Definition:Monomial of Free Commutative Monoid

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


 * $\prod_{k_j\neq 0}X_j^{k_j}$

over $$j\in J$$.

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

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


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

Notation
For brevity, for a mononomial $$m=\prod_{j\in J}X_j^{k_j}$$ we sometimes let $$k=(k_j)_{j\in J}$$ be the family on indices, which we call a multiindex, that is, a family of non-negative integers indexed by $$J$$, with only finitely many non-zero.

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


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