add Semiring and Ring instances for Perm

[michaelficarra]

Untested, since I don't see any tests. 😅

[hdgarrood]

This is very pleasing. :)

Fair enough re tests. I think it should be ok to wait until psc 0.7 and then use https://github.com/purescript/purescript-quickcheck-laws.

Do you think it would make sense to remove the Semigroup instance, and get people to use (+) where they would currently use (<>)? eg mkPerms (read + write) read read. Then people could use Data.Monoid.Additive or Data.Monoid.Multiplicative if they want to use Perm with something like foldMap.

[hdgarrood]

Also, I'm not sure the Ring instance is valid. Considering a Boolean Ring instance with (-) as implication, I think we need negate a to be the additive inverse of a for all a, that is, a + (negate a) == zero. However:

true + (negate true)
= true + (zero - true)
= true + (zero && not true)
= true + (false && not true)
= true + false
= true || false
= true
/= zero

edit: sorry, I've just realised that a && not b isn't implication, it's the negation of implication. I think the above still holds, however.

[hdgarrood]

I think a Ring instance for Boolean is impossible - we would need (zero - a) + a = zero for all a. If we take true as a, we then need (false - true) + true = false. Then, it doesn't matter how we define (-), because (following from the properties of boolean or) b + true = true holds for all b.

[michaelficarra]

I removed the Ring instance.

[hdgarrood]

How do you feel about removing the Semigroup instance?