Data.Field

class (EuclideanRing a) <= Field a 

The Field class is for types that are commutative fields.

Instances must satisfy the following law in addition to the EuclideanRing laws:

• Non-zero multiplicative inverse: a `mod` b = zero for all a and b

The Unit instance is provided for backwards compatibility, but it is not law-abiding, because Unit does not obey the EuclideanRing laws. This instance will be removed in a future release.

Instances

• Field Number

class (Ring a) <= CommutativeRing a 

The CommutativeRing class is for rings where multiplication is commutative.

Instances must satisfy the following law in addition to the Ring laws:

• Commutative multiplication: a * b = b * a

Instances

• CommutativeRing Int
• CommutativeRing Number
• CommutativeRing Unit
• (CommutativeRing b) => CommutativeRing (a -> b)

class (CommutativeRing a) <= EuclideanRing a  where

The EuclideanRing class is for commutative rings that support division. The mathematical structure this class is based on is sometimes also called a Euclidean domain.

Instances must satisfy the following laws in addition to the Ring laws:

• Integral domain: one /= zero, and if a and b are both nonzero then so is their product a * b
• Euclidean function degree:
  • Nonnegativity: For all nonzero a, degree a >= 0
  • Quotient/remainder: For all a and b, where b is nonzero, let q = a / b and r = a `mod` b; then a = q*b + r, and also either r = zero or degree r < degree b
• Submultiplicative euclidean function:
  • For all nonzero a and b, degree a <= degree (a * b)

The behaviour of division by zero is unconstrained by these laws, meaning that individual instances are free to choose how to behave in this case. Similarly, there are no restrictions on what the result of degree zero is; it doesn't make sense to ask for degree zero in the same way that it doesn't make sense to divide by zero, so again, individual instances may choose how to handle this case.

For any EuclideanRing which is also a Field, one valid choice for degree is simply const 1. In fact, unless there's a specific reason not to, Field types should normally use this definition of degree.

The Unit instance is provided for backwards compatibility, but it is not law-abiding, because Unit fails to form an integral domain. This instance will be removed in a future release.

Members

• degree :: a -> Int
• div :: a -> a -> a
• mod :: a -> a -> a

Instances

• EuclideanRing Int
• EuclideanRing Number

lcm :: forall a. Eq a => EuclideanRing a => a -> a -> a

The least common multiple of two values.

gcd :: forall a. Eq a => EuclideanRing a => a -> a -> a

The greatest common divisor of two values.

Operator alias for Data.EuclideanRing.div (left-associative / precedence 7)

class (Semiring a) <= Ring a  where

The Ring class is for types that support addition, multiplication, and subtraction operations.

Instances must satisfy the following law in addition to the Semiring laws:

• Additive inverse: a - a = (zero - a) + a = zero

Members

• sub :: a -> a -> a

Instances

• Ring Int
• Ring Number
• Ring Unit
• (Ring b) => Ring (a -> b)

negate :: forall a. Ring a => a -> a

negate x can be used as a shorthand for zero - x.

class Semiring a  where

The Semiring class is for types that support an addition and multiplication operation.

Instances must satisfy the following laws:

• Commutative monoid under addition:
  • Associativity: (a + b) + c = a + (b + c)
  • Identity: zero + a = a + zero = a
  • Commutative: a + b = b + a
• Monoid under multiplication:
  • Associativity: (a * b) * c = a * (b * c)
  • Identity: one * a = a * one = a
• Multiplication distributes over addition:
  • Left distributivity: a * (b + c) = (a * b) + (a * c)
  • Right distributivity: (a + b) * c = (a * c) + (b * c)
• Annihilation: zero * a = a * zero = zero

Note: The Number and Int types are not fully law abiding members of this class hierarchy due to the potential for arithmetic overflows, and in the case of Number, the presence of NaN and Infinity values. The behaviour is unspecified in these cases.

Members

• add :: a -> a -> a
• zero :: a
• mul :: a -> a -> a
• one :: a

Instances

• Semiring Int
• Semiring Number
• (Semiring b) => Semiring (a -> b)
• Semiring Unit

Operator alias for Data.Semiring.add (left-associative / precedence 6)

Operator alias for Data.Semiring.mul (left-associative / precedence 7)