\begin{definition}[Definition:Minimal (Model Theory)]
Let $\MM$ be an $\LL$-structure.
Let $M$ be the universe of $\MM$.
Let $A$ be a subset of $M$.
Let $D \subseteq M^n$ be an infinite $A$-definable set.
Let $\phi \left({\bar x, \bar a}\right)$ be an $\LL$-formula with parameters $\bar a$ from $A\subseteq M$ and  free variables $\bar x$ which defines $D$.
{{Disambiguate|Definition:Logical Formula}}
$D$ is '''minimal''' in $\MM$ {{iff}} every definable subset of $D$ is either finite or cofinite.
$\phi$ is '''minimal''' in $\MM$ if $D$ is minimal in $\MM$.
$D$ and $\phi$ are '''strongly minimal''' in $\MM$ {{iff}} $\phi$ is minimal in any elementary extension $\NN$ of $\MM$.
An $\LL$-theory $T$ is '''strongly minimal''' {{iff}} for every model $\NN$ of $T$ with universe $N$, the set $N$ is strongly minimal.
{{NoSources}}
Category:Definitions/Model Theory
\end{definition}