Model-theoretic algebraic closure ($\mathrm{acl}(A)$)

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

An element belongs to $\mathrm{dcl}(A)$ when it is the unique realization of some formula with parameters from $A$. Equivalently, it is fixed by every automorphism fixing $A$ pointwise.

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.