# Definition:Minimal (Model Theory)

## Definition

Let $\mathcal M$ be an $\mathcal L$-structure.

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 $\phi \left({\bar x, \bar a}\right)$ 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 and only 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 and only 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.