= Model-theoretic definable closure
{title2=$\operatorname{dcl}(A)$}
For a <first-order structure> $M$ and $A\subseteq M$, the <model-theoretic definable closure> $\operatorname{dcl}^M(A)$ 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 $\operatorname{dcl}^M(A)$. 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.
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