Definition:Ordering on Multiindices

Definition
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:
 * $k \preceq \ell$




 * $\paren {\forall j \in J} \paren {k_j \le \ell_j}$

By abuse of notation, it is standard to use $\le$ to denote the ordering on the set of multiindices as well.

Also see

 * Ordering on Multiindices is Partial Order