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

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 $\mathrm{dcl}(A)\subseteq \mathrm{acl}(A)$.