Safe Haskell	None
Language	Haskell2010

Data.Monoid.MSet

Description

Monoid and group actions (M-Sets and G-Sets). The category of MSets (and GSets) is monadic (unlike the category of SSets).

Synopsis

class (Monoid m, SSet m a) => MSet m a where
- mact :: m -> a -> a
class Semigroup s => SSet s a where
- act :: s -> a -> a
newtype Endo a = Endo {
- appEndo :: a -> a
}
rep :: SSet s a => s -> Endo a
fact :: (Functor f, SSet s a) => s -> f a -> f a
newtype FreeMSet m a = FreeMSet {
- runFreeMSet :: (m, a)
}
hoistFreeMSet :: (m -> n) -> FreeMSet m a -> FreeMSet n a
foldrMSet :: forall m a b. MSet m b => (a -> b -> b) -> b -> (m, a) -> b
newtype S s = S {
- runS :: s
}

Documentation

class (Monoid m, SSet m a) => MSet m a where Source #

Lawful instance should satisfy:

act mempty = id

g `act` h `act` a = g <> h `act` a

This is the same as to say that act is a monoid homomorphism from m to the monoid of endomorphisms of a (i.e. maps from a to a).

Note that if g is a Group then an MSet is simply a GSet, this is because monoids and groups share the same morphisms (a monoid homomorphis between groups necessarily preserves inverses).

Minimal complete definition

Nothing

Methods

mact :: m -> a -> a Source #

Instances

Monoid m => MSet m m Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> m -> m Source #
MSet m a => MSet m (IO a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> IO a -> IO a Source #
MSet m a => MSet m (Down a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> Down a -> Down a Source #
MSet m a => MSet m (Maybe a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> Maybe a -> Maybe a Source #
MSet m a => MSet m (Identity a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> Identity a -> Identity a Source #
(MSet m a, Ord a) => MSet m (Set a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> Set a -> Set a Source #
MSet m a => MSet m (NonEmpty a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> NonEmpty a -> NonEmpty a Source #
MSet m a => MSet m [a] Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> [a] -> [a] Source #
Monoid m => MSet m (FreeMSet m a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> FreeMSet m a -> FreeMSet m a Source #
MSet m b => MSet m (a -> b) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> (a -> b) -> a -> b Source #
MSet m b => MSet m (Either a b) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> Either a b -> Either a b Source #
(MSet m a, MSet m b) => MSet m (a, b) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> (a, b) -> (a, b) Source #
MSet m a => MSet m (Const a b) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> Const a b -> Const a b Source #
(MSet m a, MSet m b, MSet m c) => MSet m (a, b, c) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> (a, b, c) -> (a, b, c) Source #
(Functor f, Functor h, MSet m a) => MSet m (Sum f h a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> Sum f h a -> Sum f h a Source #
(Functor f, Functor h, MSet m a) => MSet m (Product f h a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> Product f h a -> Product f h a Source #
(MSet m a, MSet m b, MSet m c, MSet m d) => MSet m (a, b, c, d) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> (a, b, c, d) -> (a, b, c, d) Source #
(MSet m a, MSet m b, MSet m c, MSet m d, MSet m e) => MSet m (a, b, c, d, e) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> (a, b, c, d, e) -> (a, b, c, d, e) Source #
(MSet m a, MSet m b, MSet m c, MSet m d, MSet m e, MSet m f) => MSet m (a, b, c, d, e, f) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source #
(MSet m a, MSet m b, MSet m c, MSet m d, MSet m e, MSet m f, MSet m h) => MSet m (a, b, c, d, e, f, h) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> (a, b, c, d, e, f, h) -> (a, b, c, d, e, f, h) Source #
(MSet m a, MSet m b, MSet m c, MSet m d, MSet m e, MSet m f, MSet m h, MSet m i) => MSet m (a, b, c, d, e, f, h, i) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> (a, b, c, d, e, f, h, i) -> (a, b, c, d, e, f, h, i) Source #
MSet m a => MSet (Identity m) a Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: Identity m -> a -> a Source #
MSet (Endo a) a Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: Endo a -> a -> a Source #
Num s => MSet (Sum s) s Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: Sum s -> s -> s Source #
Monoid m => MSet (Sum Natural) m Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: Sum Natural -> m -> m Source #
Num s => MSet (Product s) s Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: Product s -> s -> s Source #
MSet m b => MSet (S m) (Endo b) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: S m -> Endo b -> Endo b Source #

class Semigroup s => SSet s a where Source #

A lawful instance should satisfy:

g `act` h `act` a = g <> h `act` a

This is the same as to say that act is a semigroup homomorphism from s to the monoid of endomorphisms of a (i.e. maps from a to a).

Note that if g is a Group then MAct g is simply a GSet, this is because monoids and groups share the same morphisms (a monoid homomorphis between groups necessarily preserves inverses).

Methods

act :: s -> a -> a Source #

Instances

Semigroup s => SSet s s Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> s -> s Source #
SSet s a => SSet s (IO a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> IO a -> IO a Source #
SSet s a => SSet s (Down a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> Down a -> Down a Source #
SSet s a => SSet s (Maybe a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> Maybe a -> Maybe a Source #
SSet s a => SSet s (Identity a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> Identity a -> Identity a Source #
(SSet s a, Ord a) => SSet s (Set a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> Set a -> Set a Source #
SSet s a => SSet s (NonEmpty a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> NonEmpty a -> NonEmpty a Source #
SSet s a => SSet s [a] Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> [a] -> [a] Source #
SSet s b => SSet s (a -> b) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> (a -> b) -> a -> b Source #
SSet s b => SSet s (Either a b) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> Either a b -> Either a b Source #
(SSet s a, SSet s b) => SSet s (a, b) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> (a, b) -> (a, b) Source #
Semigroup m => SSet m (FreeMSet m a) Source #
Instance details Defined in Data.Monoid.MSet Methods act :: m -> FreeMSet m a -> FreeMSet m a Source #
SSet s a => SSet s (Const a b) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> Const a b -> Const a b Source #
(SSet s a, SSet s b, SSet s c) => SSet s (a, b, c) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> (a, b, c) -> (a, b, c) Source #
(Functor f, Functor h, SSet s a) => SSet s (Sum f h a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> Sum f h a -> Sum f h a Source #
(Functor f, Functor h, SSet s a) => SSet s (Product f h a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> Product f h a -> Product f h a Source #
(SSet s a, SSet s b, SSet s c, SSet s d) => SSet s (a, b, c, d) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> (a, b, c, d) -> (a, b, c, d) Source #
(SSet s a, SSet s b, SSet s c, SSet s d, SSet s e) => SSet s (a, b, c, d, e) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> (a, b, c, d, e) -> (a, b, c, d, e) Source #
(SSet s a, SSet s b, SSet s c, SSet s d, SSet s e, SSet s f) => SSet s (a, b, c, d, e, f) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source #
(SSet s a, SSet s b, SSet s c, SSet s d, SSet s e, SSet s f, SSet s h) => SSet s (a, b, c, d, e, f, h) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> (a, b, c, d, e, f, h) -> (a, b, c, d, e, f, h) Source #
(SSet s a, SSet s b, SSet s c, SSet s d, SSet s e, SSet s f, SSet s h, SSet s i) => SSet s (a, b, c, d, e, f, h, i) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: s -> (a, b, c, d, e, f, h, i) -> (a, b, c, d, e, f, h, i) Source #
SSet s a => SSet (Identity s) a Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: Identity s -> a -> a Source #
SSet (Endo a) a Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: Endo a -> a -> a Source #
Group g => SSet (Sum Integer) g Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: Sum Integer -> g -> g Source #
Num s => SSet (Sum s) s Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: Sum s -> s -> s Source #
Monoid s => SSet (Sum Natural) s Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: Sum Natural -> s -> s Source #
Num s => SSet (Product s) s Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: Product s -> s -> s Source #
SSet s a => SSet (S s) (Endo a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: S s -> Endo a -> Endo a Source #

newtype Endo a #

The monoid of endomorphisms under composition.

>>> let computation = Endo ("Hello, " ++) <> Endo (++ "!")
>>> appEndo computation "Haskell"
"Hello, Haskell!"

Constructors

Endo
Fields appEndo :: a -> a

Instances

Generic (Endo a)
Instance details Defined in Data.Semigroup.Internal Associated Types type Rep (Endo a) :: Type -> Type # Methods from :: Endo a -> Rep (Endo a) x # to :: Rep (Endo a) x -> Endo 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 #
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 #
SSet (Endo a) a Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: Endo a -> a -> a Source #
MSet (Endo a) a Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: Endo a -> a -> a Source #
SSet s a => SSet (S s) (Endo a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: S s -> Endo a -> Endo a Source #
MSet m b => MSet (S m) (Endo b) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: S m -> Endo b -> Endo b Source #
type Rep (Endo a)	Since: base-4.7.0.0
Instance details Defined in Data.Semigroup.Internal type Rep (Endo a) = D1 (MetaData "Endo" "Data.Semigroup.Internal" "base" True) (C1 (MetaCons "Endo" PrefixI True) (S1 (MetaSel (Just "appEndo") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (a -> a))))

rep :: SSet s a => s -> Endo a Source #

fact :: (Functor f, SSet s a) => s -> f a -> f a Source #

Any SSet wrapped in a functor is a valid SSet.

newtype FreeMSet m a Source #

Constructors

FreeMSet
Fields runFreeMSet :: (m, a)

Instances

Semigroup m => SSet m (FreeMSet m a) Source #
Instance details Defined in Data.Monoid.MSet Methods act :: m -> FreeMSet m a -> FreeMSet m a Source #
Monoid m => MSet m (FreeMSet m a) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: m -> FreeMSet m a -> FreeMSet m a Source #
Monoid m => Monad (FreeMSet m) Source #
Instance details Defined in Data.Monoid.MSet Methods (>>=) :: FreeMSet m a -> (a -> FreeMSet m b) -> FreeMSet m b # (>>) :: FreeMSet m a -> FreeMSet m b -> FreeMSet m b # return :: a -> FreeMSet m a # fail :: String -> FreeMSet m a #
Functor (FreeMSet m) Source #
Instance details Defined in Data.Monoid.MSet Methods fmap :: (a -> b) -> FreeMSet m a -> FreeMSet m b # (<$) :: a -> FreeMSet m b -> FreeMSet m a #
Monoid m => Applicative (FreeMSet m) Source #
Instance details Defined in Data.Monoid.MSet Methods pure :: a -> FreeMSet m a # (<>) :: FreeMSet m (a -> b) -> FreeMSet m a -> FreeMSet m b # liftA2 :: (a -> b -> c) -> FreeMSet m a -> FreeMSet m b -> FreeMSet m c # (>) :: FreeMSet m a -> FreeMSet m b -> FreeMSet m b # (<*) :: FreeMSet m a -> FreeMSet m b -> FreeMSet m a #
Monoid m => FreeAlgebra (FreeMSet m) Source #
Instance details Defined in Data.Monoid.MSet Methods returnFree :: a -> FreeMSet m a Source # foldMapFree :: (AlgebraType (FreeMSet m) d, AlgebraType0 (FreeMSet m) a) => (a -> d) -> FreeMSet m a -> d Source # codom :: AlgebraType0 (FreeMSet m) a => Proof (AlgebraType (FreeMSet m) (FreeMSet m a)) (FreeMSet m a) Source # forget :: AlgebraType (FreeMSet m) a => Proof (AlgebraType0 (FreeMSet m) a) (FreeMSet m a) Source #
(Eq m, Eq a) => Eq (FreeMSet m a) Source #
Instance details Defined in Data.Monoid.MSet Methods (==) :: FreeMSet m a -> FreeMSet m a -> Bool # (/=) :: FreeMSet m a -> FreeMSet m a -> Bool #
(Ord m, Ord a) => Ord (FreeMSet m a) Source #
Instance details Defined in Data.Monoid.MSet Methods compare :: FreeMSet m a -> FreeMSet m a -> Ordering # (<) :: FreeMSet m a -> FreeMSet m a -> Bool # (<=) :: FreeMSet m a -> FreeMSet m a -> Bool # (>) :: FreeMSet m a -> FreeMSet m a -> Bool # (>=) :: FreeMSet m a -> FreeMSet m a -> Bool # max :: FreeMSet m a -> FreeMSet m a -> FreeMSet m a # min :: FreeMSet m a -> FreeMSet m a -> FreeMSet m a #
(Show m, Show a) => Show (FreeMSet m a) Source #
Instance details Defined in Data.Monoid.MSet Methods showsPrec :: Int -> FreeMSet m a -> ShowS # show :: FreeMSet m a -> String # showList :: [FreeMSet m a] -> ShowS #
type AlgebraType0 (FreeMSet m :: Type -> Type) (a :: l) Source #
Instance details Defined in Data.Monoid.MSet type AlgebraType0 (FreeMSet m :: Type -> Type) (a :: l) = ()
type AlgebraType (FreeMSet m :: Type -> Type) (a :: Type) Source #
Instance details Defined in Data.Monoid.MSet type AlgebraType (FreeMSet m :: Type -> Type) (a :: Type) = MSet m a

hoistFreeMSet Source #

Arguments

:: (m -> n)	monoid homomorphism
-> FreeMSet m a
-> FreeMSet n a

foldrMSet :: forall m a b. MSet m b => (a -> b -> b) -> b -> (m, a) -> b Source #

foldrFree for FreeMSet

newtype S s Source #

A newtype wrapper to avoid overlapping instances.

Constructors

S
Fields runS :: s

Instances

Eq s => Eq (S s) Source #
Instance details Defined in Data.Semigroup.SSet Methods (==) :: S s -> S s -> Bool # (/=) :: S s -> S s -> Bool #
Ord s => Ord (S s) Source #
Instance details Defined in Data.Semigroup.SSet Methods compare :: S s -> S s -> Ordering # (<) :: S s -> S s -> Bool # (<=) :: S s -> S s -> Bool # (>) :: S s -> S s -> Bool # (>=) :: S s -> S s -> Bool # max :: S s -> S s -> S s # min :: S s -> S s -> S s #
Show s => Show (S s) Source #
Instance details Defined in Data.Semigroup.SSet Methods showsPrec :: Int -> S s -> ShowS # show :: S s -> String # showList :: [S s] -> ShowS #
Semigroup m => Semigroup (S m) Source #
Instance details Defined in Data.Semigroup.SSet Methods (<>) :: S m -> S m -> S m # sconcat :: NonEmpty (S m) -> S m # stimes :: Integral b => b -> S m -> S m #
Monoid m => Monoid (S m) Source #
Instance details Defined in Data.Semigroup.SSet Methods mempty :: S m # mappend :: S m -> S m -> S m # mconcat :: [S m] -> S m #
SSet s a => SSet (S s) (Endo a) Source #
Instance details Defined in Data.Semigroup.SSet Methods act :: S s -> Endo a -> Endo a Source #
MSet m b => MSet (S m) (Endo b) Source #
Instance details Defined in Data.Monoid.MSet Methods mact :: S m -> Endo b -> Endo b Source #