Working to implement binary adders in Haskell

Adders.hs

module Adders (
  Binary(..)
  , halfAdder
  , adder
  ) where

import Data.Monoid

data Binary = Zero
            | One
              deriving (Show, Eq)

newtype BinaryNumber = BinaryNumber { runBinary :: [Binary] }

type Carry = Binary
type Result = Binary

instance Monoid Binary where
  mempty = Zero
  mappend = andAdd

andAdd :: Binary -> Binary -> Carry
andAdd Zero Zero = Zero
andAdd Zero One = Zero
andAdd One Zero = Zero
andAdd One One = One

orAdd :: Binary -> Binary -> Binary
orAdd Zero Zero = Zero
orAdd Zero One = One
orAdd One Zero = One
orAdd One One = One

halfAdder :: Binary -> Binary -> Binary
halfAdder Zero Zero = Zero
halfAdder Zero One = One
halfAdder One Zero = One
halfAdder One One = Zero

adder :: Binary -> Binary -> Carry -> (Result, Carry)
adder a b c = (res, car)
  where res = halfAdder c (halfAdder a b)
        car = orAdd (a <> b) (halfAdder res c)

I don't remember why I was working on this but I found it in my computer and wanted to save it and come back to it.