Copyright	(C) 2010-2015 Maximilian Bolingbroke
License	BSD-3-Clause (see the file LICENSE)
Maintainer	Oleg Grenrus <oleg.grenrus@iki.fi>
Safe Haskell	Safe
Language	Haskell2010

Algebra.PartialOrd

Contents

Partial orderings
Fixed points of chains in partial orders

Description

Synopsis

Partial orderings

class Eq a => PartialOrd a where Source #

A partial ordering on sets (http://en.wikipedia.org/wiki/Partially_ordered_set) is a set equipped with a binary relation, leq, that obeys the following laws

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

Two elements of the set are said to be comparable when they are are ordered with respect to the leq relation. So

comparable a b ==> a `leq` b || b `leq` a

If comparable always returns true then the relation leq defines a total ordering (and an Ord instance may be defined). Any Ord instance is trivially an instance of PartialOrd. Ordered provides a convenient wrapper to satisfy PartialOrd given Ord.

As an example consider the partial ordering on sets induced by set inclusion. Then for sets a and b,

a `leq` b

is true when a is a subset of b. Two sets are comparable if one is a subset of the other. Concretely

a = {1, 2, 3}
b = {1, 3, 4}
c = {1, 2}

a `leq` a = True
a `leq` b = False
a `leq` c = False
b `leq` a = False
b `leq` b = True
b `leq` c = False
c `leq` a = True
c `leq` b = False
c `leq` c = True

comparable a b = False
comparable a c = True
comparable b c = False

Minimal complete definition

leq

Methods

leq :: a -> a -> Bool Source #

The relation that induces the partial ordering

comparable :: a -> a -> Bool Source #

Whether two elements are ordered with respect to the relation. A default implementation is given by

comparable x y = leq x y || leq y x

Instances

PartialOrd IntSet Source #
Methods leq :: IntSet -> IntSet -> Bool Source # comparable :: IntSet -> IntSet -> Bool Source #
PartialOrd v => PartialOrd (IntMap v) Source #
Methods leq :: IntMap v -> IntMap v -> Bool Source # comparable :: IntMap v -> IntMap v -> Bool Source #
Ord a => PartialOrd (Set a) Source #
Methods leq :: Set a -> Set a -> Bool Source # comparable :: Set a -> Set a -> Bool Source #
(Eq a, Integral a) => PartialOrd (Divisibility a) Source #
Methods leq :: Divisibility a -> Divisibility a -> Bool Source # comparable :: Divisibility a -> Divisibility a -> Bool Source #
PartialOrd a => PartialOrd (Op a) Source #
Methods leq :: Op a -> Op a -> Bool Source # comparable :: Op a -> Op a -> Bool Source #
Ord a => PartialOrd (Ordered a) Source #
Methods leq :: Ordered a -> Ordered a -> Bool Source # comparable :: Ordered a -> Ordered a -> Bool Source #
(PartialOrd a, PartialOrd b) => PartialOrd (a, b) Source #
Methods leq :: (a, b) -> (a, b) -> Bool Source # comparable :: (a, b) -> (a, b) -> Bool Source #
(Ord k, PartialOrd v) => PartialOrd (Map k v) Source #
Methods leq :: Map k v -> Map k v -> Bool Source # comparable :: Map k v -> Map k v -> Bool Source #
(PartialOrd k, PartialOrd v) => PartialOrd (Lexicographic k v) Source #
Methods leq :: Lexicographic k v -> Lexicographic k v -> Bool Source # comparable :: Lexicographic k v -> Lexicographic k v -> Bool Source #

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

The equality relation induced by the partial-order structure. It must obey the laws Reflexive: a == a Transitive: a == b && b == c ==> a == c

Fixed points of chains in partial orders

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

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

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

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

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

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

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

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