Category:Monomials

From ProofWiki
Jump to navigation Jump to search

This category contains results about Monomials.
Definitions specific to this category can be found in Definitions/Monomials.


Monomial of Polynomial Ring

Multiple Variables

Let $I$ be a set.

Let $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ be a polynomial ring in $I$ variables $\family {X_i}_{i \mathop \in I}$.

Let $y \in R \sqbrk {\family {X_i}_{i \mathop \in I} }$.


A monomial of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ is an element that is a product of variables; specifically:


Definition 1

The element $y$ is a monomial of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ if and only if there exists a mapping $a: I \to \N$ with finite support such that:

$y = \ds \prod_{i \mathop \in I} X_i^{a_i}$

where:

$\prod$ denotes the product with finite support over $I$
$X_i^{a_i}$ denotes the $a_i$th power of $X_i$.


Definition 2

The element $y$ is a monomial of $R \sqbrk {\family {X_i}_{i \mathop \in I} }$ if and only if there exists a finite set $S$ and a mapping $f: S \to \set {X_i : i \in I}$ such that it equals

$y = \ds \prod_{s \mathop \in S} \map f s$

where $\prod$ denotes the product over the finite set $S$.


Monomial of Free Commutative Monoid

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.