Copyright	(c) 2011 diagrams-core team (see LICENSE)
License	BSD-style (see LICENSE)
Maintainer	diagrams-discuss@googlegroups.com
Safe Haskell	Safe-Inferred
Language	Haskell2010

Data.Monoid.Action

Description

Monoid and semigroup actions.

Synopsis

class Action m s where
- act :: m -> s -> s
newtype Regular m = Regular {
- getRegular :: m
}
newtype Conjugate m = Conjugate {
- getConjugate :: m
}
class Group m => Torsor m s where
- difference :: s -> s -> m

Documentation

class Action m s where Source #

Type class for monoid (and semigroup) actions, where monoidal values of type m "act" on values of another type s. Instances are required to satisfy the laws

```
act mempty = id
```
```
act (m1 `mappend` m2) = act m1 . act m2
```

Semigroup instances are required to satisfy the second law but with (<>) instead of mappend. Additionally, if the type s has any algebraic structure, act m should be a homomorphism. For example, if s is also a monoid we should have act m mempty = mempty and act m (s1 `mappend` s2) = (act m s1) `mappend` (act m s2).

By default, act = const id, so for a type M which should have no action on anything, it suffices to write

instance Action M s

with no method implementations.

It is a bit awkward dealing with instances of Action, since it is a multi-parameter type class but we can't add any functional dependencies---the relationship between monoids and the types on which they act is truly many-to-many. In practice, this library has chosen to have instance selection for Action driven by the first type parameter. That is, you should never write an instance of the form Action m SomeType since it will overlap with instances of the form Action SomeMonoid t. Newtype wrappers can be used to (awkwardly) get around this.

Minimal complete definition

Nothing

Methods

act :: m -> s -> s Source #

Convert a value of type m to an action on s values.

Instances

Instances details

Action Void a Source #
Instance details Defined in Data.Monoid.Action Methods act :: Void -> a -> a Source #
Action () l Source #	`()` acts as the identity.
Instance details Defined in Data.Monoid.Action Methods act :: () -> l -> l Source #
Fractional a => Action Rational (Product a) Source #
Instance details Defined in Data.Monoid.Action Methods act :: Rational -> Product a -> Product a Source #
Fractional a => Action Rational (Sum a) Source #
Instance details Defined in Data.Monoid.Action Methods act :: Rational -> Sum a -> Sum a Source #
Num a => Action Integer (Product a) Source #
Instance details Defined in Data.Monoid.Action Methods act :: Integer -> Product a -> Product a Source #
Num a => Action Integer (Sum a) Source #
Instance details Defined in Data.Monoid.Action Methods act :: Integer -> Sum a -> Sum a Source #
Group m => Action m (Conjugate m) Source #
Instance details Defined in Data.Monoid.Action Methods act :: m -> Conjugate m -> Conjugate m Source #
Semigroup m => Action m (Regular m) Source #
Instance details Defined in Data.Monoid.Action Methods act :: m -> Regular m -> Regular m Source #
Action m s => Action (Identity m) s Source #
Instance details Defined in Data.Monoid.Action Methods act :: Identity m -> s -> s Source #
Action (First a) a Source #
Instance details Defined in Data.Monoid.Action Methods act :: First a -> a -> a Source #
Action (First a) a Source #
Instance details Defined in Data.Monoid.Action Methods act :: First a -> a -> a Source #
Action (Endo a) a Source #	`Endo` acts by application. Note that in order for this instance to satisfy the `Action` laws, whenever the type `a` has some sort of algebraic structure, the type `Endo a` must be considered to represent homomorphisms (structure-preserving maps) on `a`, even though there is no way to enforce this in the type system. For example, if `a` is an instance of `Monoid`, then one should only use `Endo a` values `f` with the property that `f mempty = mempty` and `f (a <> b) = f a <> f b`.
Instance details Defined in Data.Monoid.Action Methods act :: Endo a -> a -> a Source #
Action (SM a) () Source #
Instance details Defined in Data.Monoid.MList Methods act :: SM a -> () -> () Source #
Action m n => Action (Split m) n Source #	By default, the action of a split monoid is the same as for the underlying monoid, as if the split were removed.
Instance details Defined in Data.Monoid.Split Methods act :: Split m -> n -> n Source #
Action m s => Action (Maybe m) s Source #	`Nothing` acts as the identity; `Just m` acts as `m`.
Instance details Defined in Data.Monoid.Action Methods act :: Maybe m -> s -> s Source #
Action m s => Action [m] s Source #	act [a,b,c,...] = act a . act b . act c . ...
Instance details Defined in Data.Monoid.Action Methods act :: [m] -> s -> s Source #
(Action a a', Action (SM a) l) => Action (SM a) (Maybe a', l) Source #
Instance details Defined in Data.Monoid.MList Methods act :: SM a -> (Maybe a', l) -> (Maybe a', l) Source #
(Action m r, Action n r) => Action (m :+: n) r Source #	Coproducts act on other things by having each of the components act individually.
Instance details Defined in Data.Monoid.Coproduct Methods act :: (m :+: n) -> r -> r Source #
(Action m n, Action m r, Action n r, Semigroup n) => Action (m :+: n) r Source #	Coproducts act on other things by having each of the components act individually.
Instance details Defined in Data.Monoid.Coproduct.Strict Methods act :: (m :+: n) -> r -> r Source #
(Action (SM a) l2, Action l1 l2) => Action (a, l1) l2 Source #
Instance details Defined in Data.Monoid.MList Methods act :: (a, l1) -> l2 -> l2 Source #

newtype Regular m Source #

Any monoid acts on itself by left multiplication. This newtype witnesses this action: getRegular $ Regular m1 `act' Regular m2 = m1 <> m2

Constructors

Regular
Fields getRegular :: m

Instances

Instances details

Semigroup m => Action m (Regular m) Source #
Instance details Defined in Data.Monoid.Action Methods act :: m -> Regular m -> Regular m Source #
Group m => Torsor m (Regular m) Source #
Instance details Defined in Data.Monoid.Action Methods difference :: Regular m -> Regular m -> m Source #

newtype Conjugate m Source #

Any group acts on itself by conjugation.

Constructors

Conjugate
Fields getConjugate :: m

Instances

Instances details

Group m => Action m (Conjugate m) Source #
Instance details Defined in Data.Monoid.Action Methods act :: m -> Conjugate m -> Conjugate m Source #

class Group m => Torsor m s where Source #

An action of a group is "free transitive", "regular", or a "torsor" iff it is invertible.

Given an original value sOrig, and a value sActed that is the result of acting on sOrig by some m, it is possible to recover this m. This is encoded in the laws:

```
(m `act' s) `difference' s = m
```

(sActed `difference' sOrig) `act' sOrig = sActed

Methods

difference :: s -> s -> m Source #

difference sActed sOrig is the element m such that sActed = m `act' sOrig.

Instances

Instances details

Group m => Torsor m (Regular m) Source #
Instance details Defined in Data.Monoid.Action Methods difference :: Regular m -> Regular m -> m Source #