displaystyle A\subseteq _{\omega }G,A\in \mathbf {C} .}$

• For ${\displaystyle A\leq G,}$ if ${\displaystyle A\leq B}$ then there is a strong embedding of ${\displaystyle B}$ into ${\displaystyle G}$ over ${\displaystyle A.}$

• ${\displaystyle G}$ has finite closures: for every ${\displaystyle A\subseteq _{\omega }G,\operatorname {cl} _{G}(A)}$ is finite.

Theorem. If ${\displaystyle (\mathbf {C} ,\leq )}$ has the amalgamation property, then there is a unique ${\displaystyle (\mathbf {C} ,\leq )}$-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.