Algebra.Lattice

In mathematics, a lattice is a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).

In this module lattices are defined using meet and join operators, as it's constructive one.

Unbounded lattices

class JoinSemiLattice a where

A algebraic structure with element joins: http://en.wikipedia.org/wiki/Semilattice

Associativity: x join (y join z) == (x join y) join z
Commutativity: x join y == y join x
Idempotency:   x join x == x

Methods

join :: a -> a -> a

Instances

JoinSemiLattice Bool

JoinSemiLattice IntSet

JoinSemiLattice v => JoinSemiLattice (IntMap v)

Ord a => JoinSemiLattice (Set a)

(Eq a, Hashable a) => JoinSemiLattice (HashSet a)

JoinSemiLattice a => JoinSemiLattice (Dropped a)

JoinSemiLattice a => JoinSemiLattice (Levitated a)

JoinSemiLattice a => JoinSemiLattice (Lifted a)

JoinSemiLattice v => JoinSemiLattice (k -> v)

(JoinSemiLattice a, JoinSemiLattice b) => JoinSemiLattice (a, b)

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

(Eq k, Hashable k) => JoinSemiLattice (HashMap k v)

class MeetSemiLattice a where

A algebraic structure with element meets: http://en.wikipedia.org/wiki/Semilattice

Associativity: x meet (y meet z) == (x meet y) meet z
Commutativity: x meet y == y meet x
Idempotency:   x meet x == x

Methods

meet :: a -> a -> a

Instances

MeetSemiLattice Bool

(Ord a, Enumerable a) => MeetSemiLattice (Set (Enumerated a))

(Eq a, Hashable a) => MeetSemiLattice (HashSet a)

MeetSemiLattice a => MeetSemiLattice (Dropped a)

MeetSemiLattice a => MeetSemiLattice (Levitated a)

MeetSemiLattice a => MeetSemiLattice (Lifted a)

MeetSemiLattice v => MeetSemiLattice (k -> v)

(MeetSemiLattice a, MeetSemiLattice b) => MeetSemiLattice (a, b)

(Ord k, Enumerable k, MeetSemiLattice v) => MeetSemiLattice (Map (Enumerated k) v)

(Eq k, Hashable k) => MeetSemiLattice (HashMap k v)

class (JoinSemiLattice a, MeetSemiLattice a) => Lattice a

The combination of two semi lattices makes a lattice if the absorption law holds: see http://en.wikipedia.org/wiki/Absorption_law and http://en.wikipedia.org/wiki/Lattice_(order)

Absorption: a join (a meet b) == a meet (a join b) == a

Instances

Lattice Bool

(Ord a, Enumerable a) => Lattice (Set (Enumerated a))

Lattice a => Lattice (Dropped a)

Lattice a => Lattice (Levitated a)

Lattice a => Lattice (Lifted a)

Lattice v => Lattice (k -> v)

(Lattice a, Lattice b) => Lattice (a, b)

(Ord k, Enumerable k, Lattice v) => Lattice (Map (Enumerated k) v)

joinLeq :: (Eq a, JoinSemiLattice a) => a -> a -> Bool

The partial ordering induced by the join-semilattice structure

joins1 :: JoinSemiLattice a => [a] -> a

The join of at a list of join-semilattice elements (of length at least one)

meetLeq :: (Eq a, MeetSemiLattice a) => a -> a -> Bool

The partial ordering induced by the meet-semilattice structure

meets1 :: MeetSemiLattice a => [a] -> a

The meet of at a list of meet-semilattice elements (of length at least one)

Bounded lattices

class JoinSemiLattice a => BoundedJoinSemiLattice a where

A join-semilattice with some element |bottom| that join approaches.

Identity: x join bottom == x

Methods

bottom :: a

Instances

BoundedJoinSemiLattice Bool

BoundedJoinSemiLattice IntSet

JoinSemiLattice v => BoundedJoinSemiLattice (IntMap v)

Ord a => BoundedJoinSemiLattice (Set a)

(Eq a, Hashable a) => BoundedJoinSemiLattice (HashSet a)

BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Dropped a)

JoinSemiLattice a => BoundedJoinSemiLattice (Levitated a)

JoinSemiLattice a => BoundedJoinSemiLattice (Lifted a)

BoundedJoinSemiLattice v => BoundedJoinSemiLattice (k -> v)

(BoundedJoinSemiLattice a, BoundedJoinSemiLattice b) => BoundedJoinSemiLattice (a, b)

(Ord k, JoinSemiLattice v) => BoundedJoinSemiLattice (Map k v)

(Eq k, Hashable k) => BoundedJoinSemiLattice (HashMap k v)

class MeetSemiLattice a => BoundedMeetSemiLattice a where

A meet-semilattice with some element |top| that meet approaches.

Identity: x meet top == x

Methods

top :: a

Instances

BoundedMeetSemiLattice Bool

(Ord a, Enumerable a) => BoundedMeetSemiLattice (Set (Enumerated a))

MeetSemiLattice a => BoundedMeetSemiLattice (Dropped a)

MeetSemiLattice a => BoundedMeetSemiLattice (Levitated a)

BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Lifted a)

BoundedMeetSemiLattice v => BoundedMeetSemiLattice (k -> v)

(BoundedMeetSemiLattice a, BoundedMeetSemiLattice b) => BoundedMeetSemiLattice (a, b)

(Ord k, Enumerable k, BoundedMeetSemiLattice v) => BoundedMeetSemiLattice (Map (Enumerated k) v)

class (Lattice a, BoundedJoinSemiLattice a, BoundedMeetSemiLattice a) => BoundedLattice a

Lattices with both bounds

Instances

BoundedLattice Bool

(Ord a, Enumerable a) => BoundedLattice (Set (Enumerated a))

BoundedLattice a => BoundedLattice (Dropped a)

Lattice a => BoundedLattice (Levitated a)

BoundedLattice a => BoundedLattice (Lifted a)

BoundedLattice v => BoundedLattice (k -> v)

(BoundedLattice a, BoundedLattice b) => BoundedLattice (a, b)

(Ord k, Enumerable k, BoundedLattice v) => BoundedLattice (Map (Enumerated k) v)

joins :: BoundedJoinSemiLattice a => [a] -> a

The join of a list of join-semilattice elements

meets :: BoundedMeetSemiLattice a => [a] -> a

The meet of a list of meet-semilattice elements

Fixed points of chains in lattices

lfp :: (Eq a, BoundedJoinSemiLattice a) => (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Forces the function to be monotone.

lfpFrom :: (Eq a, BoundedJoinSemiLattice a) => a -> (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Forces the function to be monotone.

unsafeLfp :: (Eq a, BoundedJoinSemiLattice a) => (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Assumes that the function is monotone and does not check if that is correct.

gfp :: (Eq a, BoundedMeetSemiLattice a) => (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Forces the function to be antinone.

gfpFrom :: (Eq a, BoundedMeetSemiLattice a) => a -> (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Forces the function to be antinone.

unsafeGfp :: (Eq a, BoundedMeetSemiLattice a) => (a -> a) -> a

Implementation of Kleene fixed-point theorem http://en.wikipedia.org/wiki/Kleene_fixed-point_theorem. Assumes that the function is antinone and does not check if that is correct.