Data.Ring

Documentation

(<<) :: Group a => a -> a -> a infixl 6

(><) :: Semiring a => a -> a -> a infixl 7

Multiplicative operation.

(<>) :: Semigroup a => a -> a -> a infixr 6

An associative operation.

negate :: Group a => a -> a

class (Group a, Semiring a) => Ring a where

Rings.

A ring R is a commutative group with a second monoidal operation >< that distributes over <>.

The basic properties of a ring follow immediately from the axioms:

 r >< mempty ≡ mempty ≡ mempty >< r

 negate sunit >< r ≡ negate r

Furthermore, the binomial formula holds for any commuting pair of elements (that is, any a and b such that a >= b< a).

If mempty = sunit in a ring R, then R has only one element, and is called the zero ring. Otherwise the additive identity, the additive inverse of each element, and the multiplicative identity are unique.

See https://en.wikipedia.org/wiki/Ring_(mathematics).

If the ring is ordered (i.e. has an Ord instance), then the following additional properties must hold:

 a b == a <> c b < c

 mempty a && mempty <= b == mempty a< b

See the properties module for a detailed specification of the laws.

Minimal complete definition

fromInteger

Methods

fromInteger :: Integer -> a

A ring homomorphism from the integers to a.

abs :: Ord a => a -> a

Absolute value of an element.

 abs r ≡ r >< signum r

signum :: Ord a => a -> a

Instances

Ring Int

Defined in Data.Ring

Methods

fromInteger :: Integer -> Int

abs :: Int -> Int

signum :: Int -> Int

Ring Int8

Defined in Data.Ring

Methods

fromInteger :: Integer -> Int8

abs :: Int8 -> Int8

signum :: Int8 -> Int8

Ring Int16

Defined in Data.Ring

Methods

fromInteger :: Integer -> Int16

abs :: Int16 -> Int16

signum :: Int16 -> Int16

Ring Int32

Defined in Data.Ring

Methods

fromInteger :: Integer -> Int32

abs :: Int32 -> Int32

signum :: Int32 -> Int32

Ring Int64

Defined in Data.Ring

Methods

fromInteger :: Integer -> Int64

abs :: Int64 -> Int64

signum :: Int64 -> Int64

Ring Integer

Defined in Data.Ring

Methods

fromInteger :: Integer -> Integer

abs :: Integer -> Integer

signum :: Integer -> Integer

Ring Rational

Defined in Data.Ring

Methods

fromInteger :: Integer -> Rational

abs :: Rational -> Rational

signum :: Rational -> Rational

Ring Uni

Defined in Data.Ring

Methods

fromInteger :: Integer -> Uni

abs :: Uni -> Uni

signum :: Uni -> Uni

Ring Deci

Defined in Data.Ring

Methods

fromInteger :: Integer -> Deci

abs :: Deci -> Deci

signum :: Deci -> Deci

Ring Centi

Defined in Data.Ring

Methods

fromInteger :: Integer -> Centi

abs :: Centi -> Centi

signum :: Centi -> Centi

Ring Milli

Defined in Data.Ring

Methods

fromInteger :: Integer -> Milli

abs :: Milli -> Milli

signum :: Milli -> Milli

Ring Micro

Defined in Data.Ring

Methods

fromInteger :: Integer -> Micro

abs :: Micro -> Micro

signum :: Micro -> Micro

Ring Nano

Defined in Data.Ring

Methods

fromInteger :: Integer -> Nano

abs :: Nano -> Nano

signum :: Nano -> Nano

Ring Pico

Defined in Data.Ring

Methods

fromInteger :: Integer -> Pico

abs :: Pico -> Pico

signum :: Pico -> Pico

Ring a => Ring (Complex a)

Defined in Data.Ring

Methods

fromInteger :: Integer -> Complex a

abs :: Complex a -> Complex a

signum :: Complex a -> Complex a

Ring a => Ring (Quaternion a)

Defined in Data.Semifield.Quaternion

Methods

fromInteger :: Integer -> Quaternion a

abs :: Quaternion a -> Quaternion a

signum :: Quaternion a -> Quaternion a