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1. type CommutativeMonoid m = (Monoid m, Commutative m)

   monoid-subclasses	Data.Monoid.Cancellative

   No documentation available.

2. type LeftCancellativeMonoid m = (Monoid m, LeftCancellative m)

   monoid-subclasses	Data.Monoid.Cancellative

   No documentation available.

3. type LeftReductiveMonoid m = (Monoid m, LeftReductive m)

   monoid-subclasses	Data.Monoid.Cancellative

   No documentation available.

4. type ReductiveMonoid m = (Monoid m, Reductive m)

   monoid-subclasses	Data.Monoid.Cancellative

   No documentation available.

5. type RightCancellativeMonoid m = (Monoid m, RightCancellative m)

   monoid-subclasses	Data.Monoid.Cancellative

   No documentation available.

6. type RightReductiveMonoid m = (Monoid m, RightReductive m)

   monoid-subclasses	Data.Monoid.Cancellative

   No documentation available.

7. class (Factorial m, MonoidNull m) => FactorialMonoid m

   monoid-subclasses	Data.Monoid.Factorial

   Class of monoids that can be split into irreducible (i.e., atomic or prime) factors in a unique way. Note that mempty is not considered a factor. Factors of a Product are literally its prime factors:

   factors (Product 12) == [Product 2, Product 2, Product 3]
   

   Factors of a list are not its elements but all its single-item sublists:

   factors "abc" == ["a", "b", "c"]
   

   The methods of this class satisfy the following laws in addition to those of Factorial:

   null == List.null . factors
   factors == unfoldr splitPrimePrefix == List.reverse . unfoldr (fmap swap . splitPrimeSuffix)
   reverse == mconcat . List.reverse . factors
   primePrefix == maybe mempty fst . splitPrimePrefix
   primeSuffix == maybe mempty snd . splitPrimeSuffix
   inits == List.map mconcat . List.inits . factors
   tails == List.map mconcat . List.tails . factors
   span p m == (mconcat l, mconcat r) where (l, r) = List.span p (factors m)
   List.all (List.all (not . pred) . factors) . split pred
   mconcat . intersperse prime . split (== prime) == id
   splitAt i m == (mconcat l, mconcat r) where (l, r) = List.splitAt i (factors m)
   spanMaybe () (const $ bool Nothing (Maybe ()) . p) m == (takeWhile p m, dropWhile p m, ())
   spanMaybe s0 (\s m-> Just $ f s m) m0 == (m0, mempty, foldl f s0 m0)
   let (prefix, suffix, s') = spanMaybe s f m
   foldMaybe = foldl g (Just s)
   g s m = s >>= flip f m
   in all ((Nothing ==) . foldMaybe) (inits prefix)
   && prefix == last (filter (isJust . foldMaybe) $ inits m)
   && Just s' == foldMaybe prefix
   && m == prefix <> suffix
   

   A minimal instance definition should implement splitPrimePrefix for performance reasons, and other methods where beneficial.

8. type StableFactorialMonoid m = (StableFactorial m, FactorialMonoid m, PositiveMonoid m)

   monoid-subclasses	Data.Monoid.Factorial

   Deprecated: Use Data.Semigroup.Factorial.StableFactorial instead.

9. class (LeftDistributiveGCDMonoid m, RightDistributiveGCDMonoid m, GCDMonoid m) => DistributiveGCDMonoid m

   monoid-subclasses	Data.Monoid.GCD

   Class of commutative GCD monoids with symmetric distributivity. In addition to the general GCDMonoid laws, instances of this class must also satisfy the following laws:

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

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

10. class (Monoid m, Commutative m, Reductive m, LeftGCDMonoid m, RightGCDMonoid m, OverlappingGCDMonoid m) => GCDMonoid m

    monoid-subclasses	Data.Monoid.GCD

    Class of Abelian monoids that allow the greatest common divisor to be found for any two given values. The operations must satisfy the following laws:

    gcd a b == commonPrefix a b == commonSuffix a b
    Just a' = a </> p && Just b' = b </> p
    where p = gcd a b
    

    In addition, the gcd operation must satisfy the following properties: Uniqueness

    all isJust
    [ a </> c
    , b </> c
    , c </> gcd a b
    ]
    ==>
    (c == gcd a b)
    

    Idempotence

    gcd a a == a
    

    Identity

    gcd mempty a == mempty
    

    gcd a mempty == mempty
    

    Commutativity

    gcd a b == gcd b a
    

    Associativity

    gcd (gcd a b) c == gcd a (gcd b c)
    

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