Definition:Saturated Model

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Let $T$ be an $\LL$-theory.

Let $\kappa$ be an infinite cardinal.

A model $\MM$ of $T$ is $\kappa$-saturated if and only if for every subset $A$ of the universe of $\MM$ of cardinality strictly less than $\kappa$, and for every $n \in \N$, every complete $n$-type $p$ over $A$ is realized in $\MM$.

That is, $\MM$ is $\kappa$-saturated if and only if for all $A \subseteq \MM$ with $\card A < \kappa$, and for all $n \in \N$, each $p \in \map {S_n^\MM} A$ is realized in $\MM$.

We say $\MM$ is saturated if and only if it is $\kappa$-saturated where $\kappa$ is the cardinality of the universe of $\MM$.