Precursor.Algebra.Semiring

Documentation

class Semiring a where

A Semiring is like the the combination of two Monoids. The first is called +; it has the identity element zero, and it is commutative. The second is called *; it has identity element one, and it must distribute over +.

Laws

Normal Monoid laws

• (a + b) + c = a + (b + c)

• zero + a = a + zero = a

• (a * b) * c = a * (b * c)

• one * a = a * one = a

Commutativity of +

• a + b = b + a

Distribution of * over +

• a*(b + c) = (a*b) + (a*c)

• (a + b)*c = (a*c) + (b*c)

Another useful law, annihilation, may be deduced from the axioms above:

• zero * a = a * zero = zero

Methods

one :: a

The identity of *.

zero :: a

The identity of +.

(*) :: a -> a -> a infixl 7

An associative binary operation, which distributes over +.

(+) :: a -> a -> a infixl 6

An associative, commutative binary operation.

one :: Num a => a

The identity of *.

zero :: Num a => a

The identity of +.

(+) :: Num a => a -> a -> a infixl 6

An associative, commutative binary operation.

(*) :: Num a => a -> a -> a infixl 7

An associative binary operation, which distributes over +.

Instances

Semiring Bool

Methods

one :: Bool

zero :: Bool

(*) :: Bool -> Bool -> Bool

(+) :: Bool -> Bool -> Bool

Semiring Double

Methods

one :: Double

zero :: Double

(*) :: Double -> Double -> Double

(+) :: Double -> Double -> Double

Semiring Float

Methods

one :: Float

zero :: Float

(*) :: Float -> Float -> Float

(+) :: Float -> Float -> Float

Semiring Int

Methods

one :: Int

zero :: Int

(*) :: Int -> Int -> Int

(+) :: Int -> Int -> Int

Semiring Int8

Methods

one :: Int8

zero :: Int8

(*) :: Int8 -> Int8 -> Int8

(+) :: Int8 -> Int8 -> Int8

Semiring Int16

Methods

one :: Int16

zero :: Int16

(*) :: Int16 -> Int16 -> Int16

(+) :: Int16 -> Int16 -> Int16

Semiring Int32

Methods

one :: Int32

zero :: Int32

(*) :: Int32 -> Int32 -> Int32

(+) :: Int32 -> Int32 -> Int32

Semiring Int64

Methods

one :: Int64

zero :: Int64

(*) :: Int64 -> Int64 -> Int64

(+) :: Int64 -> Int64 -> Int64

Semiring Integer

Methods

one :: Integer

zero :: Integer

(*) :: Integer -> Integer -> Integer

(+) :: Integer -> Integer -> Integer

Semiring b => Semiring (a -> b)

Methods

one :: a -> b

zero :: a -> b

(*) :: (a -> b) -> (a -> b) -> a -> b

(+) :: (a -> b) -> (a -> b) -> a -> b