 Model-theoretic definable closure

For a first-order structure $M$ and $A\subseteq M$, the model-theoretic definable closure ${\mathrm{dcl}}^M(A)$ 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 ${\mathrm{dcl}}^M(A)$. 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.