Definition:Monomial of Free Commutative Monoid/Multiplication

Definition

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:

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