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 $j$.

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


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

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

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


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

which using multiindex notation reads:


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

Degree
The degree of a mononomial is $\displaystyle \sum_{j \in J} k_j$, that is, the modulus of the corresponding 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.