Ideas.Common.Algebra.Group

Monoids

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.

Minimal complete definition: mempty and mappend.

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.

Methods

mempty :: a

Identity of mappend

mappend :: a -> a -> a

An associative operation

mconcat :: [a] -> a

Fold a list using the monoid. For most types, the default definition for mconcat will be used, but the function is included in the class definition so that an optimized version can be provided for specific types.

Instances

Monoid Ordering

Monoid ()

Monoid All

Monoid Any

Monoid ByteString

Monoid ByteString

Monoid IntSet

Monoid XMLBuilder

Monoid TestSuiteResult

Monoid Id

Monoid Environment

Monoid Substitution

Monoid Location

Monoid Text

Monoid Script

Monoid DomainReasoner

Monoid [a]

Monoid a => Monoid (Dual a)

Monoid (Endo a)

Num a => Monoid (Sum a)

Num a => Monoid (Product a)

Monoid (First a)

Monoid (Last 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 e*e = e and e*s = s = s*e for all s ∈ S." Since there is no "Semigroup" typeclass providing just mappend, we use Monoid instead.

Monoid (Seq a)

Monoid (IntMap a)

Ord a => Monoid (Set a)

Monoid (ArbGen a)

Monoid a => Monoid (WithZero a)

SemiRing a => Monoid (Multiplicative a)

SemiRing a => Monoid (Additive a)

Monoid (Recognizer a)

Monoid (Option a)

Boolean a => Monoid (Or a)

Boolean a => Monoid (And a)

(CoGroup a, Group a) => Monoid (SmartGroup a)

(CoMonoidZero a, MonoidZero a) => Monoid (SmartZero a)

(CoMonoid a, Monoid a) => Monoid (Smart a)

Monoid b => Monoid (a -> b)

(Monoid a, Monoid b) => Monoid (a, b)

Ord k => Monoid (Map k v)

Monoid (Trans a b)

(Monoid a, Monoid b, Monoid c) => Monoid (a, b, c)

Monoid b => Monoid (EncoderState st a b)

(Monoid a, Monoid b, Monoid c, Monoid d) => Monoid (a, b, c, d)

(Monoid a, Monoid b, Monoid c, Monoid d, Monoid e) => Monoid (a, b, c, d, e)

(<>) :: Monoid m => m -> m -> m

An infix synonym for mappend.

Groups

class Monoid a => Group a where

Minimal complete definition: inverse or appendInverse

Methods

inverse :: a -> a

appendInv :: a -> a -> a

Instances

Field a => Group (Multiplicative a)

Ring a => Group (Additive a)

(CoGroup a, Group a) => Group (SmartGroup a)

(<>-) :: Group a => a -> a -> a

Monoids with a zero element

class Monoid a => MonoidZero a where

Methods

mzero :: a

Instances

Monoid a => MonoidZero (WithZero a)

SemiRing a => MonoidZero (Multiplicative a)

Boolean a => MonoidZero (Or a)

Boolean a => MonoidZero (And a)

(CoGroup a, MonoidZero a, Group a) => MonoidZero (SmartGroup a)

(CoMonoidZero a, MonoidZero a) => MonoidZero (SmartZero a)

(CoMonoid a, MonoidZero a) => MonoidZero (Smart a)

data WithZero a

Instances

Functor WithZero

Applicative WithZero

Foldable WithZero

Traversable WithZero

Eq a => Eq (WithZero a)

Ord a => Ord (WithZero a)

Monoid a => Monoid (WithZero a)

CoMonoid a => CoMonoidZero (WithZero a)

CoMonoid a => CoMonoid (WithZero a)

Monoid a => MonoidZero (WithZero a)

fromWithZero :: WithZero a -> Maybe a

CoMonoid, CoGroup, and CoMonoidZero (for matching)

class CoMonoid a where

Methods

isEmpty :: a -> Bool

isAppend :: a -> Maybe (a, a)

Instances

CoMonoid [a]

CoMonoid (Set a)

CoMonoid a => CoMonoid (WithZero a)

CoSemiRing a => CoMonoid (Multiplicative a)

CoSemiRing a => CoMonoid (Additive a)

CoBoolean a => CoMonoid (Or a)

CoBoolean a => CoMonoid (And a)

CoMonoid a => CoMonoid (SmartGroup a)

CoMonoid a => CoMonoid (SmartZero a)

CoMonoid a => CoMonoid (Smart a)

class CoMonoid a => CoGroup a where

Methods

isInverse :: a -> Maybe a

isAppendInv :: a -> Maybe (a, a)

Instances

CoField a => CoGroup (Multiplicative a)

CoRing a => CoGroup (Additive a)

CoGroup a => CoGroup (SmartGroup a)

class CoMonoid a => CoMonoidZero a where

Methods

isMonoidZero :: a -> Bool

Instances

CoMonoid a => CoMonoidZero (WithZero a)

CoSemiRing a => CoMonoidZero (Multiplicative a)

CoBoolean a => CoMonoidZero (Or a)

CoBoolean a => CoMonoidZero (And a)

CoMonoidZero a => CoMonoidZero (SmartGroup a)

CoMonoidZero a => CoMonoidZero (SmartZero a)

CoMonoidZero a => CoMonoidZero (Smart a)

associativeList :: CoMonoid a => a -> [a]