Is fromInteger supposed to be a ring homomorphism?

Question

Until today I had the assumption that fromInteger in the Num class was a ring homomorphism. I had assumed this because integer is is coterminal so every ring must have a unique homomorphism from integer, so it made sense that Num, which is basically the ring class of the standard library, would include that homomorphism.

However today I was reading the laws for Num and saw that fromInteger is not required to be a homomorphism, but only required to preserve identities. So for example we can implement the Klein 4 group, and have fromInteger map 1 to multiplicative identity and everything else to the addative identity, and the result is a legal Num instance but not a homomorphism.

type Klein4
  = ( Bool
    , Bool
    )

instance
  (
  )
    => Num Klein4
  where
  ( a, b ) + ( c, d )
    = ( a /= c
      , b /= d
      )
  ( a, b ) * ( c, d )
    = ( a && b
      , b && d
      )
  negate
    = id
  fromInteger x
    = ( x == 1
      , x == 1
      )

This is a little surprising to me. So my question is why is this the case? Is the intention that fromInteger is a ring homomorphism and I'm just being pedantic, or is there a good use case where you might want to define fromInteger in a way that it is not a homomorphism (and still follows the Num laws)?

Answer 1

Personally, I expect fromInteger to be a ring homomorphism and would be very surprised and annoyed to find an instance that didn't have that property.

Of course, Float and Double have to be the exception to this, as to every good property. Suffice it to say that I am very surprised and annoyed by this:

> fromInteger (2^64) + fromInteger (-1) == (fromInteger (2^64-1) :: Double)
False

Most of the other proposed Num properties are broken by Float and Double, too.

I don't know of any practical types that satisfy all the current Num laws but not fromInteger x + fromInteger y == fromInteger (x+y).