\begin{definition}[Definition:Saturated Model]
Let $T$ be an $\LL$-theory.
Let $\kappa$ be an infinite cardinal.
A model $\MM$ of $T$ is $\kappa$-'''saturated''' {{iff}} 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 {{iff}} 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''' {{iff}} it is $\kappa$-saturated where $\kappa$ is the cardinality of the universe of $\MM$.
Category:Definitions/Model Theory
\end{definition}