# 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.

### Multiplication

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

- $\ds \paren {\prod_{j \mathop \in J} X_j^{k_j} } \circ \paren {\prod_{j \mathop \in J} X_j^{k_j'} } = \paren {\prod_{j \mathop \in J} X_j^{k_j + k_j'} }$

which using multiindex addition notation reads:

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

### Degree

The **degree** of a monomial is defined as:

- $\ds \sum_{j \mathop \in J} k_j$

that is, the modulus of the corresponding multiindex.

## Linguistic Note

Because the word **monomial** derives ultimately from the late Latin word **binomium** (**binomial**), by changing the prefix **bi** (**two** in Latin), it should theoretically be **mononomial**.

**Monomial** is a syncope by haplology of **mononomial**.

The word **mononomial** does exist in English: it means **having a single name**.