Strongly minimal theory

A complete theory with infinite models is strongly minimal when, in every model, each definable subset in one variable, with parameters allowed, is finite or cofinite. The requirement applies to all models and all parameter choices. The infinite-universe condition excludes trivial finite theories.

For an infinite set in the language of equality, a one-variable formula with finitely many parameters can only distinguish those named elements from the remaining elements, so every definable subset is finite or cofinite. In an algebraically closed field, quantifier elimination likewise makes every one-variable definable subset finite or cofinite, since a nonzero polynomial has finitely many roots. A dense linear order without endpoints fails this property: a proper open interval and its complement are both infinite. Strong minimality is a particularly restrictive source of omega-stability.

An element belongs to $\mathrm{acl}(A)$ when it lies in a finite set definable with parameters from $A$.

For a first-order structure $M$ and $A\subseteq M$, the model-theoretic definable closure ${\mathrm{dcl}}^M(A)$ consists of the elements uniquely specified in $M$ by some first-order formula with finitely many parameters from $A$. For example, in a field, the inverse of a nonzero parameter $a$ is uniquely defined by $ax=1$, so it belongs to the definable closure of $\{a\}$.

Every automorphism of $M$ fixing $A$ pointwise fixes ${\mathrm{dcl}}^M(A)$. The converse uses a sufficiently saturated and strongly homogeneous ambient model with $A$ small: if a type over $A$ has another realization, homogeneity moves the element, while saturation ensures that failure of unique definability gives another realization. In an arbitrary structure the converse can fail; having few automorphisms does not by itself produce defining formulas. This distinction prevents automorphism rigidity from being confused with definability.

A pregeometry is a finitary closure structure with the exchange axiom, extending the closure description of a matroid to possibly infinite underlying sets. A pregeometry is a closure operator with finite character and exchange. Algebraic closure in a strongly minimal theory is a pregeometry.

A basis of a pregeometry is a maximal independent set. Every element lies in the closure of a basis, and the exchange property implies that all bases have the same cardinality.

The dimension of a closed set over a smaller closed set is the cardinality of any basis of the larger set over the smaller one. The exchange property makes this cardinality independent of the chosen basis.

Over an algebraically closed parameter set in a strongly minimal theory, there is a unique nonalgebraic one-type. It is called the generic type: its realizations are precisely the elements outside the algebraic closure of the parameters.

A set is independent when no element lies in the algebraic closure of the others. Every bijection between independent sets in a strongly minimal monster model is elementary because successive independent elements realize the unique generic type over the algebraic closure of their predecessors.

In a countable strongly minimal theory, every model has finite-tuple homogeneity. Extend a finite partial elementary map over algebraic closures, choose bases of equal dimension for the remaining pregeometry, map one basis bijectively to the other, and extend over algebraic closure.