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Language	Haskell2010

Data.Semigroup.Quantale

Description

Synopsis

type UnitalQuantale a = (Monoid a, Quantale a)
class Semigroup a => Quantale a where
- residL :: a -> ConnL a a
- residR :: a -> ConnR a a
- (//) :: a -> a -> a
- (\\) :: a -> a -> a
(<==) :: (Meet - Quantale) a => a -> a -> a
(==>) :: (Meet - Quantale) a => a -> a -> a

Documentation

type UnitalQuantale a = (Monoid a, Quantale a) Source #

class Semigroup a => Quantale a where Source #

A residuated, partially ordered semigroup.

In the interest of broader usability we relax the common definition slightly and use the term quantale to describe any residuated, partially ordered semigroup. This admits instances of hoops and triangular (co)-norms.

Laws:

x \\ x = mempty  
x <~ y iff mempty = x \\ y
x <> (x \\ y) = y <> (y \\ x)  
(x <> y) \\ z = y \\ (x \\ z) (currying)
x <> y <~ z iff y <~ x \\ z iff x <~ z // y.

See https://ncatlab.org/nlab/show/quantale.

TODO: There are several additional properties that apply when the poset structure is lattice-ordered (i.e. a residuated lattice) or when the semigroup is a monoid or semiring.

Minimal complete definition

Nothing

Methods

residL :: a -> ConnL a a Source #

residR :: a -> ConnR a a Source #

(//) :: a -> a -> a infixl 5 Source #

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

Instances

Quantale All Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: All -> ConnL All All Source # residR :: All -> ConnR All All Source # (//) :: All -> All -> All Source # (\\) :: All -> All -> All Source #
(TotalOrder a, Bounded a) => Quantale (Min a) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Min a -> ConnL (Min a) (Min a) Source # residR :: Min a -> ConnR (Min a) (Min a) Source # (//) :: Min a -> Min a -> Min a Source # (\\) :: Min a -> Min a -> Min a Source #
Quantale (Meet Bool) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Bool -> ConnL (Meet Bool) (Meet Bool) Source # residR :: Meet Bool -> ConnR (Meet Bool) (Meet Bool) Source # (//) :: Meet Bool -> Meet Bool -> Meet Bool Source # (\\) :: Meet Bool -> Meet Bool -> Meet Bool Source #
Quantale (Meet a) => Quantale (Meet (r -> a)) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet (r -> a) -> ConnL (Meet (r -> a)) (Meet (r -> a)) Source # residR :: Meet (r -> a) -> ConnR (Meet (r -> a)) (Meet (r -> a)) Source # (//) :: Meet (r -> a) -> Meet (r -> a) -> Meet (r -> a) Source # (\\) :: Meet (r -> a) -> Meet (r -> a) -> Meet (r -> a) Source #
Quantale (Meet Int) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Int -> ConnL (Meet Int) (Meet Int) Source # residR :: Meet Int -> ConnR (Meet Int) (Meet Int) Source # (//) :: Meet Int -> Meet Int -> Meet Int Source # (\\) :: Meet Int -> Meet Int -> Meet Int Source #
Quantale (Meet Int8) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Int8 -> ConnL (Meet Int8) (Meet Int8) Source # residR :: Meet Int8 -> ConnR (Meet Int8) (Meet Int8) Source # (//) :: Meet Int8 -> Meet Int8 -> Meet Int8 Source # (\\) :: Meet Int8 -> Meet Int8 -> Meet Int8 Source #
Quantale (Meet Int16) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Int16 -> ConnL (Meet Int16) (Meet Int16) Source # residR :: Meet Int16 -> ConnR (Meet Int16) (Meet Int16) Source # (//) :: Meet Int16 -> Meet Int16 -> Meet Int16 Source # (\\) :: Meet Int16 -> Meet Int16 -> Meet Int16 Source #
Quantale (Meet Int32) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Int32 -> ConnL (Meet Int32) (Meet Int32) Source # residR :: Meet Int32 -> ConnR (Meet Int32) (Meet Int32) Source # (//) :: Meet Int32 -> Meet Int32 -> Meet Int32 Source # (\\) :: Meet Int32 -> Meet Int32 -> Meet Int32 Source #
Quantale (Meet Int64) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Int64 -> ConnL (Meet Int64) (Meet Int64) Source # residR :: Meet Int64 -> ConnR (Meet Int64) (Meet Int64) Source # (//) :: Meet Int64 -> Meet Int64 -> Meet Int64 Source # (\\) :: Meet Int64 -> Meet Int64 -> Meet Int64 Source #
UnitalQuantale (Meet a) => Quantale (Meet (Maybe a)) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet (Maybe a) -> ConnL (Meet (Maybe a)) (Meet (Maybe a)) Source # residR :: Meet (Maybe a) -> ConnR (Meet (Maybe a)) (Meet (Maybe a)) Source # (//) :: Meet (Maybe a) -> Meet (Maybe a) -> Meet (Maybe a) Source # (\\) :: Meet (Maybe a) -> Meet (Maybe a) -> Meet (Maybe a) Source #
Quantale (Meet Ordering) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Ordering -> ConnL (Meet Ordering) (Meet Ordering) Source # residR :: Meet Ordering -> ConnR (Meet Ordering) (Meet Ordering) Source # (//) :: Meet Ordering -> Meet Ordering -> Meet Ordering Source # (\\) :: Meet Ordering -> Meet Ordering -> Meet Ordering Source #
Quantale (Meet Word) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Word -> ConnL (Meet Word) (Meet Word) Source # residR :: Meet Word -> ConnR (Meet Word) (Meet Word) Source # (//) :: Meet Word -> Meet Word -> Meet Word Source # (\\) :: Meet Word -> Meet Word -> Meet Word Source #
Quantale (Meet Word8) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Word8 -> ConnL (Meet Word8) (Meet Word8) Source # residR :: Meet Word8 -> ConnR (Meet Word8) (Meet Word8) Source # (//) :: Meet Word8 -> Meet Word8 -> Meet Word8 Source # (\\) :: Meet Word8 -> Meet Word8 -> Meet Word8 Source #
Quantale (Meet Word16) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Word16 -> ConnL (Meet Word16) (Meet Word16) Source # residR :: Meet Word16 -> ConnR (Meet Word16) (Meet Word16) Source # (//) :: Meet Word16 -> Meet Word16 -> Meet Word16 Source # (\\) :: Meet Word16 -> Meet Word16 -> Meet Word16 Source #
Quantale (Meet Word32) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Word32 -> ConnL (Meet Word32) (Meet Word32) Source # residR :: Meet Word32 -> ConnR (Meet Word32) (Meet Word32) Source # (//) :: Meet Word32 -> Meet Word32 -> Meet Word32 Source # (\\) :: Meet Word32 -> Meet Word32 -> Meet Word32 Source #
Quantale (Meet Word64) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet Word64 -> ConnL (Meet Word64) (Meet Word64) Source # residR :: Meet Word64 -> ConnR (Meet Word64) (Meet Word64) Source # (//) :: Meet Word64 -> Meet Word64 -> Meet Word64 Source # (\\) :: Meet Word64 -> Meet Word64 -> Meet Word64 Source #
(Preorder a, UnitalQuantale (Meet a)) => Quantale (Meet (Either a a)) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet (Either a a) -> ConnL (Meet (Either a a)) (Meet (Either a a)) Source # residR :: Meet (Either a a) -> ConnR (Meet (Either a a)) (Meet (Either a a)) Source # (//) :: Meet (Either a a) -> Meet (Either a a) -> Meet (Either a a) Source # (\\) :: Meet (Either a a) -> Meet (Either a a) -> Meet (Either a a) Source #
Quantale (Meet ()) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet () -> ConnL (Meet ()) (Meet ()) Source # residR :: Meet () -> ConnR (Meet ()) (Meet ()) Source # (//) :: Meet () -> Meet () -> Meet () Source # (\\) :: Meet () -> Meet () -> Meet () Source #
Quantale (Meet (Predicate a)) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet (Predicate a) -> ConnL (Meet (Predicate a)) (Meet (Predicate a)) Source # residR :: Meet (Predicate a) -> ConnR (Meet (Predicate a)) (Meet (Predicate a)) Source # (//) :: Meet (Predicate a) -> Meet (Predicate a) -> Meet (Predicate a) Source # (\\) :: Meet (Predicate a) -> Meet (Predicate a) -> Meet (Predicate a) Source #
Quantale (Join a) => Quantale (Meet (Down a)) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet (Down a) -> ConnL (Meet (Down a)) (Meet (Down a)) Source # residR :: Meet (Down a) -> ConnR (Meet (Down a)) (Meet (Down a)) Source # (//) :: Meet (Down a) -> Meet (Down a) -> Meet (Down a) Source # (\\) :: Meet (Down a) -> Meet (Down a) -> Meet (Down a) Source #
(TotalOrder k, Finite k, UnitalQuantale (Meet a)) => Quantale (Meet (Map k a)) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet (Map k a) -> ConnL (Meet (Map k a)) (Meet (Map k a)) Source # residR :: Meet (Map k a) -> ConnR (Meet (Map k a)) (Meet (Map k a)) Source # (//) :: Meet (Map k a) -> Meet (Map k a) -> Meet (Map k a) Source # (\\) :: Meet (Map k a) -> Meet (Map k a) -> Meet (Map k a) Source #
TotalOrder a => Quantale (Meet (Set a)) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Meet (Set a) -> ConnL (Meet (Set a)) (Meet (Set a)) Source # residR :: Meet (Set a) -> ConnR (Meet (Set a)) (Meet (Set a)) Source # (//) :: Meet (Set a) -> Meet (Set a) -> Meet (Set a) Source # (\\) :: Meet (Set a) -> Meet (Set a) -> Meet (Set a) Source #
Quantale (Meet a) => Quantale (Join (Down a)) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: Join (Down a) -> ConnL (Join (Down a)) (Join (Down a)) Source # residR :: Join (Down a) -> ConnR (Join (Down a)) (Join (Down a)) Source # (//) :: Join (Down a) -> Join (Down a) -> Join (Down a) Source # (\\) :: Join (Down a) -> Join (Down a) -> Join (Down a) Source #
Quantale a => Quantale (r -> a) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: (r -> a) -> ConnL (r -> a) (r -> a) Source # residR :: (r -> a) -> ConnR (r -> a) (r -> a) Source # (//) :: (r -> a) -> (r -> a) -> r -> a Source # (\\) :: (r -> a) -> (r -> a) -> r -> a Source #
(Applicative f, Quantale a) => Quantale (F1 f a) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: F1 f a -> ConnL (F1 f a) (F1 f a) Source # residR :: F1 f a -> ConnR (F1 f a) (F1 f a) Source # (//) :: F1 f a -> F1 f a -> F1 f a Source # (\\) :: F1 f a -> F1 f a -> F1 f a Source #
(Applicative f, Applicative g, Quantale a) => Quantale (F2 f g a) Source #
Instance details Defined in Data.Semigroup.Quantale Methods residL :: F2 f g a -> ConnL (F2 f g a) (F2 f g a) Source # residR :: F2 f g a -> ConnR (F2 f g a) (F2 f g a) Source # (//) :: F2 f g a -> F2 f g a -> F2 f g a Source # (\\) :: F2 f g a -> F2 f g a -> F2 f g a Source #

(<==) :: (Meet - Quantale) a => a -> a -> a infixl 1 Source #

(==>) :: (Meet - Quantale) a => a -> a -> a infixr 1 Source #

Logical implication.

x ==> x           = top
x /\ (x ==> y)  = x /\ y
y /\ (x ==> y)  = y
x ==> (y ==> z) = (x /\ y) ==> z
x ==> (y /\ z)  = (x ==> y) /\ (x ==> z)
meetLe ((x ==> y) \/ x) y
meetLe y (x ==> x /\ y)
meetLe x y => meetLe (z ==> x) (z ==> y)
meetLe x y => meetLe (x ==> z) (y ==> z)
meetLe x y = x ==> y == top
meetLe (x /\ y) z = meetLe x (y ==> z)

See Heyting for the laws.