\begin{definition}[Definition:Free Commutative Monoid]
The '''free commutative monoid''' on an indexed set $X = \family {X_j: j \in J}$ is the set $M$ of all monomials under the standard multiplication.
{{questionable|$M$ as above is most definitely not the set of finite sequences}}
That is, it is the set $M$ of all finite sequences of $X$.
\end{definition}