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:
 * $\displaystyle \prod_{j \mathop \in J} X_j^{k_j}$

with integer exponents $k_j \ge 0$ such that $k_j = 0$ for all but finitely many $j$.

Let $\mathbf X = \left({X_j}\right)_{j \in J}$ and for a multiindex $k = \left({k_j}\right)_{j \in J}$ over $J$ define:


 * $\displaystyle \mathbf X^k = \prod_{j \mathop \in J}X_j^{k_j}$

Then a mononomial is an object of the form $\mathbf X^k$, where $k$ is a multiindex.

Linguistic Note
Some sources give this as monomial which, although not technically correct (the breakdown is mono-nomial: mono for one and nomial for number), is shorter to write and say.