Functional Programming Refresher

A Magma is a set with a (closed) binary operation. A Semigroup is a magma where the operation is associative. class Semigroup a where ( <> ) :: a -> a -> a -- ^ read as "append" Must satisfy: ( x <> y ) <> z == x <> ( y <> z ) -- associativity Example: [ 1 , 2 ] <> [ 3 , 4 ] <> [ 5 ] == [ 1 , 2 , 3 , 4 , 5 ] A Monoid is a semigroup with an identity element. class Semigroup a => Monoid a where mempty :: a -- ^ identity element of <> Must satisfy: mempty <> x == x -- left identity x <> mempty == x -- right identity ( x <> y ) <> z == x <> ( y <> z ) -- associativity (from Semigroup) Note: mappend is a historical name for <> and is now deprecated. Example: mempty <> [ 1 , 2 ] == [ 1 , 2 ] singleton 1 <> mempty == singleton 1 A Group is a monoid where every element has an inverse. class Monoid a => Group a where invert :: a -> a -- ^ inverse of <> Must satisfy: x <> invert x == mempty == invert x <> x A Functor represents a type that can be mapped over. class Functor f where fmap :: ( a ->