\begin{definition}[Definition:Ordering on Multiindices]
Let $Z$ be the set of multiindices.
We define an ordering on $Z$ as follows.
Let:
:$k = \sequence {k_j}_{j \mathop \in J}$
:$\ell = \sequence {\ell_j}_{j \mathop \in J}$
be multiindices.
Then we write:
:$k \preceq \ell$
{{iff}}:
:$\paren {\forall j \in J} \paren {k_j \le \ell_j}$
By abuse of notation, it is standard to use $\le$ to denote the ordering on the set of multiindices as well.
\end{definition}