Safe Haskell	Safe
Language	Haskell2010

Algebra

Synopsis

class Semigroup a where
- (<>) :: a -> a -> a
- sconcat :: NonEmpty a -> a
- stimes :: Integral b => b -> a -> a
class Semigroup a => Monoid a where
- mempty :: a
class Monoid a => Group a where
- invert :: a -> a
class Semigroup a => Abelian a
class Semigroup a => Idempotent a
(+) :: Semigroup (Sum a) => a -> a -> a
(-) :: Group (Sum a) => a -> a -> a
(*) :: Semigroup (Product a) => a -> a -> a
(/) :: (Semigroup (Product a), Group (Product a)) => a -> a -> a
(×) :: Semigroup (Product a) => a -> a -> a
commuteWith :: Group b => (a -> a -> b) -> a -> a -> b

Documentation

class Semigroup a where #

The class of semigroups (types with an associative binary operation).

Instances should satisfy the associativity law:

```
x <> (y <> z) = (x <> y) <> z
```

Since: base-4.9.0.0

Minimal complete definition

(<>)

Methods

(<>) :: a -> a -> a infixr 6 #

An associative operation.

sconcat :: NonEmpty a -> a #

Reduce a non-empty list with <>

The default definition should be sufficient, but this can be overridden for efficiency.

stimes :: Integral b => b -> a -> a #

Repeat a value n times.

Given that this works on a Semigroup it is allowed to fail if you request 0 or fewer repetitions, and the default definition will do so.

By making this a member of the class, idempotent semigroups and monoids can upgrade this to execute in O(1) by picking stimes = stimesIdempotent or stimes = stimesIdempotentMonoid respectively.

Instances

Semigroup Ordering	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: Ordering -> Ordering -> Ordering # sconcat :: NonEmpty Ordering -> Ordering # stimes :: Integral b => b -> Ordering -> Ordering #
Semigroup ()	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: () -> () -> () # sconcat :: NonEmpty () -> () # stimes :: Integral b => b -> () -> () #
Semigroup All	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup.Internal Methods (<>) :: All -> All -> All # sconcat :: NonEmpty All -> All # stimes :: Integral b => b -> All -> All #
Semigroup Any	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup.Internal Methods (<>) :: Any -> Any -> Any # sconcat :: NonEmpty Any -> Any # stimes :: Integral b => b -> Any -> Any #
Semigroup [a]	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: [a] -> [a] -> [a] # sconcat :: NonEmpty [a] -> [a] # stimes :: Integral b => b -> [a] -> [a] #
Semigroup a => Semigroup (Maybe a)	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: Maybe a -> Maybe a -> Maybe a # sconcat :: NonEmpty (Maybe a) -> Maybe a # stimes :: Integral b => b -> Maybe a -> Maybe a #
Semigroup a => Semigroup (IO a)	Since: base-4.10.0.0
Instance details Defined in GHC.Base Methods (<>) :: IO a -> IO a -> IO a # sconcat :: NonEmpty (IO a) -> IO a # stimes :: Integral b => b -> IO a -> IO a #
Semigroup p => Semigroup (Par1 p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods (<>) :: Par1 p -> Par1 p -> Par1 p # sconcat :: NonEmpty (Par1 p) -> Par1 p # stimes :: Integral b => b -> Par1 p -> Par1 p #
Ord a => Semigroup (Min a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods (<>) :: Min a -> Min a -> Min a # sconcat :: NonEmpty (Min a) -> Min a # stimes :: Integral b => b -> Min a -> Min a #
Ord a => Semigroup (Max a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods (<>) :: Max a -> Max a -> Max a # sconcat :: NonEmpty (Max a) -> Max a # stimes :: Integral b => b -> Max a -> Max a #
Semigroup (First a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods (<>) :: First a -> First a -> First a # sconcat :: NonEmpty (First a) -> First a # stimes :: Integral b => b -> First a -> First a #
Semigroup (Last a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods (<>) :: Last a -> Last a -> Last a # sconcat :: NonEmpty (Last a) -> Last a # stimes :: Integral b => b -> Last a -> Last a #
Monoid m => Semigroup (WrappedMonoid m)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods (<>) :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # sconcat :: NonEmpty (WrappedMonoid m) -> WrappedMonoid m # stimes :: Integral b => b -> WrappedMonoid m -> WrappedMonoid m #
Semigroup a => Semigroup (Option a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods (<>) :: Option a -> Option a -> Option a # sconcat :: NonEmpty (Option a) -> Option a # stimes :: Integral b => b -> Option a -> Option a #
Semigroup a => Semigroup (Identity a)	Since: base-4.9.0.0
Instance details Defined in Data.Functor.Identity Methods (<>) :: Identity a -> Identity a -> Identity a # sconcat :: NonEmpty (Identity a) -> Identity a # stimes :: Integral b => b -> Identity a -> Identity a #
Semigroup (First a)	Since: base-4.9.0.0
Instance details Defined in Data.Monoid Methods (<>) :: First a -> First a -> First a # sconcat :: NonEmpty (First a) -> First a # stimes :: Integral b => b -> First a -> First a #
Semigroup (Last a)	Since: base-4.9.0.0
Instance details Defined in Data.Monoid Methods (<>) :: Last a -> Last a -> Last a # sconcat :: NonEmpty (Last a) -> Last a # stimes :: Integral b => b -> Last a -> Last a #
Semigroup a => Semigroup (Dual a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup.Internal Methods (<>) :: Dual a -> Dual a -> Dual a # sconcat :: NonEmpty (Dual a) -> Dual a # stimes :: Integral b => b -> Dual a -> Dual a #
Semigroup (Endo a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup.Internal Methods (<>) :: Endo a -> Endo a -> Endo a # sconcat :: NonEmpty (Endo a) -> Endo a # stimes :: Integral b => b -> Endo a -> Endo a #
Num a => Semigroup (Sum a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup.Internal Methods (<>) :: Sum a -> Sum a -> Sum a # sconcat :: NonEmpty (Sum a) -> Sum a # stimes :: Integral b => b -> Sum a -> Sum a #
Num a => Semigroup (Product a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup.Internal Methods (<>) :: Product a -> Product a -> Product a # sconcat :: NonEmpty (Product a) -> Product a # stimes :: Integral b => b -> Product a -> Product a #
Semigroup a => Semigroup (Down a)	Since: base-4.11.0.0
Instance details Defined in Data.Ord Methods (<>) :: Down a -> Down a -> Down a # sconcat :: NonEmpty (Down a) -> Down a # stimes :: Integral b => b -> Down a -> Down a #
Semigroup (NonEmpty a)	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: NonEmpty a -> NonEmpty a -> NonEmpty a # sconcat :: NonEmpty (NonEmpty a) -> NonEmpty a # stimes :: Integral b => b -> NonEmpty a -> NonEmpty a #
Ord a => Semigroup (Min a) Source #
Instance details Defined in Relation.Binary.Comparison Methods (<>) :: Min a -> Min a -> Min a # sconcat :: NonEmpty (Min a) -> Min a # stimes :: Integral b => b -> Min a -> Min a #
Bits a => Semigroup (Min (BitSet a)) Source #
Instance details Defined in Data.BitSet Methods (<>) :: Min (BitSet a) -> Min (BitSet a) -> Min (BitSet a) # sconcat :: NonEmpty (Min (BitSet a)) -> Min (BitSet a) # stimes :: Integral b => b -> Min (BitSet a) -> Min (BitSet a) #
Ord a => Semigroup (Max a) Source #
Instance details Defined in Relation.Binary.Comparison Methods (<>) :: Max a -> Max a -> Max a # sconcat :: NonEmpty (Max a) -> Max a # stimes :: Integral b => b -> Max a -> Max a #
Bits a => Semigroup (Max (BitSet a)) Source #
Instance details Defined in Data.BitSet Methods (<>) :: Max (BitSet a) -> Max (BitSet a) -> Max (BitSet a) # sconcat :: NonEmpty (Max (BitSet a)) -> Max (BitSet a) # stimes :: Integral b => b -> Max (BitSet a) -> Max (BitSet a) #
Semigroup a => Semigroup (Lexical a) Source #
Instance details Defined in Relation.Binary.Comparison Methods (<>) :: Lexical a -> Lexical a -> Lexical a # sconcat :: NonEmpty (Lexical a) -> Lexical a # stimes :: Integral b => b -> Lexical a -> Lexical a #
Bits a => Semigroup (BitSet a) Source #
Instance details Defined in Data.BitSet Methods (<>) :: BitSet a -> BitSet a -> BitSet a # sconcat :: NonEmpty (BitSet a) -> BitSet a # stimes :: Integral b => b -> BitSet a -> BitSet a #
Semigroup b => Semigroup (a -> b)	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: (a -> b) -> (a -> b) -> a -> b # sconcat :: NonEmpty (a -> b) -> a -> b # stimes :: Integral b0 => b0 -> (a -> b) -> a -> b #
Semigroup (Either a b)	Since: base-4.9.0.0
Instance details Defined in Data.Either Methods (<>) :: Either a b -> Either a b -> Either a b # sconcat :: NonEmpty (Either a b) -> Either a b # stimes :: Integral b0 => b0 -> Either a b -> Either a b #
Semigroup (V1 p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods (<>) :: V1 p -> V1 p -> V1 p # sconcat :: NonEmpty (V1 p) -> V1 p # stimes :: Integral b => b -> V1 p -> V1 p #
Semigroup (U1 p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods (<>) :: U1 p -> U1 p -> U1 p # sconcat :: NonEmpty (U1 p) -> U1 p # stimes :: Integral b => b -> U1 p -> U1 p #
(Semigroup a, Semigroup b) => Semigroup (a, b)	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: (a, b) -> (a, b) -> (a, b) # sconcat :: NonEmpty (a, b) -> (a, b) # stimes :: Integral b0 => b0 -> (a, b) -> (a, b) #
Semigroup (Proxy s)	Since: base-4.9.0.0
Instance details Defined in Data.Proxy Methods (<>) :: Proxy s -> Proxy s -> Proxy s # sconcat :: NonEmpty (Proxy s) -> Proxy s # stimes :: Integral b => b -> Proxy s -> Proxy s #
(Applicative p, Semigroup a) => Semigroup (Ap p a)
Instance details Defined in Util Methods (<>) :: Ap p a -> Ap p a -> Ap p a # sconcat :: NonEmpty (Ap p a) -> Ap p a # stimes :: Integral b => b -> Ap p a -> Ap p a #
Semigroup (f p) => Semigroup (Rec1 f p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods (<>) :: Rec1 f p -> Rec1 f p -> Rec1 f p # sconcat :: NonEmpty (Rec1 f p) -> Rec1 f p # stimes :: Integral b => b -> Rec1 f p -> Rec1 f p #
(Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c)	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: (a, b, c) -> (a, b, c) -> (a, b, c) # sconcat :: NonEmpty (a, b, c) -> (a, b, c) # stimes :: Integral b0 => b0 -> (a, b, c) -> (a, b, c) #
Semigroup a => Semigroup (Const a b)	Since: base-4.9.0.0
Instance details Defined in Data.Functor.Const Methods (<>) :: Const a b -> Const a b -> Const a b # sconcat :: NonEmpty (Const a b) -> Const a b # stimes :: Integral b0 => b0 -> Const a b -> Const a b #
(Applicative f, Semigroup a) => Semigroup (Ap f a)	Since: base-4.12.0.0
Instance details Defined in Data.Monoid Methods (<>) :: Ap f a -> Ap f a -> Ap f a # sconcat :: NonEmpty (Ap f a) -> Ap f a # stimes :: Integral b => b -> Ap f a -> Ap f a #
Alternative f => Semigroup (Alt f a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup.Internal Methods (<>) :: Alt f a -> Alt f a -> Alt f a # sconcat :: NonEmpty (Alt f a) -> Alt f a # stimes :: Integral b => b -> Alt f a -> Alt f a #
Semigroup c => Semigroup (K1 i c p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods (<>) :: K1 i c p -> K1 i c p -> K1 i c p # sconcat :: NonEmpty (K1 i c p) -> K1 i c p # stimes :: Integral b => b -> K1 i c p -> K1 i c p #
(Semigroup (f p), Semigroup (g p)) => Semigroup ((f :*: g) p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods (<>) :: (f :: g) p -> (f :: g) p -> (f :: g) p # sconcat :: NonEmpty ((f :: g) p) -> (f :: g) p # stimes :: Integral b => b -> (f :: g) p -> (f :*: g) p #
(Semigroup a, Semigroup b, Semigroup c, Semigroup d) => Semigroup (a, b, c, d)	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) # sconcat :: NonEmpty (a, b, c, d) -> (a, b, c, d) # stimes :: Integral b0 => b0 -> (a, b, c, d) -> (a, b, c, d) #
Semigroup (f p) => Semigroup (M1 i c f p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods (<>) :: M1 i c f p -> M1 i c f p -> M1 i c f p # sconcat :: NonEmpty (M1 i c f p) -> M1 i c f p # stimes :: Integral b => b -> M1 i c f p -> M1 i c f p #
Semigroup (f (g p)) => Semigroup ((f :.: g) p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods (<>) :: (f :.: g) p -> (f :.: g) p -> (f :.: g) p # sconcat :: NonEmpty ((f :.: g) p) -> (f :.: g) p # stimes :: Integral b => b -> (f :.: g) p -> (f :.: g) p #
(Semigroup a, Semigroup b, Semigroup c, Semigroup d, Semigroup e) => Semigroup (a, b, c, d, e)	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods (<>) :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) # sconcat :: NonEmpty (a, b, c, d, e) -> (a, b, c, d, e) # stimes :: Integral b0 => b0 -> (a, b, c, d, e) -> (a, b, c, d, e) #
Semigroup (k3 b a) => Semigroup (Dual k3 a b)
Instance details Defined in Control.Category.Dual Methods (<>) :: Dual k3 a b -> Dual k3 a b -> Dual k3 a b # sconcat :: NonEmpty (Dual k3 a b) -> Dual k3 a b # stimes :: Integral b0 => b0 -> Dual k3 a b -> Dual k3 a b #

class Semigroup a => Monoid a where #

The class of monoids (types with an associative binary operation that has an identity). Instances should satisfy the following laws:

```
x <> mempty = x
```
```
mempty <> x = x
```
x <> (y <> z) = (x <> y) <> z (Semigroup law)
```
mconcat = foldr '(<>)' mempty
```

The method names refer to the monoid of lists under concatenation, but there are many other instances.

Some types can be viewed as a monoid in more than one way, e.g. both addition and multiplication on numbers. In such cases we often define newtypes and make those instances of Monoid, e.g. Sum and Product.

NOTE: Semigroup is a superclass of Monoid since base-4.11.0.0.

Methods

mempty :: a #

Identity of mappend

Instances

Monoid Ordering	Since: base-2.1
Instance details Defined in GHC.Base Methods mempty :: Ordering # mappend :: Ordering -> Ordering -> Ordering # mconcat :: [Ordering] -> Ordering #
Monoid ()	Since: base-2.1
Instance details Defined in GHC.Base Methods mempty :: () # mappend :: () -> () -> () # mconcat :: [()] -> () #
Monoid All	Since: base-2.1
Instance details Defined in Data.Semigroup.Internal Methods mempty :: All # mappend :: All -> All -> All # mconcat :: [All] -> All #
Monoid Any	Since: base-2.1
Instance details Defined in Data.Semigroup.Internal Methods mempty :: Any # mappend :: Any -> Any -> Any # mconcat :: [Any] -> Any #
Monoid [a]	Since: base-2.1
Instance details Defined in GHC.Base Methods mempty :: [a] # mappend :: [a] -> [a] -> [a] # mconcat :: [[a]] -> [a] #
Semigroup a => Monoid (Maybe a)	Lift a semigroup into `Maybe` forming a `Monoid` according to http://en.wikipedia.org/wiki/Monoid: "Any semigroup `S` may be turned into a monoid simply by adjoining an element `e` not in `S` and defining `ee = e` and `es = s = se` for all `s ∈ S`." Since 4.11.0: constraint on inner `a` value generalised from `Monoid` to `Semigroup`. Since: base-2.1*
Instance details Defined in GHC.Base Methods mempty :: Maybe a # mappend :: Maybe a -> Maybe a -> Maybe a # mconcat :: [Maybe a] -> Maybe a #
Monoid a => Monoid (IO a)	Since: base-4.9.0.0
Instance details Defined in GHC.Base Methods mempty :: IO a # mappend :: IO a -> IO a -> IO a # mconcat :: [IO a] -> IO a #
Monoid p => Monoid (Par1 p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods mempty :: Par1 p # mappend :: Par1 p -> Par1 p -> Par1 p # mconcat :: [Par1 p] -> Par1 p #
(Ord a, Bounded a) => Monoid (Min a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods mempty :: Min a # mappend :: Min a -> Min a -> Min a # mconcat :: [Min a] -> Min a #
(Ord a, Bounded a) => Monoid (Max a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods mempty :: Max a # mappend :: Max a -> Max a -> Max a # mconcat :: [Max a] -> Max a #
Monoid m => Monoid (WrappedMonoid m)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods mempty :: WrappedMonoid m # mappend :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # mconcat :: [WrappedMonoid m] -> WrappedMonoid m #
Semigroup a => Monoid (Option a)	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods mempty :: Option a # mappend :: Option a -> Option a -> Option a # mconcat :: [Option a] -> Option a #
Monoid a => Monoid (Identity a)	Since: base-4.9.0.0
Instance details Defined in Data.Functor.Identity Methods mempty :: Identity a # mappend :: Identity a -> Identity a -> Identity a # mconcat :: [Identity a] -> Identity a #
Monoid (First a)	Since: base-2.1
Instance details Defined in Data.Monoid Methods mempty :: First a # mappend :: First a -> First a -> First a # mconcat :: [First a] -> First a #
Monoid (Last a)	Since: base-2.1
Instance details Defined in Data.Monoid Methods mempty :: Last a # mappend :: Last a -> Last a -> Last a # mconcat :: [Last a] -> Last a #
Monoid a => Monoid (Dual a)	Since: base-2.1
Instance details Defined in Data.Semigroup.Internal Methods mempty :: Dual a # mappend :: Dual a -> Dual a -> Dual a # mconcat :: [Dual a] -> Dual a #
Monoid (Endo a)	Since: base-2.1
Instance details Defined in Data.Semigroup.Internal Methods mempty :: Endo a # mappend :: Endo a -> Endo a -> Endo a # mconcat :: [Endo a] -> Endo a #
Num a => Monoid (Sum a)	Since: base-2.1
Instance details Defined in Data.Semigroup.Internal Methods mempty :: Sum a # mappend :: Sum a -> Sum a -> Sum a # mconcat :: [Sum a] -> Sum a #
Num a => Monoid (Product a)	Since: base-2.1
Instance details Defined in Data.Semigroup.Internal Methods mempty :: Product a # mappend :: Product a -> Product a -> Product a # mconcat :: [Product a] -> Product a #
Monoid a => Monoid (Down a)	Since: base-4.11.0.0
Instance details Defined in Data.Ord Methods mempty :: Down a # mappend :: Down a -> Down a -> Down a # mconcat :: [Down a] -> Down a #
Bits a => Monoid (Min (BitSet a)) Source #
Instance details Defined in Data.BitSet Methods mempty :: Min (BitSet a) # mappend :: Min (BitSet a) -> Min (BitSet a) -> Min (BitSet a) # mconcat :: [Min (BitSet a)] -> Min (BitSet a) #
Bits a => Monoid (Max (BitSet a)) Source #
Instance details Defined in Data.BitSet Methods mempty :: Max (BitSet a) # mappend :: Max (BitSet a) -> Max (BitSet a) -> Max (BitSet a) # mconcat :: [Max (BitSet a)] -> Max (BitSet a) #
Monoid a => Monoid (Lexical a) Source #
Instance details Defined in Relation.Binary.Comparison Methods mempty :: Lexical a # mappend :: Lexical a -> Lexical a -> Lexical a # mconcat :: [Lexical a] -> Lexical a #
Bits a => Monoid (BitSet a) Source #
Instance details Defined in Data.BitSet Methods mempty :: BitSet a # mappend :: BitSet a -> BitSet a -> BitSet a # mconcat :: [BitSet a] -> BitSet a #
Monoid b => Monoid (a -> b)	Since: base-2.1
Instance details Defined in GHC.Base Methods mempty :: a -> b # mappend :: (a -> b) -> (a -> b) -> a -> b # mconcat :: [a -> b] -> a -> b #
Monoid (U1 p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods mempty :: U1 p # mappend :: U1 p -> U1 p -> U1 p # mconcat :: [U1 p] -> U1 p #
(Monoid a, Monoid b) => Monoid (a, b)	Since: base-2.1
Instance details Defined in GHC.Base Methods mempty :: (a, b) # mappend :: (a, b) -> (a, b) -> (a, b) # mconcat :: [(a, b)] -> (a, b) #
Monoid (Proxy s)	Since: base-4.7.0.0
Instance details Defined in Data.Proxy Methods mempty :: Proxy s # mappend :: Proxy s -> Proxy s -> Proxy s # mconcat :: [Proxy s] -> Proxy s #
(Applicative p, Semigroup a, Monoid a) => Monoid (Ap p a)
Instance details Defined in Util Methods mempty :: Ap p a # mappend :: Ap p a -> Ap p a -> Ap p a # mconcat :: [Ap p a] -> Ap p a #
Monoid (f p) => Monoid (Rec1 f p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods mempty :: Rec1 f p # mappend :: Rec1 f p -> Rec1 f p -> Rec1 f p # mconcat :: [Rec1 f p] -> Rec1 f p #
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)	Since: base-2.1
Instance details Defined in GHC.Base Methods mempty :: (a, b, c) # mappend :: (a, b, c) -> (a, b, c) -> (a, b, c) # mconcat :: [(a, b, c)] -> (a, b, c) #
Monoid a => Monoid (Const a b)	Since: base-4.9.0.0
Instance details Defined in Data.Functor.Const Methods mempty :: Const a b # mappend :: Const a b -> Const a b -> Const a b # mconcat :: [Const a b] -> Const a b #
(Applicative f, Monoid a) => Monoid (Ap f a)	Since: base-4.12.0.0
Instance details Defined in Data.Monoid Methods mempty :: Ap f a # mappend :: Ap f a -> Ap f a -> Ap f a # mconcat :: [Ap f a] -> Ap f a #
Alternative f => Monoid (Alt f a)	Since: base-4.8.0.0
Instance details Defined in Data.Semigroup.Internal Methods mempty :: Alt f a # mappend :: Alt f a -> Alt f a -> Alt f a # mconcat :: [Alt f a] -> Alt f a #
Monoid c => Monoid (K1 i c p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods mempty :: K1 i c p # mappend :: K1 i c p -> K1 i c p -> K1 i c p # mconcat :: [K1 i c p] -> K1 i c p #
(Monoid (f p), Monoid (g p)) => Monoid ((f :*: g) p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods mempty :: (f :: g) p # mappend :: (f :: g) p -> (f :: g) p -> (f :: g) p # mconcat :: [(f :: g) p] -> (f :: g) p #
(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d)	Since: base-2.1
Instance details Defined in GHC.Base Methods mempty :: (a, b, c, d) # mappend :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) # mconcat :: [(a, b, c, d)] -> (a, b, c, d) #
Monoid (f p) => Monoid (M1 i c f p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods mempty :: M1 i c f p # mappend :: M1 i c f p -> M1 i c f p -> M1 i c f p # mconcat :: [M1 i c f p] -> M1 i c f p #
Monoid (f (g p)) => Monoid ((f :.: g) p)	Since: base-4.12.0.0
Instance details Defined in GHC.Generics Methods mempty :: (f :.: g) p # mappend :: (f :.: g) p -> (f :.: g) p -> (f :.: g) p # mconcat :: [(f :.: g) p] -> (f :.: g) p #
(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)	Since: base-2.1
Instance details Defined in GHC.Base Methods mempty :: (a, b, c, d, e) # mappend :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) # mconcat :: [(a, b, c, d, e)] -> (a, b, c, d, e) #
Monoid (k3 b a) => Monoid (Dual k3 a b)
Instance details Defined in Control.Category.Dual Methods mempty :: Dual k3 a b # mappend :: Dual k3 a b -> Dual k3 a b -> Dual k3 a b # mconcat :: [Dual k3 a b] -> Dual k3 a b #

class Monoid a => Group a where Source #

Methods

invert :: a -> a Source #

Instances

Group () Source #
Instance details Defined in Algebra Methods invert :: () -> () Source #
Group a => Group (Identity a) Source #
Instance details Defined in Algebra Methods invert :: Identity a -> Identity a Source #
Group a => Group (Dual a) Source #
Instance details Defined in Algebra Methods invert :: Dual a -> Dual a Source #
Num a => Group (Sum a) Source #
Instance details Defined in Algebra Methods invert :: Sum a -> Sum a Source #
Fractional a => Group (Product a) Source #
Instance details Defined in Algebra Methods invert :: Product a -> Product a Source #
Group a => Group (Lexical a) Source #
Instance details Defined in Relation.Binary.Comparison Methods invert :: Lexical a -> Lexical a Source #
Bits a => Group (BitSet a) Source #
Instance details Defined in Data.BitSet Methods invert :: BitSet a -> BitSet a Source #
Group b => Group (a -> b) Source #
Instance details Defined in Algebra Methods invert :: (a -> b) -> a -> b Source #
(Group a, Group b) => Group (a, b) Source #
Instance details Defined in Algebra Methods invert :: (a, b) -> (a, b) Source #
Group (Proxy a) Source #
Instance details Defined in Algebra Methods invert :: Proxy a -> Proxy a Source #
(Group a, Group b, Group c) => Group (a, b, c) Source #
Instance details Defined in Algebra Methods invert :: (a, b, c) -> (a, b, c) Source #
Group a => Group (Const a b) Source #
Instance details Defined in Algebra Methods invert :: Const a b -> Const a b Source #
(Group a, Group b, Group c, Group d) => Group (a, b, c, d) Source #
Instance details Defined in Algebra Methods invert :: (a, b, c, d) -> (a, b, c, d) Source #
(Group a, Group b, Group c, Group d, Group e) => Group (a, b, c, d, e) Source #
Instance details Defined in Algebra Methods invert :: (a, b, c, d, e) -> (a, b, c, d, e) Source #
Group (k3 b a) => Group (Dual k3 a b) Source #
Instance details Defined in Algebra Methods invert :: Dual k3 a b -> Dual k3 a b Source #

class Semigroup a => Abelian a Source #

Instances

Abelian () Source #
Instance details Defined in Algebra
Abelian All Source #
Instance details Defined in Algebra
Abelian Any Source #
Instance details Defined in Algebra
Abelian (Min Int) Source #
Instance details Defined in Algebra
Abelian (Min Integer) Source #
Instance details Defined in Algebra
Abelian (Min Natural) Source #
Instance details Defined in Algebra
Abelian (Min Word) Source #
Instance details Defined in Algebra
Abelian (Max Int) Source #
Instance details Defined in Algebra
Abelian (Max Integer) Source #
Instance details Defined in Algebra
Abelian (Max Natural) Source #
Instance details Defined in Algebra
Abelian (Max Word) Source #
Instance details Defined in Algebra
Abelian a => Abelian (Identity a) Source #
Instance details Defined in Algebra
Abelian a => Abelian (Dual a) Source #
Instance details Defined in Algebra
Abelian (Sum Int) Source #
Instance details Defined in Algebra
Abelian (Sum Integer) Source #
Instance details Defined in Algebra
Abelian (Sum Natural) Source #
Instance details Defined in Algebra
Abelian (Sum Word) Source #
Instance details Defined in Algebra
Abelian (Product Int) Source #
Instance details Defined in Algebra
Abelian (Product Integer) Source #
Instance details Defined in Algebra
Abelian (Product Natural) Source #
Instance details Defined in Algebra
Abelian (Product Word) Source #
Instance details Defined in Algebra
Bits a => Abelian (Min (BitSet a)) Source #
Instance details Defined in Data.BitSet
Bits a => Abelian (Max (BitSet a)) Source #
Instance details Defined in Data.BitSet
Bits a => Abelian (BitSet a) Source #
Instance details Defined in Data.BitSet
Abelian b => Abelian (a -> b) Source #
Instance details Defined in Algebra
(Abelian a, Abelian b) => Abelian (a, b) Source #
Instance details Defined in Algebra
Abelian (Proxy a) Source #
Instance details Defined in Algebra
(Abelian a, Abelian b, Abelian c) => Abelian (a, b, c) Source #
Instance details Defined in Algebra
Abelian a => Abelian (Const a b) Source #
Instance details Defined in Algebra
(Abelian a, Abelian b, Abelian c, Abelian d) => Abelian (a, b, c, d) Source #
Instance details Defined in Algebra
(Abelian a, Abelian b, Abelian c, Abelian d, Abelian e) => Abelian (a, b, c, d, e) Source #
Instance details Defined in Algebra

class Semigroup a => Idempotent a Source #

Instances

Idempotent () Source #
Instance details Defined in Algebra
Ord a => Idempotent (Min a) Source #
Instance details Defined in Algebra
Ord a => Idempotent (Max a) Source #
Instance details Defined in Algebra
Idempotent a => Idempotent (Identity a) Source #
Instance details Defined in Algebra
Idempotent a => Idempotent (Dual a) Source #
Instance details Defined in Algebra
Bits a => Idempotent (Min (BitSet a)) Source #
Instance details Defined in Data.BitSet
Bits a => Idempotent (Max (BitSet a)) Source #
Instance details Defined in Data.BitSet
Idempotent b => Idempotent (a -> b) Source #
Instance details Defined in Algebra
(Idempotent a, Idempotent b) => Idempotent (a, b) Source #
Instance details Defined in Algebra
Idempotent (Proxy a) Source #
Instance details Defined in Algebra
(Idempotent a, Idempotent b, Idempotent c) => Idempotent (a, b, c) Source #
Instance details Defined in Algebra
Idempotent a => Idempotent (Const a b) Source #
Instance details Defined in Algebra
(Idempotent a, Idempotent b, Idempotent c, Idempotent d) => Idempotent (a, b, c, d) Source #
Instance details Defined in Algebra
(Idempotent a, Idempotent b, Idempotent c, Idempotent d, Idempotent e) => Idempotent (a, b, c, d, e) Source #
Instance details Defined in Algebra

(+) :: Semigroup (Sum a) => a -> a -> a infixl 6 Source #

(-) :: Group (Sum a) => a -> a -> a infixl 6 Source #

(*) :: Semigroup (Product a) => a -> a -> a infixl 7 Source #

(/) :: (Semigroup (Product a), Group (Product a)) => a -> a -> a infixl 7 Source #

(×) :: Semigroup (Product a) => a -> a -> a infixl 7 Source #

commuteWith :: Group b => (a -> a -> b) -> a -> a -> b Source #