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Suppose $a\in \mathrm{acl}(A)$ in the sense of model-theoretic algebraic closure. Some formula $\phi (x,\bar{c})$ over $A$ has exactly $n<\infty$ realizations in the monster model and includes $a$. Every model $M$ containing $A$ is an elementary substructure of the monster and therefore contains $n$ distinct realizations. Since there are only $n$ in the monster, it contains all of them, including $a$. Hence

The model-theoretic algebraic closure isEquivalently, it consists of elements with finite orbit under $\mathrm{Aut}(\mathcal{U}/A)$. The model-theoretic definable closure isequivalently the set fixed pointwise by every automorphism fixing $A$. Consequently $\mathrm{dcl}(A)\subseteq \mathrm{acl}(A)$.

In a strongly minimal theory, model-theoretic algebraic closure is a pregeometry. If $C$ is algebraically closed, every element outside $C$ realizes the generic type in a strongly minimal theory: every one-variable definable set is finite or cofinite, and a point outside $C$ belongs to none of the finite ones.

Enumerate a finite tuple $a_1,\dots ,a_n$ of distinct elements of the independent set $A$. Independence saysIts image tuple under the bijection has the same property. Inductively, $a_i$ and $f(a_i)$ realize the same generic type over the algebraic closures of the preceding tuples, so every finite restriction of $f$ is elementary. First-order formulas involve only finitely many parameters, hence the entire bijection is an elementary map. This is the independent set in a strongly minimal theory principle.