- newtype Linear r a = Linear {
- ($*) :: (a -> r) -> r
- (.*) :: LeftModule r m => r -> m -> m
- (*.) :: RightModule r m => m -> r -> m
- type Vector = (->)
- unitVector :: FreeUnitalAlgebra r a => a -> r
- type Covector a r = Linear r a
- counitCovector :: FreeCounitalCoalgebra r c => Linear r c
- embedCovector :: (Unital m, FreeCounitalCoalgebra r m) => r -> Linear r m
- augmentCovector :: Unital s => Linear s a -> s
Documentation
Linear functionals from elements of a free module to a scalar
RightModule r s => RightModule r (Linear s m) | |
LeftModule r s => LeftModule r (Linear s m) | |
Monad (Linear r) | |
Functor (Linear r) | |
AdditiveMonoid r => MonadPlus (Linear r) | |
Applicative (Linear r) | |
AdditiveMonoid r => Alternative (Linear r) | |
AdditiveMonoid r => Plus (Linear r) | |
Additive r => Alt (Linear r) | |
Apply (Linear r) | |
Bind (Linear r) | |
Additive r => Additive (Linear r a) | |
Abelian s => Abelian (Linear s a) | |
FreeCoalgebra r m => Semiring (Linear r m) | |
FreeCoalgebra r m => Multiplicative (Linear r m) | |
FreeCounitalCoalgebra r m => Unital (Linear r m) | |
(Commutative m, FreeCoalgebra r m) => Commutative (Linear r m) | |
AdditiveMonoid s => AdditiveMonoid (Linear s a) | |
(Rig r, FreeCounitalCoalgebra r m) => Rig (Linear r m) | |
AdditiveGroup s => AdditiveGroup (Linear s a) | |
(Rng r, FreeCounitalCoalgebra r m) => Rng (Linear r m) | |
(Ring r, FreeCounitalCoalgebra r m) => Ring (Linear r m) | |
FreeCoalgebra r m => RightModule (Linear r m) (Linear r m) | |
FreeCoalgebra r m => LeftModule (Linear r m) (Linear r m) |
(.*) :: LeftModule r m => r -> m -> mSource
(*.) :: RightModule r m => m -> r -> mSource
Vectors
unitVector :: FreeUnitalAlgebra r a => a -> rSource
Covectors as linear functionals
counitCovector :: FreeCounitalCoalgebra r c => Linear r cSource
embedCovector :: (Unital m, FreeCounitalCoalgebra r m) => r -> Linear r mSource
augmentCovector :: Unital s => Linear s a -> sSource
The augmentation ring homomorphism from r^a -> r, generalizes the augmentation homomorphism from a monoid semiring to the underlying semiring