Definition:Multiindex

Definition
Let $\displaystyle m = \prod_{j \in J} X_j^{k_j}$ be a mononomial in the indexed set $\{X_j:j \in J\}$.

Such a mononomial can be expressed implicitly and more compactly by referring only to the sequence of indices:
 * $k = \left \langle {k_j}\right \rangle_{j \in J}$

and write $m = \mathbf X^k$ without explicit reference to the indexing set.

Such an expression is called a multiindex (or multi-index).

Its plural form is multiindices (or multi-indices).

Formal Definition
A multiindex is an element of $\Z^J$, the free $\Z$-module on $J$, an abelian group of rank over $\Z$ equal to the cardinality of $J$.

Addition
Addition of multiindices is defined as:
 * $\left \langle {k+k'}\right \rangle_j := k_j + k_j'$

With this notation, multiplication of mononomials $m = \mathbf X^k$, $m' = \mathbf X^{k'}$ is written:


 * $m \circ m' = \mathbf X^{k+k'}$.