| Safe Haskell | None |
|---|---|
| Language | Haskell2010 |
NumHask.Algebra.Ring
Description
Ring classes
Synopsis
- class (Additive a, Multiplicative a) => Distributive a
- class (Distributive a, Subtractive a) => Ring a
- class Distributive a => StarSemiring a where
- class (StarSemiring a, Idempotent a) => KleeneAlgebra a
- class Distributive a => InvolutiveRing a where
- adj :: a -> a
- two :: (Multiplicative a, Additive a) => a
Documentation
class (Additive a, Multiplicative a) => Distributive a Source #
\a b c -> a * (b + c) == a * b + a * c
\a b c -> (a + b) * c == a * c + b * c
\a -> zero * a == zero
\a -> a * zero == zero
The sneaking in of the Absorption laws here glosses over the possibility that the multiplicative zero element does not have to correspond with the additive unital zero.
Instances
class (Distributive a, Subtractive a) => Ring a Source #
A Ring is an abelian group under addition (Unital, Associative, Commutative, Invertible) and monoidal under multiplication (Unital, Associative), and where multiplication distributes over addition.
\a -> zero + a == a \a -> a + zero == a \a b c -> (a + b) + c == a + (b + c) \a b -> a + b == b + a \a -> a - a == zero \a -> negate a == zero - a \a -> negate a + a == zero \a -> a + negate a == zero \a -> one * a == a \a -> a * one == a \a b c -> (a * b) * c == a * (b * c) \a b c -> a * (b + c) == a * b + a * c \a b c -> (a + b) * c == a * c + b * c \a -> zero * a == zero \a -> a * zero == zero
Instances
| (Distributive a, Subtractive a) => Ring a Source # | |
Defined in NumHask.Algebra.Ring | |
class Distributive a => StarSemiring a where Source #
A StarSemiring is a semiring with an additional unary operator (star) satisfying:
\a -> star a == one + a * star a
Minimal complete definition
Nothing
Instances
| StarSemiring b => StarSemiring (a -> b) Source # | |
class (StarSemiring a, Idempotent a) => KleeneAlgebra a Source #
A Kleene Algebra is a Star Semiring with idempotent addition.
a * x + x = a ==> star a * x + x = x x * a + x = a ==> x * star a + x = x
Instances
| KleeneAlgebra b => KleeneAlgebra (a -> b) Source # | |
Defined in NumHask.Algebra.Ring | |
class Distributive a => InvolutiveRing a where Source #
Involutive Ring
adj (a + b) ==> adj a + adj b adj (a * b) ==> adj a * adj b adj one ==> one adj (adj a) ==> a
Note: elements for which adj a == a are called "self-adjoint".
Minimal complete definition
Nothing