Definition:Monomial of Free Commutative Monoid

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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.


Also see