# Definition:Homogeneous (Model Theory)

## Definition

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

Let $\kappa$ be an infinite cardinal.

A model $\MM$ of $T$ is $\kappa$-homogeneous if and only if for every subset $A$ and element $b$ in the universe of $\MM$ with the cardinality of $A$ strictly less than $\kappa$, if $f: A \to \MM$ is partial elementary, then $f$ extends to an elementary map $f^*: A \cup \set b \to \MM$.

That is, $\MM$ is $\kappa$-homogeneous if and only if for all $A \subseteq \MM$ with $\card A < \kappa$ and all $b \in \MM$, every elementary $f: A \to \MM$ extends to an elementary $f^*: A \cup \set b \to \MM$.

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

## Equivalent Definition

$\MM$ is homogeneous if and only if it has some infinite cardinality $\kappa$ and for every $A \subseteq \MM$ with $\card A < \kappa$, each partial elementary map $f: A \to \MM$ extends to an automorphism.

The equivalence of this definition is proved in Homogeneous iff Partial Elementary Maps Extend to Automorphisms.