Definition:Monomial of Free Commutative Monoid

Definition
A mononomial in the indexed set $\{X_j:j\in J\}$ 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:
 * $\displaystyle \prod_{\substack{j\in J\\k_j\neq 0}}X_j^{k_j}$

over $j\in J$.

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

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


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

Notation
For brevity, for a mononomial $\displaystyle m=\prod_{j\in J}X_j^{k_j}$ we sometimes let $k=(k_j)_{j\in J}$ be the family of indices, which we call a multiindex.

Formally speaking, a multiindex is an element of $\Z^J$, the free $\Z$-module on $J$, an abelian group of rank over $\Z$ equal to the cardinality of $J$.

We define addition of multiindices by $(k+k')_j=k_j+k_j'$, and write $m=\mathbf{X}^k$ without explicit reference to the indexing set.

With this notation, multiplication of mononomials $m=\mathbf X^k$, $m'=\mathbf X^{k'}$ is written:


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