As a toy project, I would like to understand how to model mathematical groups in Haskell in general.
To start us off, we begin by observing that a to be defined Group is just a Monoid with inversion.
class (Monoid m) => Group m where
minvert :: m -> m
Next, we first restrict ourselves to cyclic groups and start by defining the cyclic group of order 12.
data Cyclic12 = Cyclic12 Int deriving (Show, Eq)
Finally, we instantiate both classes for Cyclic12.
instance Monoid Cyclic12 where
mempty = Cyclic12 0
mappend (Cyclic12 x) (Cyclic12 y) = Cyclic12 ((x + y) `mod` 12)
instance Group Cyclic12 where
minvert (Cyclic12 x) = Cyclic12 ((0 - x) `mod` 12)
How do I abstract the previous definition from the specific value of 12 to allow a more generic definition of different cyclic groups?
Ideally, I would like to write definitions like
instance Monoid (Cyclic k) where
mempty = Cyclic k 0
mappend (Cyclic k x) (Cyclic k y) = Cyclic k ((x + y) `mod` k)
instance Group (Cyclic k) where
minvert (Cyclic k x) = Cyclic k ((0 - x) `mod` k)
But with a data definition like
data Cyclic = Cyclic Int Int deriving (Show, Eq)
we still don't get very far, because k is "not in scope". Regarding its apparent triviality I have a feeling to be missing out on some fundamental concept here. Thanks in advance for your help.