Definition:Minimal (Model Theory)

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Definition

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 $D$.

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$D$ is minimal in $\MM$ if and only if 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 $\MM$ if and only if $\phi$ is minimal in any elementary extension $\NN$ of $\MM$.

An $\LL$-theory $T$ is strongly minimal if and only if for every model $\NN$ of $T$ with universe $N$, the set $N$ is strongly minimal.

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