Monoid over XOR or XNOR, and Monoid lifted by Applicative.

[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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