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

2 May 2018

      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

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

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