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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