The <model-theoretic algebraic closure> is
$$
\operatorname{acl}(A)=
\{c:\text{$c$ belongs to a finite set definable over $A$}\}.
$$
Equivalently, it consists of elements with finite orbit under $\operatorname{Aut}(\mathcal U/A)$. The <model-theoretic definable closure> is
$$
\operatorname{dcl}(A)=
\{c:\text{$c$ is uniquely definable over $A$}\},
$$
equivalently the set fixed pointwise by every automorphism fixing $A$. Consequently $\operatorname{dcl}(A)\subseteq\operatorname{acl}(A)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 2 / c / Solution

{title2=$\operatorname{dcl}(A)$}

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


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 Incoming links: Model-theoretic definable closure