I agree that the AMonoid type you describe is useful. In fact, it's going
to be in base when GHC 8.6 is released: https://github.com/ghc/ghc/pull/123
On Wed, May 2, 2018 at 10:31 AM, 박신환
Logical XOR is associative, also is XNOR. And they both have an identity element.
Hence:
newtype Even = Even {getEven :: Bool}
newtype Odd = Odd {getOdd :: Bool}
instance Semigroup Even where
(<>) = (==) -- (==) over Bool == Logical XNOR
instance Semigroup Odd where
(<>) = (/=) -- (/=) over Bool == Logical XOR
instance Monoid Even where
mempty = True
instance Monoid Odd where
mempty = False
So foldMap would determine the parity of the number of Trues.
Also, Monoid lifted by Applicative is also Monoid. This might be useful:
newtype AMonoid f a = AMonoid {getAMonoid :: f a}
instance (Applicative f, Semigroup a) => Semigroup (AMonoid f a) where
(<>) = liftA2 (<>)
instance (Applicative f, Monoid a) => Monoid (AMonoid f a) where
mempty = pure mempty
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