Safe Haskell	Safe
Language	Haskell2010

Algebra

Synopsis

Documentation

class Semigroup a where #

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

Since: 4.9.0.0

Methods

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

An associative operation.

(a <> b) <> c = a <> (b <> c)

If a is also a Monoid we further require

(<>) = mappend

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
Methods (<>) :: Ordering -> Ordering -> Ordering # sconcat :: NonEmpty Ordering -> Ordering # stimes :: Integral b => b -> Ordering -> Ordering #
Semigroup ()
Methods (<>) :: () -> () -> () # sconcat :: NonEmpty () -> () # stimes :: Integral b => b -> () -> () #
Semigroup Void
Methods (<>) :: Void -> Void -> Void # sconcat :: NonEmpty Void -> Void # stimes :: Integral b => b -> Void -> Void #
Semigroup All
Methods (<>) :: All -> All -> All # sconcat :: NonEmpty All -> All # stimes :: Integral b => b -> All -> All #
Semigroup Any
Methods (<>) :: Any -> Any -> Any # sconcat :: NonEmpty Any -> Any # stimes :: Integral b => b -> Any -> Any #
Semigroup [a]
Methods (<>) :: [a] -> [a] -> [a] # sconcat :: NonEmpty [a] -> [a] # stimes :: Integral b => b -> [a] -> [a] #
Semigroup a => Semigroup (Maybe a)
Methods (<>) :: Maybe a -> Maybe a -> Maybe a # sconcat :: NonEmpty (Maybe a) -> Maybe a # stimes :: Integral b => b -> Maybe a -> Maybe a #
Semigroup a => Semigroup (Identity a)
Methods (<>) :: Identity a -> Identity a -> Identity a # sconcat :: NonEmpty (Identity a) -> Identity a # stimes :: Integral b => b -> Identity a -> Identity a #
Ord a => Semigroup (Min a)
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)
Methods (<>) :: Max a -> Max a -> Max a # sconcat :: NonEmpty (Max a) -> Max a # stimes :: Integral b => b -> Max a -> Max a #
Semigroup (First a)
Methods (<>) :: First a -> First a -> First a # sconcat :: NonEmpty (First a) -> First a # stimes :: Integral b => b -> First a -> First a #
Semigroup (Last a)
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)
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)
Methods (<>) :: Option a -> Option a -> Option a # sconcat :: NonEmpty (Option a) -> Option a # stimes :: Integral b => b -> Option a -> Option a #
Semigroup (NonEmpty a)
Methods (<>) :: NonEmpty a -> NonEmpty a -> NonEmpty a # sconcat :: NonEmpty (NonEmpty a) -> NonEmpty a # stimes :: Integral b => b -> NonEmpty a -> NonEmpty a #
Semigroup a => Semigroup (Dual a)
Methods (<>) :: Dual a -> Dual a -> Dual a # sconcat :: NonEmpty (Dual a) -> Dual a # stimes :: Integral b => b -> Dual a -> Dual a #
Semigroup (Endo a)
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)
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)
Methods (<>) :: Product a -> Product a -> Product a # sconcat :: NonEmpty (Product a) -> Product a # stimes :: Integral b => b -> Product a -> Product a #
Semigroup (First a)
Methods (<>) :: First a -> First a -> First a # sconcat :: NonEmpty (First a) -> First a # stimes :: Integral b => b -> First a -> First a #
Semigroup (Last a)
Methods (<>) :: Last a -> Last a -> Last a # sconcat :: NonEmpty (Last a) -> Last a # stimes :: Integral b => b -> Last a -> Last a #
Semigroup b => Semigroup (a -> b)
Methods (<>) :: (a -> b) -> (a -> b) -> a -> b # sconcat :: NonEmpty (a -> b) -> a -> b # stimes :: Integral b => b -> (a -> b) -> a -> b #
Semigroup (Either a b)
Methods (<>) :: Either a b -> Either a b -> Either a b # sconcat :: NonEmpty (Either a b) -> Either a b # stimes :: Integral b => b -> Either a b -> Either a b #
(Semigroup a, Semigroup b) => Semigroup (a, b)
Methods (<>) :: (a, b) -> (a, b) -> (a, b) # sconcat :: NonEmpty (a, b) -> (a, b) # stimes :: Integral b => b -> (a, b) -> (a, b) #
Semigroup (Proxy k s)
Methods (<>) :: Proxy k s -> Proxy k s -> Proxy k s # sconcat :: NonEmpty (Proxy k s) -> Proxy k s # stimes :: Integral b => b -> Proxy k s -> Proxy k s #
(Semigroup a, Semigroup b, Semigroup c) => Semigroup (a, b, c)
Methods (<>) :: (a, b, c) -> (a, b, c) -> (a, b, c) # sconcat :: NonEmpty (a, b, c) -> (a, b, c) # stimes :: Integral b => b -> (a, b, c) -> (a, b, c) #
Semigroup a => Semigroup (Const k a b)
Methods (<>) :: Const k a b -> Const k a b -> Const k a b # sconcat :: NonEmpty (Const k a b) -> Const k a b # stimes :: Integral b => b -> Const k a b -> Const k a b #
Alternative f => Semigroup (Alt * f a)
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 a, Semigroup b, Semigroup c, Semigroup d) => Semigroup (a, b, c, d)
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 b => b -> (a, b, c, d) -> (a, b, c, d) #
(Semigroup a, Semigroup b, Semigroup c, Semigroup d, Semigroup e) => Semigroup (a, b, c, d, e)
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 b => b -> (a, b, c, d, e) -> (a, b, c, d, e) #

class Monoid a where #

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

```
mappend mempty x = x
```
```
mappend x mempty = x
```

mappend x (mappend y z) = mappend (mappend x y) z

```
mconcat = foldr mappend 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.

Minimal complete definition

mempty, mappend

Methods

mempty :: a #

Identity of mappend

Instances

Monoid Ordering
Methods mempty :: Ordering # mappend :: Ordering -> Ordering -> Ordering # mconcat :: [Ordering] -> Ordering #
Monoid ()
Methods mempty :: () # mappend :: () -> () -> () # mconcat :: [()] -> () #
Monoid All
Methods mempty :: All # mappend :: All -> All -> All # mconcat :: [All] -> All #
Monoid Any
Methods mempty :: Any # mappend :: Any -> Any -> Any # mconcat :: [Any] -> Any #
Monoid [a]
Methods mempty :: [a] # mappend :: [a] -> [a] -> [a] # mconcat :: [[a]] -> [a] #
Monoid 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 = s*e` for all `s ∈ S`." Since there is no "Semigroup" typeclass providing just `mappend`, we use `Monoid` instead.
Methods mempty :: Maybe a # mappend :: Maybe a -> Maybe a -> Maybe a # mconcat :: [Maybe a] -> Maybe a #
Monoid a => Monoid (IO a)
Methods mempty :: IO a # mappend :: IO a -> IO a -> IO a # mconcat :: [IO a] -> IO a #
Monoid a => Monoid (Identity a)
Methods mempty :: Identity a # mappend :: Identity a -> Identity a -> Identity a # mconcat :: [Identity a] -> Identity a #
(Ord a, Bounded a) => Monoid (Min a)
Methods mempty :: Min a # mappend :: Min a -> Min a -> Min a # mconcat :: [Min a] -> Min a #
(Ord a, Bounded a) => Monoid (Max a)
Methods mempty :: Max a # mappend :: Max a -> Max a -> Max a # mconcat :: [Max a] -> Max a #
Monoid m => Monoid (WrappedMonoid m)
Methods mempty :: WrappedMonoid m # mappend :: WrappedMonoid m -> WrappedMonoid m -> WrappedMonoid m # mconcat :: [WrappedMonoid m] -> WrappedMonoid m #
Semigroup a => Monoid (Option a)
Methods mempty :: Option a # mappend :: Option a -> Option a -> Option a # mconcat :: [Option a] -> Option a #
Monoid a => Monoid (Dual a)
Methods mempty :: Dual a # mappend :: Dual a -> Dual a -> Dual a # mconcat :: [Dual a] -> Dual a #
Monoid (Endo a)
Methods mempty :: Endo a # mappend :: Endo a -> Endo a -> Endo a # mconcat :: [Endo a] -> Endo a #
Num a => Monoid (Sum a)
Methods mempty :: Sum a # mappend :: Sum a -> Sum a -> Sum a # mconcat :: [Sum a] -> Sum a #
Num a => Monoid (Product a)
Methods mempty :: Product a # mappend :: Product a -> Product a -> Product a # mconcat :: [Product a] -> Product a #
Monoid (First a)
Methods mempty :: First a # mappend :: First a -> First a -> First a # mconcat :: [First a] -> First a #
Monoid (Last a)
Methods mempty :: Last a # mappend :: Last a -> Last a -> Last a # mconcat :: [Last a] -> Last a #
Monoid b => Monoid (a -> b)
Methods mempty :: a -> b # mappend :: (a -> b) -> (a -> b) -> a -> b # mconcat :: [a -> b] -> a -> b #
(Monoid a, Monoid b) => Monoid (a, b)
Methods mempty :: (a, b) # mappend :: (a, b) -> (a, b) -> (a, b) # mconcat :: [(a, b)] -> (a, b) #
Monoid (Proxy k s)
Methods mempty :: Proxy k s # mappend :: Proxy k s -> Proxy k s -> Proxy k s # mconcat :: [Proxy k s] -> Proxy k s #
(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)
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 k a b)
Methods mempty :: Const k a b # mappend :: Const k a b -> Const k a b -> Const k a b # mconcat :: [Const k a b] -> Const k a b #
Alternative f => Monoid (Alt * f a)
Methods mempty :: Alt * f a # mappend :: Alt * f a -> Alt * f a -> Alt * f a # mconcat :: [Alt * f a] -> Alt * f a #
(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d)
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 a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)
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) #

class Monoid a => Group a where Source #

Minimal complete definition

invert

Methods

invert :: a -> a Source #

Instances

Group () Source #
Methods invert :: () -> () Source #
Group a => Group (Identity a) Source #
Methods invert :: Identity a -> Identity a Source #
Group a => Group (Dual a) Source #
Methods invert :: Dual a -> Dual a Source #
Group (Sum Int) Source #
Methods invert :: Sum Int -> Sum Int Source #
Group (Sum Integer) Source #
Methods invert :: Sum Integer -> Sum Integer Source #
Group (Sum Word) Source #
Methods invert :: Sum Word -> Sum Word Source #
Group b => Group (a -> b) Source #
Methods invert :: (a -> b) -> a -> b Source #
(Group a, Group b) => Group (a, b) Source #
Methods invert :: (a, b) -> (a, b) Source #
Group (Proxy k a) Source #
Methods invert :: Proxy k a -> Proxy k a Source #
(Group a, Group b, Group c) => Group (a, b, c) Source #
Methods invert :: (a, b, c) -> (a, b, c) Source #
Group a => Group (Const k a b) Source #
Methods invert :: Const k a b -> Const k a b Source #
(Group a, Group b, Group c, Group d) => Group (a, b, c, d) Source #
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 #
Methods invert :: (a, b, c, d, e) -> (a, b, c, d, e) Source #

class Semigroup a => Abelian a Source #

Instances

Abelian () Source #

Abelian All Source #

Abelian Any Source #

Abelian a => Abelian (Identity a) Source #

Abelian (Min Int) Source #

Abelian (Min Integer) Source #

Abelian (Min Word) Source #

Abelian (Min Natural) Source #

Abelian (Max Int) Source #

Abelian (Max Integer) Source #

Abelian (Max Word) Source #

Abelian (Max Natural) Source #

Abelian a => Abelian (Dual a) Source #

Abelian (Sum Int) Source #

Abelian (Sum Integer) Source #

Abelian (Sum Word) Source #

Abelian (Sum Natural) Source #

Abelian (Product Int) Source #

Abelian (Product Integer) Source #

Abelian (Product Word) Source #

Abelian (Product Natural) Source #

Abelian b => Abelian (a -> b) Source #

(Abelian a, Abelian b) => Abelian (a, b) Source #

Abelian (Proxy k a) Source #

(Abelian a, Abelian b, Abelian c) => Abelian (a, b, c) Source #

Abelian a => Abelian (Const k a b) Source #

(Abelian a, Abelian b, Abelian c, Abelian d) => Abelian (a, b, c, d) Source #

(Abelian a, Abelian b, Abelian c, Abelian d, Abelian e) => Abelian (a, b, c, d, e) Source #

class Semigroup a => Idempotent a Source #

Instances

Idempotent () Source #

Idempotent a => Idempotent (Identity a) Source #

Ord a => Idempotent (Min a) Source #

Ord a => Idempotent (Max a) Source #

Idempotent a => Idempotent (Dual a) Source #

Idempotent b => Idempotent (a -> b) Source #

(Idempotent a, Idempotent b) => Idempotent (a, b) Source #

Idempotent (Proxy k a) Source #

(Idempotent a, Idempotent b, Idempotent c) => Idempotent (a, b, c) Source #

Idempotent a => Idempotent (Const k a b) Source #

(Idempotent a, Idempotent b, Idempotent c, Idempotent d) => Idempotent (a, b, c, d) Source #

(Idempotent a, Idempotent b, Idempotent c, Idempotent d, Idempotent e) => Idempotent (a, b, c, d, e) Source #

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

(-) :: (Semigroup (Sum a), Group (Sum a)) => a -> a -> a Source #

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

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