Definition:Monomial of Free Commutative Monoid

Definition
A mononomial in the indexed set $\left\{{X_j: j \in J}\right\}$ is a possibly infinite product $\displaystyle \prod_{j \in J} X_j^{k_j}$ with integer exponents $k_j \geq 0$ such that $k_j = 0$ for all but finitely many $j$.

Then $\displaystyle \prod_{j\in J}X_j^{k_j}$ is the finite product:
 * $\displaystyle \prod_{\substack {j \in J \\ k_j \ne 0}} X_j^{k_j}$

over $j\in J$.

Multiplication
The set of mononomials over $\left\{{X_j: j \in J}\right\}$ has multiplication $\circ$ defined by:


 * $\displaystyle \left({\prod_{j \in J} X_j^{k_j}}\right) \circ \left({\prod_{j \in J} X_j^{k_j'}}\right) = \left({\prod_{j \in J} X_j^{k_j + k_j'}}\right)$

Degree
The degree of a mononomial is $\displaystyle \sum_{j \in J} k_j$.

Also see

 * Multiindex