Safe Haskell	None
Language	Haskell2010

Data.Semilattice.Join

Description

Join semilattices, related to Lower and Upper.

Synopsis

class Join s where
- (\/) :: s -> s -> s
newtype Joining a = Joining {
- getJoining :: a
}
newtype LessThan a = LessThan {
- getLessThan :: a
}

Documentation

class Join s where Source #

A join semilattice is an idempotent commutative semigroup.

Methods

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

The join operation.

Laws:

Idempotence:

x \/ x = x

Associativity:

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

Commutativity:

a \/ b = b \/ a

Additionally, if s has a Lower bound, then lowerBound must be its identity:

lowerBound \/ a = a
a \/ lowerBound = a

If s has an Upper bound, then upperBound must be its absorbing element:

upperBound \/ a = upperBound
a \/ upperBound = upperBound

Instances

Instances details

Join IntSet Source #	IntSet union forms a semilattice. Idempotence: x \/ x == (x :: IntSet) Associativity: a \/ (b \/ c) == (a \/ b) \/ (c :: IntSet) Commutativity: a \/ b == b \/ (a :: IntSet) Identity: lowerBound \/ a == (a :: IntSet)
Instance details Defined in Data.Semilattice.Join Methods (\/) :: IntSet -> IntSet -> IntSet Source #
Join Ordering Source #	Orderings form a semilattice. Idempotence: x \/ x == (x :: Ordering) Associativity: a \/ (b \/ c) == (a \/ b) \/ (c :: Ordering) Commutativity: a \/ b == b \/ (a :: Ordering) Identity: lowerBound \/ a == (a :: Ordering) Absorption: upperBound \/ a == (upperBound :: Ordering)
Instance details Defined in Data.Semilattice.Join Methods (\/) :: Ordering -> Ordering -> Ordering Source #
Join () Source #
Instance details Defined in Data.Semilattice.Join Methods (\/) :: () -> () -> () Source #
Join Bool Source #	Boolean disjunction forms a semilattice. Idempotence: x \/ x == (x :: Bool) Associativity: a \/ (b \/ c) == (a \/ b) \/ (c :: Bool) Commutativity: a \/ b == b \/ (a :: Bool) Identity: lowerBound \/ a == (a :: Bool) Absorption: upperBound \/ a == (upperBound :: Bool)
Instance details Defined in Data.Semilattice.Join Methods (\/) :: Bool -> Bool -> Bool Source #
Ord a => Join (Max a) Source #	The least upperBound bound gives rise to a join semilattice. Idempotence: x \/ x == (x :: Max Int) Associativity: a \/ (b \/ c) == (a \/ b) \/ (c :: Max Int) Commutativity: a \/ b == b \/ (a :: Max Int) Identity: lowerBound \/ a == (a :: Max Int) Absorption: upperBound \/ a == (upperBound :: Max Int)
Instance details Defined in Data.Semilattice.Join Methods (\/) :: Max a -> Max a -> Max a Source #
Join a => Join (IntMap a) Source #	IntMap union with `Join`able values forms a semilattice. Idempotence: x \/ x == (x :: IntMap (Set Char)) Associativity: a \/ (b \/ c) == (a \/ b) \/ (c :: IntMap (Set Char)) Commutativity: a \/ b == b \/ (a :: IntMap (Set Char)) Identity: lowerBound \/ a == (a :: IntMap (Set Char))
Instance details Defined in Data.Semilattice.Join Methods (\/) :: IntMap a -> IntMap a -> IntMap a Source #
Ord a => Join (Set a) Source #	Set union forms a semilattice. Idempotence: x \/ x == (x :: Set Char) Associativity: a \/ (b \/ c) == (a \/ b) \/ (c :: Set Char) Commutativity: a \/ b == b \/ (a :: Set Char) Identity: lowerBound \/ a == (a :: Set Char)
Instance details Defined in Data.Semilattice.Join Methods (\/) :: Set a -> Set a -> Set a Source #
Eq a => Join (HashSet a) Source #	HashSet union forms a semilattice. Idempotence: x \/ x == (x :: HashSet Char) Associativity: a \/ (b \/ c) == (a \/ b) \/ (c :: HashSet Char) Commutativity: a \/ b == b \/ (a :: HashSet Char) Identity: lowerBound \/ a == (a :: HashSet Char)
Instance details Defined in Data.Semilattice.Join Methods (\/) :: HashSet a -> HashSet a -> HashSet a Source #
Join a => Join (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods (\/) :: Joining a -> Joining a -> Joining a Source #
Join a => Join (LessThan a) Source #
Instance details Defined in Data.Semilattice.Join Methods (\/) :: LessThan a -> LessThan a -> LessThan a Source #
Ord a => Join (Order a) Source #	Total `Ord`erings give rise to a join semilattice satisfying: Idempotence: Order x \/ Order x == Order x Associativity: Order a \/ (Order b \/ Order c) == (Order a \/ Order b) \/ Order c Commutativity: Order a \/ Order b == Order b \/ Order a Identity: lowerBound \/ Order a == Order (a :: Int) Absorption: upperBound \/ Order a == (upperBound :: Order Int) Distributivity: Order a \/ Order b /\ Order c == (Order a \/ Order b) /\ (Order a \/ Order c)
Instance details Defined in Data.Semilattice.Order Methods (\/) :: Order a -> Order a -> Order a Source #
Meet a => Join (Tumble a) Source #
Instance details Defined in Data.Semilattice.Tumble Methods (\/) :: Tumble a -> Tumble a -> Tumble a Source #
(Ord k, Join a) => Join (Map k a) Source #	Map union with `Join`able values forms a semilattice. Idempotence: x \/ x == (x :: Map Char (Set Char)) Associativity: a \/ (b \/ c) == (a \/ b) \/ (c :: Map Char (Set Char)) Commutativity: a \/ b == b \/ (a :: Map Char (Set Char)) Identity: lowerBound \/ a == (a :: Map Char (Set Char))
Instance details Defined in Data.Semilattice.Join Methods (\/) :: Map k a -> Map k a -> Map k a Source #
(Eq k, Join a) => Join (HashMap k a) Source #	HashMap union with `Join`able values forms a semilattice. Idempotence: x \/ x == (x :: HashMap Char (Set Char)) Associativity: a \/ (b \/ c) == (a \/ b) \/ (c :: HashMap Char (Set Char)) Commutativity: a \/ b == b \/ (a :: HashMap Char (Set Char)) Identity: lowerBound \/ a == (a :: HashMap Char (Set Char))
Instance details Defined in Data.Semilattice.Join Methods (\/) :: HashMap k a -> HashMap k a -> HashMap k a Source #
Join b => Join (a -> b) Source #	Functions with semilattice codomains form a semilattice. Idempotence: \ (Fn x) -> x \/ x ~= (x :: Int -> Bool) Associativity: \ (Fn a) (Fn b) (Fn c) -> a \/ (b \/ c) ~= (a \/ b) \/ (c :: Int -> Bool) Commutativity: \ (Fn a) (Fn b) -> a \/ b ~= b \/ (a :: Int -> Bool) Identity: \ (Fn a) -> lowerBound \/ a ~= (a :: Int -> Bool) Absorption: \ (Fn a) -> upperBound \/ a ~= (upperBound :: Int -> Bool)
Instance details Defined in Data.Semilattice.Join Methods (\/) :: (a -> b) -> (a -> b) -> a -> b Source #

newtype Joining a Source #

A Semigroup for any Join semilattice.

If the semilattice has a Lower bound, there is additionally a Monoid instance.

Constructors

Joining
Fields getJoining :: a

Instances

Instances details

Functor Joining Source #
Instance details Defined in Data.Semilattice.Join Methods fmap :: (a -> b) -> Joining a -> Joining b # (<$) :: a -> Joining b -> Joining a #
Foldable Joining Source #
Instance details Defined in Data.Semilattice.Join Methods fold :: Monoid m => Joining m -> m # foldMap :: Monoid m => (a -> m) -> Joining a -> m # foldMap' :: Monoid m => (a -> m) -> Joining a -> m # foldr :: (a -> b -> b) -> b -> Joining a -> b # foldr' :: (a -> b -> b) -> b -> Joining a -> b # foldl :: (b -> a -> b) -> b -> Joining a -> b # foldl' :: (b -> a -> b) -> b -> Joining a -> b # foldr1 :: (a -> a -> a) -> Joining a -> a # foldl1 :: (a -> a -> a) -> Joining a -> a # toList :: Joining a -> [a] # null :: Joining a -> Bool # length :: Joining a -> Int # elem :: Eq a => a -> Joining a -> Bool # maximum :: Ord a => Joining a -> a # minimum :: Ord a => Joining a -> a # sum :: Num a => Joining a -> a # product :: Num a => Joining a -> a #
Traversable Joining Source #
Instance details Defined in Data.Semilattice.Join Methods traverse :: Applicative f => (a -> f b) -> Joining a -> f (Joining b) # sequenceA :: Applicative f => Joining (f a) -> f (Joining a) # mapM :: Monad m => (a -> m b) -> Joining a -> m (Joining b) # sequence :: Monad m => Joining (m a) -> m (Joining a) #
(Lower a, Join a) => Monoid (Joining a) Source #	`Joining` `mempty` is the left- and right-identity. \ x -> let (l, r) = (mappend mempty (Joining x), mappend (Joining x) mempty) in l == Joining x && r == Joining (x :: IntSet)
Instance details Defined in Data.Semilattice.Join Methods mempty :: Joining a # mappend :: Joining a -> Joining a -> Joining a # mconcat :: [Joining a] -> Joining a #
Join a => Semigroup (Joining a) Source #	`Joining` `<>` is associative. \ a b c -> Joining a <> (Joining b <> Joining c) == (Joining a <> Joining b) <> Joining (c :: IntSet)
Instance details Defined in Data.Semilattice.Join Methods (<>) :: Joining a -> Joining a -> Joining a # sconcat :: NonEmpty (Joining a) -> Joining a # stimes :: Integral b => b -> Joining a -> Joining a #
Bounded a => Bounded (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods minBound :: Joining a # maxBound :: Joining a #
Enum a => Enum (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods succ :: Joining a -> Joining a # pred :: Joining a -> Joining a # toEnum :: Int -> Joining a # fromEnum :: Joining a -> Int # enumFrom :: Joining a -> [Joining a] # enumFromThen :: Joining a -> Joining a -> [Joining a] # enumFromTo :: Joining a -> Joining a -> [Joining a] # enumFromThenTo :: Joining a -> Joining a -> Joining a -> [Joining a] #
Num a => Num (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods (+) :: Joining a -> Joining a -> Joining a # (-) :: Joining a -> Joining a -> Joining a # (*) :: Joining a -> Joining a -> Joining a # negate :: Joining a -> Joining a # abs :: Joining a -> Joining a # signum :: Joining a -> Joining a # fromInteger :: Integer -> Joining a #
Read a => Read (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods readsPrec :: Int -> ReadS (Joining a) # readList :: ReadS [Joining a] # readPrec :: ReadPrec (Joining a) # readListPrec :: ReadPrec [Joining a] #
Show a => Show (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods showsPrec :: Int -> Joining a -> ShowS # show :: Joining a -> String # showList :: [Joining a] -> ShowS #
Eq a => Eq (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods (==) :: Joining a -> Joining a -> Bool # (/=) :: Joining a -> Joining a -> Bool #
Ord a => Ord (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods compare :: Joining a -> Joining a -> Ordering # (<) :: Joining a -> Joining a -> Bool # (<=) :: Joining a -> Joining a -> Bool # (>) :: Joining a -> Joining a -> Bool # (>=) :: Joining a -> Joining a -> Bool # max :: Joining a -> Joining a -> Joining a # min :: Joining a -> Joining a -> Joining a #
Join a => Join (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods (\/) :: Joining a -> Joining a -> Joining a Source #
Lower a => Lower (Joining a) Source #
Instance details Defined in Data.Semilattice.Join Methods lowerBound :: Joining a Source #

newtype LessThan a Source #

Join semilattices give rise to a partial Ordering.

Constructors

LessThan
Fields getLessThan :: a

Instances

Instances details

Functor LessThan Source #
Instance details Defined in Data.Semilattice.Join Methods fmap :: (a -> b) -> LessThan a -> LessThan b # (<$) :: a -> LessThan b -> LessThan a #
Foldable LessThan Source #
Instance details Defined in Data.Semilattice.Join Methods fold :: Monoid m => LessThan m -> m # foldMap :: Monoid m => (a -> m) -> LessThan a -> m # foldMap' :: Monoid m => (a -> m) -> LessThan a -> m # foldr :: (a -> b -> b) -> b -> LessThan a -> b # foldr' :: (a -> b -> b) -> b -> LessThan a -> b # foldl :: (b -> a -> b) -> b -> LessThan a -> b # foldl' :: (b -> a -> b) -> b -> LessThan a -> b # foldr1 :: (a -> a -> a) -> LessThan a -> a # foldl1 :: (a -> a -> a) -> LessThan a -> a # toList :: LessThan a -> [a] # null :: LessThan a -> Bool # length :: LessThan a -> Int # elem :: Eq a => a -> LessThan a -> Bool # maximum :: Ord a => LessThan a -> a # minimum :: Ord a => LessThan a -> a # sum :: Num a => LessThan a -> a # product :: Num a => LessThan a -> a #
Traversable LessThan Source #
Instance details Defined in Data.Semilattice.Join Methods traverse :: Applicative f => (a -> f b) -> LessThan a -> f (LessThan b) # sequenceA :: Applicative f => LessThan (f a) -> f (LessThan a) # mapM :: Monad m => (a -> m b) -> LessThan a -> m (LessThan b) # sequence :: Monad m => LessThan (m a) -> m (LessThan a) #
Enum a => Enum (LessThan a) Source #
Instance details Defined in Data.Semilattice.Join Methods succ :: LessThan a -> LessThan a # pred :: LessThan a -> LessThan a # toEnum :: Int -> LessThan a # fromEnum :: LessThan a -> Int # enumFrom :: LessThan a -> [LessThan a] # enumFromThen :: LessThan a -> LessThan a -> [LessThan a] # enumFromTo :: LessThan a -> LessThan a -> [LessThan a] # enumFromThenTo :: LessThan a -> LessThan a -> LessThan a -> [LessThan a] #
Num a => Num (LessThan a) Source #
Instance details Defined in Data.Semilattice.Join Methods (+) :: LessThan a -> LessThan a -> LessThan a # (-) :: LessThan a -> LessThan a -> LessThan a # (*) :: LessThan a -> LessThan a -> LessThan a # negate :: LessThan a -> LessThan a # abs :: LessThan a -> LessThan a # signum :: LessThan a -> LessThan a # fromInteger :: Integer -> LessThan a #
Read a => Read (LessThan a) Source #
Instance details Defined in Data.Semilattice.Join Methods readsPrec :: Int -> ReadS (LessThan a) # readList :: ReadS [LessThan a] # readPrec :: ReadPrec (LessThan a) # readListPrec :: ReadPrec [LessThan a] #
Show a => Show (LessThan a) Source #
Instance details Defined in Data.Semilattice.Join Methods showsPrec :: Int -> LessThan a -> ShowS # show :: LessThan a -> String # showList :: [LessThan a] -> ShowS #
Eq a => Eq (LessThan a) Source #
Instance details Defined in Data.Semilattice.Join Methods (==) :: LessThan a -> LessThan a -> Bool # (/=) :: LessThan a -> LessThan a -> Bool #
(Eq a, Join a) => Ord (LessThan a) Source #	NB: This is not in general a total ordering.
Instance details Defined in Data.Semilattice.Join Methods compare :: LessThan a -> LessThan a -> Ordering # (<) :: LessThan a -> LessThan a -> Bool # (<=) :: LessThan a -> LessThan a -> Bool # (>) :: LessThan a -> LessThan a -> Bool # (>=) :: LessThan a -> LessThan a -> Bool # max :: LessThan a -> LessThan a -> LessThan a # min :: LessThan a -> LessThan a -> LessThan a #
Join a => Join (LessThan a) Source #
Instance details Defined in Data.Semilattice.Join Methods (\/) :: LessThan a -> LessThan a -> LessThan a Source #