功能编程 – 在函数式编程中是否还有任何代数结构,而不是单一的

class Group a where
    mult :: a -> a -> a
    identity :: a
    inverse :: a -> a

instance Group Integer where
    mult = (+)
    identity = 0
    inverse = negate

-- S_3 (group of all bijections of a 3-element set)
data S3 = ABC | ACB | BAC | BCA | CAB | CBA
instance Group S3 where
    mult ABC x = x
    ... -- some boring code
    identity = ABC
    inverse ABC = ABC
    ... -- remaining cases

-- Operations on groups. Dual:
data Dual a = Dual { getDual :: a }
instance Group a => Group (Dual a) where
    mult (Dual x) (Dual y) = Dual (mult y x)
    identity = Dual identity
    inverse (Dual x) = Dual (inverse x)

-- Product:
instance (Group a,Group b) => Group (a,b) where
    mult (x,y) (z,t) = (x `mult` z,y `mult` t)
    identity = (identity,identity)
    inverse (x,y) = (inverse x,inverse y)

现在,您可以编写多(双CAB,5)(双CBA,1)并获得结果.这将是组S3 *?Z中的计算.您可以添加其他组,以任何可能的方式组合它们,并与它们进行计算.

类似的事情可以通过环,字段,排序,向量空间,类别等来完成.Haskell的数字层次结构不幸的是被模仿,但是有一个numeric prelude尝试修复它.还有DoCon把它带到极限.对于类型课程(主要由类别理论推动),有Typeclassopedia具有大量示例和应用程序.