| Safe Haskell | Unsafe |
|---|---|
| Language | Haskell2010 |
NumHask.Algebra.Multiplicative
Contents
Description
Multiplicate structure Many treatments of a numeric tower treat multiplication differently to addition. NumHask treats these two as exactly symmetrical, and thus departs from the usual mathematical terminology.
- class MultiplicativeMagma a where
- class MultiplicativeMagma a => MultiplicativeUnital a where
- class MultiplicativeMagma a => MultiplicativeAssociative a
- class MultiplicativeMagma a => MultiplicativeCommutative a
- class MultiplicativeMagma a => MultiplicativeInvertible a where
- class MultiplicativeMagma b => MultiplicativeHomomorphic a b where
- class (MultiplicativeUnital a, MultiplicativeAssociative a) => MultiplicativeMonoidal a
- class (MultiplicativeCommutative a, MultiplicativeUnital a, MultiplicativeAssociative a) => Multiplicative a where
- class (MultiplicativeUnital a, MultiplicativeAssociative a, MultiplicativeInvertible a) => MultiplicativeRightCancellative a where
- class (MultiplicativeUnital a, MultiplicativeAssociative a, MultiplicativeInvertible a) => MultiplicativeLeftCancellative a where
- class (Multiplicative a, MultiplicativeInvertible a) => MultiplicativeGroup a where
Multiplicative Structure
class MultiplicativeMagma a where Source #
times is used for the multiplicative magma to distinguish from * which, by convention, implies commutativity
Minimal complete definition
class MultiplicativeMagma a => MultiplicativeUnital a where Source #
MultiplicativeUnital
one `times` a == a a `times` one == a
Minimal complete definition
class MultiplicativeMagma a => MultiplicativeAssociative a Source #
MultiplicativeAssociative
(a `times` b) `times` c == a `times` (b `times` c)
Instances
class MultiplicativeMagma a => MultiplicativeCommutative a Source #
MultiplicativeCommutative
a `times` b == b `times` a
Instances
class MultiplicativeMagma a => MultiplicativeInvertible a where Source #
MultiplicativeInvertible
∀ a ∈ A: recip a ∈ A
law is true by construction in Haskell
Minimal complete definition
Instances
class MultiplicativeMagma b => MultiplicativeHomomorphic a b where Source #
MultiplicativeHomomorphic
∀ a ∈ A: timeshom a ∈ B
law is true by construction in Haskell
Minimal complete definition
Instances
class (MultiplicativeUnital a, MultiplicativeAssociative a) => MultiplicativeMonoidal a Source #
MultiplicativeMonoidal
class (MultiplicativeCommutative a, MultiplicativeUnital a, MultiplicativeAssociative a) => Multiplicative a where Source #
Multiplicative is commutative, associative and unital under multiplication
a * b = b * a
(a * b) * c = a * (b * c)
one * a = a
a * one = a
Instances
class (MultiplicativeUnital a, MultiplicativeAssociative a, MultiplicativeInvertible a) => MultiplicativeRightCancellative a where Source #
Non-commutative right divide
class (MultiplicativeUnital a, MultiplicativeAssociative a, MultiplicativeInvertible a) => MultiplicativeLeftCancellative a where Source #
Non-commutative left divide
class (Multiplicative a, MultiplicativeInvertible a) => MultiplicativeGroup a where Source #
MultiplicativeGroup
a / a = one
recip a = one / a
recip a * a = one
Instances