-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Algebraic structures
--   
--   Algebraic structures
@package alg
@version 0.1.1.0

module Algebra

-- | The class of semigroups (types with an associative binary operation).
class Semigroup a

-- | An associative operation.
--   
--   <pre>
--   (a <a>&lt;&gt;</a> b) <a>&lt;&gt;</a> c = a <a>&lt;&gt;</a> (b <a>&lt;&gt;</a> c)
--   </pre>
--   
--   If <tt>a</tt> is also a <a>Monoid</a> we further require
--   
--   <pre>
--   (<a>&lt;&gt;</a>) = <a>mappend</a>
--   </pre>
(<>) :: Semigroup a => a -> a -> a

-- | Reduce a non-empty list with <tt>&lt;&gt;</tt>
--   
--   The default definition should be sufficient, but this can be
--   overridden for efficiency.
sconcat :: Semigroup a => NonEmpty a -> a

-- | Repeat a value <tt>n</tt> times.
--   
--   Given that this works on a <a>Semigroup</a> 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 <i>O(1)</i> by picking
--   <tt>stimes = stimesIdempotent</tt> or <tt>stimes =
--   stimesIdempotentMonoid</tt> respectively.
stimes :: (Semigroup a, Integral b) => b -> a -> a

-- | The class of monoids (types with an associative binary operation that
--   has an identity). Instances should satisfy the following laws:
--   
--   <ul>
--   <li><pre>mappend mempty x = x</pre></li>
--   <li><pre>mappend x mempty = x</pre></li>
--   <li><pre>mappend x (mappend y z) = mappend (mappend x y) z</pre></li>
--   <li><pre>mconcat = <a>foldr</a> mappend mempty</pre></li>
--   </ul>
--   
--   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
--   <tt>newtype</tt>s and make those instances of <a>Monoid</a>, e.g.
--   <tt>Sum</tt> and <tt>Product</tt>.
class Monoid a

-- | Identity of <a>mappend</a>
mempty :: Monoid a => a
class Monoid a => Group a
invert :: Group a => a -> a
(+) :: Semigroup (Sum a) => a -> a -> a
(-) :: (Semigroup (Sum a), Group (Sum a)) => a -> a -> a
(*) :: Semigroup (Product a) => a -> a -> a
(/) :: (Semigroup (Product a), Group (Product a)) => a -> a -> a
instance Algebra.Group ()
instance (Algebra.Group a, Algebra.Group b) => Algebra.Group (a, b)
instance (Algebra.Group a, Algebra.Group b, Algebra.Group c) => Algebra.Group (a, b, c)
instance (Algebra.Group a, Algebra.Group b, Algebra.Group c, Algebra.Group d) => Algebra.Group (a, b, c, d)
instance (Algebra.Group a, Algebra.Group b, Algebra.Group c, Algebra.Group d, Algebra.Group e) => Algebra.Group (a, b, c, d, e)
instance Algebra.Group a => Algebra.Group (Data.Functor.Identity.Identity a)
instance Algebra.Group a => Algebra.Group (Data.Monoid.Dual a)
instance forall k (a :: k). Algebra.Group (Data.Proxy.Proxy a)
instance forall k a (b :: k). Algebra.Group a => Algebra.Group (Data.Functor.Const.Const a b)
instance Algebra.Group b => Algebra.Group (a -> b)
instance Algebra.Group a => Algebra.Group (GHC.Types.IO a)
instance Algebra.Group (Data.Monoid.Sum GHC.Integer.Type.Integer)