\begin{definition}[Definition:Divide (Model Theory)]
Let $T$ be a complete $\LL$-theory.
Let $\mathfrak C$ be a monster model for $T$.
Let $A$ be a subset of the universe of $\mathfrak C$.
Let $\map \phi {\bar x, \bar b}$ be an $\LL$-formula with free variables $\bar x$ and parameters $\bar b$ from the universe of $\mathfrak C$.
Let $\map {\operatorname{tp} } {\bar b / A}$ denote the type of $\bar b$ over $A$.
Then $\map \phi {\bar x, \bar b}$ '''$k$-divides in $\mathfrak C$ over $A$''' {{iff}}:
:there exists a sequence $\sequence {\bar b_i}_{i \mathop \in \N}$ such that $\map {\operatorname{tp} } {\bar b_i / A} = \map {\operatorname{tp} } {\bar b / A}$ for each $i \in \N$
and:
:for any distinct $k$-many terms $\bar b_{i_1}, \ldots, \bar b_{i_k}$ of the sequence, the set $\set {\map \phi {\bar x, \bar b_{i_1} }, \ldots, \map \phi {\bar x, \bar b_{i_k} } }$ is not satisfiable in $\mathfrak C$.
Let $\map \pi {\bar x, \bar b}$ be a set of formulas with parameters $\bar b$.
$\map \pi {\bar x, \bar b}$ '''$k$-divides''' over $A$ if it implies a formula $\map \phi {x, \bar c}$ which $k$-divides over $A$.
Formulas and sets of formulas are said to '''divide''' if they $k$-divide for some $k$.
\end{definition}