Definition:Minimal (Model Theory)

Definition
Let $\mathcal{M}$ be an $\mathcal{L}$-structure, and let $M$ be the universe of $\mathcal{M}$.

Let $A$ be a subset of $M$.

Let $D\subseteq M^n$ be an infinite $A$-definable set.

Let $\phi(\bar{x},\bar{a})$ be an $\mathcal{L}$-formula with parameters $\bar{a}$ from $A\subseteq M$ and free variables $\bar{x}$ which defines $D$.

$D$ is minimal in $\mathcal{M}$ if every definable subset of $D$ is either finite or cofinite.

$\phi$ is minimal in $\mathcal{M}$ if $D$ is minimal in $\mathcal{M}$.

$D$ and $\phi$ are strongly minimal in $\mathcal{M}$ if $\phi$ is minimal in any elementary extension $\mathcal{N}$ of $\mathcal{M}$.

An $\mathcal{L}$-theory $T$ is strongly minimal if for every model $\mathcal{N}$ of $T$ with universe $N$, the set $N$ is strongly minimal.