Definition:Multiindex Power Notation

Definition
Let $M$ be a commutative monoid.

Let $I$ be a set.

Let $\mathbf x = \family {x_i}_{i \mathop \in I}$ be a family of elements of $M$.

Let $\mathbf a = \family {a_i}_{i \mathop \in I}$ be a family of natural numbers of finite support.

We denote $\ds \mathbf x^{\mathbf a} = \prod_{i \mathop \in I} x_i^{a_i}$ where:
 * $x_i^{a_i}$ is the $a_i$th power of $x_i$
 * $\prod$ denotes product with finite support