Definition:Monomial of Free Commutative Monoid

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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.


Multiplication

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

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

which using multiindex addition notation reads:

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


Degree

The degree of a mononomial is defined as:

$\displaystyle \sum_{j \mathop \in J} k_j$

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.


Also see