Definition:Monomial of Free Commutative Monoid

Definition
A monomial in the indexed set $\family {X_j: j \in J}$ is a possibly infinite product:
 * $\ds \prod_{j \mathop \in J} X_j^{k_j}$

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

Let $\mathbf X = \family {X_j}_{j \mathop \in J}$ and for a multiindex $k = \paren {k_j}_{j \mathop \in J}$ over $J$ define:


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

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

Also see

 * Definition:Ordered Tuple