\begin{definition}[Definition:Monomial of Free Commutative Monoid/Multiplication]
The set of monomials over $\family {X_j: j \in J}$ has '''multiplication''' $\circ$ defined by:
:$\ds \paren {\prod_{j \mathop \in J} X_j^{k_j} } \circ \paren {\prod_{j \mathop \in J} X_j^{k_j'} } = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$
which using multiindex addition notation reads:
:$\mathbf X^k \circ \mathbf X^{k'} = \mathbf X^{k + k'}$
Category:Definitions/Monomials
\end{definition}