| Safe Haskell | Safe |
|---|---|
| Language | Haskell2010 |
NumHask.Algebra.Additive
Contents
Description
Additive Structure
- class AdditiveMagma a where
- class AdditiveMagma a => AdditiveUnital a where
- class AdditiveMagma a => AdditiveAssociative a
- class AdditiveMagma a => AdditiveCommutative a
- class AdditiveMagma a => AdditiveInvertible a where
- class AdditiveMagma b => AdditiveHomomorphic a b where
- class AdditiveMagma a => AdditiveIdempotent a
- class (AdditiveUnital a, AdditiveAssociative a) => AdditiveMonoidal a
- class (AdditiveCommutative a, AdditiveUnital a, AdditiveAssociative a) => Additive a where
- class (AdditiveUnital a, AdditiveAssociative a, AdditiveInvertible a) => AdditiveRightCancellative a where
- class (AdditiveUnital a, AdditiveAssociative a, AdditiveInvertible a) => AdditiveLeftCancellative a where
- class (Additive a, AdditiveInvertible a) => AdditiveGroup a where
Additive Structure
class AdditiveMagma a where Source #
plus is used for the additive magma to distinguish from + which, by convention, implies commutativity
Minimal complete definition
Instances
class AdditiveMagma a => AdditiveUnital a where Source #
AdditiveUnital
zero `plus` a == a a `plus` zero == a
Minimal complete definition
Instances
class AdditiveMagma a => AdditiveAssociative a Source #
AdditiveAssociative
(a `plus` b) `plus` c == a `plus` (b `plus` c)
class AdditiveMagma a => AdditiveCommutative a Source #
AdditiveCommutative
a `plus` b == b `plus` a
class AdditiveMagma a => AdditiveInvertible a where Source #
AdditiveInvertible
∀ a ∈ A: negate a ∈ A
law is true by construction in Haskell
Minimal complete definition
class AdditiveMagma b => AdditiveHomomorphic a b where Source #
AdditiveHomomorphic
∀ a ∈ A: plushom a ∈ B
law is true by construction in Haskell
Minimal complete definition
Instances
| AdditiveMagma a => AdditiveHomomorphic a a Source # | |
class (AdditiveUnital a, AdditiveAssociative a) => AdditiveMonoidal a Source #
AdditiveMonoidal
class (AdditiveCommutative a, AdditiveUnital a, AdditiveAssociative a) => Additive a where Source #
Additive is commutative, unital and associative under addition
a + b = b + a
(a + b) + c = a + (b + c)
zero + a = a
a + zero = a
class (AdditiveUnital a, AdditiveAssociative a, AdditiveInvertible a) => AdditiveRightCancellative a where Source #
Non-commutative right minus
class (AdditiveUnital a, AdditiveAssociative a, AdditiveInvertible a) => AdditiveLeftCancellative a where Source #
Non-commutative left minus
class (Additive a, AdditiveInvertible a) => AdditiveGroup a where Source #
AdditiveGroup
a - a = zero
negate a = zero - a
negate a + a = zero
Instances