Safe Haskell	Safe-Inferred
Language	GHC2021

NumHask.Algebra.Lattice

Description

Synopsis

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

Documentation

class Eq a => JoinSemiLattice a where Source #

A algebraic structure with element joins: See 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

Instances details

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

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

The partial ordering induced by the join-semilattice structure

(<\) :: JoinSemiLattice a => a -> a -> Bool infixr 6 Source #

The partial ordering induced by the join-semilattice structure

class Eq a => MeetSemiLattice a where Source #

A algebraic structure with element meets: See 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

Instances details

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

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

The partial ordering induced by the meet-semilattice structure

(</) :: MeetSemiLattice a => a -> a -> Bool infixr 6 Source #

The partial ordering induced by the meet-semilattice structure

class JoinSemiLattice a => BoundedJoinSemiLattice a where Source #

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

x \/ bottom == bottom

Methods

bottom :: a Source #

Instances

Instances details

BoundedJoinSemiLattice Int16 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Int16 Source #
BoundedJoinSemiLattice Int32 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Int32 Source #
BoundedJoinSemiLattice Int64 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Int64 Source #
BoundedJoinSemiLattice Int8 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Int8 Source #
BoundedJoinSemiLattice Word16 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Word16 Source #
BoundedJoinSemiLattice Word32 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Word32 Source #
BoundedJoinSemiLattice Word64 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Word64 Source #
BoundedJoinSemiLattice Word8 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Word8 Source #
BoundedJoinSemiLattice Natural Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Natural Source #
BoundedJoinSemiLattice Bool Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Bool Source #
BoundedJoinSemiLattice Double Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Double Source #
BoundedJoinSemiLattice Float Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Float Source #
BoundedJoinSemiLattice Int Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Int Source #
BoundedJoinSemiLattice Word Source #
Instance details Defined in NumHask.Algebra.Lattice Methods bottom :: Word Source #
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (EuclideanPair a) Source #
Instance details Defined in NumHask.Algebra.Metric Methods bottom :: EuclideanPair a Source #
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Complex a) Source #
Instance details Defined in NumHask.Data.Complex Methods bottom :: Complex a Source #
(Additive a, JoinSemiLattice a) => BoundedJoinSemiLattice (Positive a) Source #
Instance details Defined in NumHask.Data.Positive Methods bottom :: Positive a Source #
BoundedJoinSemiLattice a => BoundedJoinSemiLattice (Wrapped a) Source #
Instance details Defined in NumHask.Data.Wrapped Methods bottom :: Wrapped a Source #

class MeetSemiLattice a => BoundedMeetSemiLattice a where Source #

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

x /\ top == top

Methods

top :: a Source #

Instances

Instances details

BoundedMeetSemiLattice Int16 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Int16 Source #
BoundedMeetSemiLattice Int32 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Int32 Source #
BoundedMeetSemiLattice Int64 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Int64 Source #
BoundedMeetSemiLattice Int8 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Int8 Source #
BoundedMeetSemiLattice Word16 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Word16 Source #
BoundedMeetSemiLattice Word32 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Word32 Source #
BoundedMeetSemiLattice Word64 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Word64 Source #
BoundedMeetSemiLattice Word8 Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Word8 Source #
BoundedMeetSemiLattice Bool Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Bool Source #
BoundedMeetSemiLattice Double Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Double Source #
BoundedMeetSemiLattice Float Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Float Source #
BoundedMeetSemiLattice Int Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Int Source #
BoundedMeetSemiLattice Word Source #
Instance details Defined in NumHask.Algebra.Lattice Methods top :: Word Source #
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (EuclideanPair a) Source #
Instance details Defined in NumHask.Algebra.Metric Methods top :: EuclideanPair a Source #
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Complex a) Source #
Instance details Defined in NumHask.Data.Complex Methods top :: Complex a Source #
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Positive a) Source #
Instance details Defined in NumHask.Data.Positive Methods top :: Positive a Source #
BoundedMeetSemiLattice a => BoundedMeetSemiLattice (Wrapped a) Source #
Instance details Defined in NumHask.Data.Wrapped Methods top :: Wrapped a Source #

type Lattice a = (JoinSemiLattice a, MeetSemiLattice a) Source #

The combination of two semi lattices makes a lattice if the absorption law holds: see Absorption Law and Lattice

Absorption: a \/ (a /\ b) == a /\ (a \/ b) == a

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

Lattices with both bounds

x /\ bottom == x
x \/ top = x