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
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

Documentation

class (Monoid m, SSet m a) => MSet m a 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).

Instances

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

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) :: * -> * # 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
type Rep (Endo a)
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
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 # proof :: 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 :: * -> *) (a :: l) Source #
Instance details Defined in Data.Monoid.MSet type AlgebraType0 (FreeMSet m :: * -> *) (a :: l) = ()
type AlgebraType (FreeMSet m :: * -> ) (a :: ) Source #
Instance details Defined in Data.Monoid.MSet type AlgebraType (FreeMSet m :: * -> ) (a :: ) = MSet m a

hoistFreeMSet Source #

Arguments

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