Safe Haskell	None
Language	Haskell2010

NumHask.Algebra.Abstract.Lattice

Synopsis

class Eq a => JoinSemiLattice a where
- (\/) :: a -> a -> a
joinLeq :: JoinSemiLattice a => a -> a -> Bool
class Eq a => MeetSemiLattice a where
- (/\) :: a -> a -> a
meetLeq :: MeetSemiLattice a => a -> a -> Bool
class (JoinSemiLattice a, MeetSemiLattice a) => Lattice a
class JoinSemiLattice a => BoundedJoinSemiLattice a where
- bottom :: a
class MeetSemiLattice a => BoundedMeetSemiLattice a where
- top :: a
class (JoinSemiLattice a, MeetSemiLattice a, BoundedJoinSemiLattice a, BoundedMeetSemiLattice a) => BoundedLattice a

Documentation

class Eq a => JoinSemiLattice a where Source #

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

Associativity: x \/ (y \/ z) == (x \/ y) \/ z
Commutativity: x \/ y == y \/ x
Idempotency:   x \/ x == x

Methods

(\/) :: a -> a -> a infixr 5 Source #

Instances

JoinSemiLattice Bool Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Bool -> Bool -> Bool Source #
JoinSemiLattice Double Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Double -> Double -> Double Source #
JoinSemiLattice Float Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Float -> Float -> Float Source #
JoinSemiLattice Int Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Int -> Int -> Int Source #
JoinSemiLattice Int8 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Int8 -> Int8 -> Int8 Source #
JoinSemiLattice Int16 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Int16 -> Int16 -> Int16 Source #
JoinSemiLattice Int32 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Int32 -> Int32 -> Int32 Source #
JoinSemiLattice Int64 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Int64 -> Int64 -> Int64 Source #
JoinSemiLattice Integer Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Integer -> Integer -> Integer Source #
JoinSemiLattice Natural Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Natural -> Natural -> Natural Source #
JoinSemiLattice Word Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Word -> Word -> Word Source #
JoinSemiLattice Word8 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Word8 -> Word8 -> Word8 Source #
JoinSemiLattice Word16 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Word16 -> Word16 -> Word16 Source #
JoinSemiLattice Word32 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Word32 -> Word32 -> Word32 Source #
JoinSemiLattice Word64 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: Word64 -> Word64 -> Word64 Source #
JoinSemiLattice a => JoinSemiLattice (Complex a) Source #
Instance details Defined in NumHask.Data.Complex Methods (\/) :: Complex a -> Complex a -> Complex a Source #
GCDConstraints a => JoinSemiLattice (Ratio a) Source #
Instance details Defined in NumHask.Data.Rational Methods (\/) :: Ratio a -> Ratio a -> Ratio a Source #
JoinSemiLattice a => JoinSemiLattice (Pair a) Source #
Instance details Defined in NumHask.Data.Pair Methods (\/) :: Pair a -> Pair a -> Pair a Source #
Ord a => JoinSemiLattice (LogField a) Source #
Instance details Defined in NumHask.Data.LogField Methods (\/) :: LogField a -> LogField a -> LogField a Source #
JoinSemiLattice a => JoinSemiLattice (Wrapped a) Source #
Instance details Defined in NumHask.Data.Wrapped Methods (\/) :: Wrapped a -> Wrapped a -> Wrapped a Source #
JoinSemiLattice a => JoinSemiLattice (Positive a) Source #
Instance details Defined in NumHask.Data.Positive Methods (\/) :: Positive a -> Positive a -> Positive a Source #
(Eq (a -> b), JoinSemiLattice b) => JoinSemiLattice (a -> b) Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (\/) :: (a -> b) -> (a -> b) -> a -> b Source #

joinLeq :: JoinSemiLattice a => a -> a -> Bool Source #

The partial ordering induced by the join-semilattice structure

class Eq a => MeetSemiLattice a where Source #

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

Associativity: x /\ (y /\ z) == (x /\ y) /\ z
Commutativity: x /\ y == y /\ x
Idempotency:   x /\ x == x

Methods

(/\) :: a -> a -> a infixr 6 Source #

Instances

MeetSemiLattice Bool Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Bool -> Bool -> Bool Source #
MeetSemiLattice Double Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Double -> Double -> Double Source #
MeetSemiLattice Float Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Float -> Float -> Float Source #
MeetSemiLattice Int Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Int -> Int -> Int Source #
MeetSemiLattice Int8 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Int8 -> Int8 -> Int8 Source #
MeetSemiLattice Int16 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Int16 -> Int16 -> Int16 Source #
MeetSemiLattice Int32 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Int32 -> Int32 -> Int32 Source #
MeetSemiLattice Int64 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Int64 -> Int64 -> Int64 Source #
MeetSemiLattice Integer Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Integer -> Integer -> Integer Source #
MeetSemiLattice Natural Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Natural -> Natural -> Natural Source #
MeetSemiLattice Word Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Word -> Word -> Word Source #
MeetSemiLattice Word8 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Word8 -> Word8 -> Word8 Source #
MeetSemiLattice Word16 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Word16 -> Word16 -> Word16 Source #
MeetSemiLattice Word32 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Word32 -> Word32 -> Word32 Source #
MeetSemiLattice Word64 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: Word64 -> Word64 -> Word64 Source #
MeetSemiLattice a => MeetSemiLattice (Complex a) Source #
Instance details Defined in NumHask.Data.Complex Methods (/\) :: Complex a -> Complex a -> Complex a Source #
GCDConstraints a => MeetSemiLattice (Ratio a) Source #
Instance details Defined in NumHask.Data.Rational Methods (/\) :: Ratio a -> Ratio a -> Ratio a Source #
MeetSemiLattice a => MeetSemiLattice (Pair a) Source #
Instance details Defined in NumHask.Data.Pair Methods (/\) :: Pair a -> Pair a -> Pair a Source #
Ord a => MeetSemiLattice (LogField a) Source #
Instance details Defined in NumHask.Data.LogField Methods (/\) :: LogField a -> LogField a -> LogField a Source #
MeetSemiLattice a => MeetSemiLattice (Wrapped a) Source #
Instance details Defined in NumHask.Data.Wrapped Methods (/\) :: Wrapped a -> Wrapped a -> Wrapped a Source #
MeetSemiLattice a => MeetSemiLattice (Positive a) Source #
Instance details Defined in NumHask.Data.Positive Methods (/\) :: Positive a -> Positive a -> Positive a Source #
(Eq (a -> b), MeetSemiLattice b) => MeetSemiLattice (a -> b) Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods (/\) :: (a -> b) -> (a -> b) -> a -> b Source #

meetLeq :: MeetSemiLattice a => a -> a -> Bool Source #

The partial ordering induced by the meet-semilattice structure

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

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 \/ (a /\ b) == a /\ (a \/ b) == a

Instances

(JoinSemiLattice a, MeetSemiLattice a) => Lattice a Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice

class JoinSemiLattice a => BoundedJoinSemiLattice a where Source #

A join-semilattice with an identity element bottom for \/.

Identity: x \/ bottom == x

Methods

bottom :: a Source #

Instances

BoundedJoinSemiLattice Bool Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Bool Source #
BoundedJoinSemiLattice Double Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Double Source #
BoundedJoinSemiLattice Float Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Float Source #
BoundedJoinSemiLattice Int Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Int Source #
BoundedJoinSemiLattice Int8 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Int8 Source #
BoundedJoinSemiLattice Int16 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Int16 Source #
BoundedJoinSemiLattice Int32 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Int32 Source #
BoundedJoinSemiLattice Int64 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Int64 Source #
BoundedJoinSemiLattice Natural Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Natural Source #
BoundedJoinSemiLattice Word Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Word Source #
BoundedJoinSemiLattice Word8 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Word8 Source #
BoundedJoinSemiLattice Word16 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Word16 Source #
BoundedJoinSemiLattice Word32 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Word32 Source #
BoundedJoinSemiLattice Word64 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: Word64 Source #
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Complex a) Source #
Instance details Defined in NumHask.Data.Complex Methods bottom :: Complex a Source #
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Pair a) Source #
Instance details Defined in NumHask.Data.Pair Methods bottom :: Pair a Source #
(Eq (a -> b), BoundedJoinSemiLattice b) => BoundedJoinSemiLattice (a -> b) Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods bottom :: a -> b Source #

class MeetSemiLattice a => BoundedMeetSemiLattice a where Source #

A meet-semilattice with an identity element top for /\.

Identity: x /\ top == x

Methods

top :: a Source #

Instances

BoundedMeetSemiLattice Bool Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Bool Source #
BoundedMeetSemiLattice Double Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Double Source #
BoundedMeetSemiLattice Float Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Float Source #
BoundedMeetSemiLattice Int Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Int Source #
BoundedMeetSemiLattice Int8 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Int8 Source #
BoundedMeetSemiLattice Int16 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Int16 Source #
BoundedMeetSemiLattice Int32 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Int32 Source #
BoundedMeetSemiLattice Int64 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Int64 Source #
BoundedMeetSemiLattice Word Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Word Source #
BoundedMeetSemiLattice Word8 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Word8 Source #
BoundedMeetSemiLattice Word16 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Word16 Source #
BoundedMeetSemiLattice Word32 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Word32 Source #
BoundedMeetSemiLattice Word64 Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: Word64 Source #
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Complex a) Source #
Instance details Defined in NumHask.Data.Complex Methods top :: Complex a Source #
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Pair a) Source #
Instance details Defined in NumHask.Data.Pair Methods top :: Pair a Source #
(Eq (a -> b), BoundedMeetSemiLattice b) => BoundedMeetSemiLattice (a -> b) Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice Methods top :: a -> b Source #

class (JoinSemiLattice a, MeetSemiLattice a, BoundedJoinSemiLattice a, BoundedMeetSemiLattice a) => BoundedLattice a Source #

Lattices with both bounds

Instances

(JoinSemiLattice a, MeetSemiLattice a, BoundedJoinSemiLattice a, BoundedMeetSemiLattice a) => BoundedLattice a Source #
Instance details Defined in NumHask.Algebra.Abstract.Lattice