Merge pull request #1 from rightfold/master

Name laws and add Unit and Dual instances

Author: morganthomas

src/Data/Group.purs

@@ -15,26 +15,35 @@
 module Data.Group where
 
 import Data.Monoid (class Monoid)
+import Data.Monoid.Dual (Dual(..))
 import Data.Ring (class Ring, negate)
 import Data.Monoid.Additive (Additive(..))
+import Data.Unit (Unit, unit)
 
 -- | A `Group` is a `Monoid` with inverses. Instances
 -- | must satisfy the following law in addition to the monoid laws:
 -- |
--- | ```text
--- | forall x. ginverse x <> x = mempty = x <> ginverse x
--- | ```
+-- | - Inverse: `forall x. ginverse x <> x = mempty = x <> ginverse x`
 class Monoid g <= Group g where
   ginverse :: g -> g
 
--- | A `CommutativeGroup` is a `Group` with a commutative monoid operation.
--- | Instances must satisfy the following law in addition to the group laws:
+-- | A `CommutativeGroup` is a `Group` with a commutative semigroup operation.
+-- | Also known as an Abelian group. Instances must satisfy the following law
+-- | in addition to the group laws:
 -- |
--- | ```text
--- | forall x, y. x <> y = y <> x
--- | ```
+-- | - Commutativity: `forall x, y. x <> y = y <> x`
 class Group g <= CommutativeGroup g
 
+instance unitIsGroup :: Group Unit where
+  ginverse _ = unit
+
+instance unitIsCommutativeGroup :: CommutativeGroup Unit
+
+instance dualGroupsAreGroups :: (Group g) => Group (Dual g) where
+  ginverse (Dual x) = Dual (ginverse x)
+
+instance dualCommutativeGroupsAreCommutativeGroups :: (CommutativeGroup g) => CommutativeGroup (Dual g)
+
 instance ringsAreAdditiveGroups :: (Ring r) => Group (Additive r) where
   ginverse (Additive x) = Additive (negate x)