Algebra.PartialOrd

Partial orderings

class Eq a => PartialOrd a where

A partial ordering on sets: http://en.wikipedia.org/wiki/Partially_ordered_set

This can be defined using either joinLeq or meetLeq, or a more efficient definition can be derived directly.

Reflexive:     a leq a
Antisymmetric: a leq b && b leq a ==> a == b
Transitive:    a leq b && b leq c ==> a leq c

The superclass equality (which can be defined using partialOrdEq) must obey these laws:

Reflexive:  a == a
Transitive: a == b && b == c ==> a == b

Methods

leq :: a -> a -> Bool

Instances

PartialOrd IntSet

PartialOrd v => PartialOrd (IntMap v)

Ord a => PartialOrd (Set a)

(PartialOrd v, Finite k) => PartialOrd (k -> v)

(PartialOrd a, PartialOrd b) => PartialOrd (a, b)

(Ord k, PartialOrd v) => PartialOrd (Map k v)

partialOrdEq :: PartialOrd a => a -> a -> Bool

The equality relation induced by the partial-order structure

Fixed points of chains in partial orders

lfpFrom :: PartialOrd a => a -> (a -> a) -> a

Least point of a partially ordered monotone function. Checks that the function is monotone.

unsafeLfpFrom :: Eq a => a -> (a -> a) -> a

Least point of a partially ordered monotone function. Does not checks that the function is monotone.

gfpFrom :: PartialOrd a => a -> (a -> a) -> a

Greatest fixed point of a partially ordered antinone function. Checks that the function is antinone.

unsafeGfpFrom :: Eq a => a -> (a -> a) -> a

Greatest fixed point of a partially ordered antinone function. Does not check that the function is antinone.