Category:Reducible
This category contains results about Reducible.
Definitions specific to this category can be found in Definitions/Reducible.
Reducible Fraction
Let $q = \dfrac a b$ be a vulgar fraction.
Then $q$ is defined as being reducible if and only if $q$ is not in canonical form.
That is, if and only if there exists $r \in \Z: r \ne 1$ such that $r$ is a divisor of both $a$ and $b$.
Such a fraction can therefore be reduced by dividing both $a$ and $b$ by $r$.
Reducible Polynomial
Let $K$ be a field.
A reducible polynomial over $K$ is a nonconstant polynomial over $K$ that can be expressed as the product of two polynomials over $K$ of smaller degree.
Reducible Linear Representation
Let $\rho: G \to \GL V$ be a linear representation.
$\rho$ is reducible if and only if there exists a non-trivial proper vector subspace $W$ of $V$ such that:
- $\forall g \in G: \map {\map \rho g} W \subseteq W$
That is, such that $W$ is invariant for every linear operator in the set $\set {\map \rho g: g \in G}$.
Reducible $G$-Module
Let $M$ be a $G$-module.
Then $M$ is reducible if and only if the corresponding linear representation is reducible.
Mapping Reducible, also known as Many-One Reducible
Let $\Sigma, \Sigma'$ be finite sets.
Let:
\(\ds L\) | \(\subseteq\) | \(\ds \Sigma^*\) | ||||||||||||
\(\ds L'\) | \(\subseteq\) | \(\ds \Sigma'^*\) |
be sets of finite strings over $\Sigma$ and $\Sigma'$ respectively, where:
- $\Sigma^*$ denotes the set of all finite strings over the alphabet $\Sigma$.
Let $f : \Sigma^* \to \Sigma'^*$ be a computable function such that, for all $w \in \Sigma^*$:
- $w \in L \iff \map f w \in L'$
Then, $f$ is a mapping reduction from $L$ to $L'$.
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