\begin{definition}[Definition:Monomial of Free Commutative Monoid]
{{handwaving|infinite product is hopelessly inadequate because we are dealing with symbols. Instead, use mappings $X \to \N$ and proceed carefully. Getting this definition formal is tricky business}}
A '''monomial''' in the indexed set $\family {X_j: j \in J}$ is a possibly infinite product:
:$\ds \prod_{j \mathop \in J} X_j^{k_j}$
with integer exponents $k_j \ge 0$ such that $k_j = 0$ for all but finitely many $j$.
Let $\mathbf X = \family {X_j}_{j \mathop \in J}$ and for a multiindex $k = \paren {k_j}_{j \mathop \in J}$ over $J$ define:
:$\ds \mathbf X^k = \prod_{j \mathop \in J} X_j^{k_j}$
Then a '''monomial''' is an object of the form $\mathbf X^k$, where $k$ is a multiindex.
\end{definition}