Data.Semilattice.Join

Join semilattices, related to Lower and Upper.

Documentation

class Join s where

A join semilattice is an idempotent commutative semigroup.

Minimal complete definition

(\/)

Methods

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

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

Join Bool

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)

Methods

(\/) :: Bool -> Bool -> Bool

Join Ordering

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)

Methods

(\/) :: Ordering -> Ordering -> Ordering

Join ()

Methods

(\/) :: () -> () -> ()

Join IntSet

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)

Methods

(\/) :: IntSet -> IntSet -> IntSet

Ord a => Join (Max a)

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)

Methods

(\/) :: Max a -> Max a -> Max a

Join a => Join (IntMap a)

IntMap union with Joinable 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))

Methods

(\/) :: IntMap a -> IntMap a -> IntMap a

Ord a => Join (Set a)

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)

Methods

(\/) :: Set a -> Set a -> Set a

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

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)

Methods

(\/) :: HashSet a -> HashSet a -> HashSet a

Join a => Join (LessThan a)

Methods

(\/) :: LessThan a -> LessThan a -> LessThan a

Join a => Join (Joining a)

Methods

(\/) :: Joining a -> Joining a -> Joining a

Meet a => Join (Tumble a)

Methods

(\/) :: Tumble a -> Tumble a -> Tumble a

Ord a => Join (Order a)

Total Orderings 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)

Methods

(\/) :: Order a -> Order a -> Order a

Join b => Join (a -> b)

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)

Methods

(\/) :: (a -> b) -> (a -> b) -> a -> b

(Ord k, Join a) => Join (Map k a)

Map union with Joinable 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))

Methods

(\/) :: Map k a -> Map k a -> Map k a

(Eq k, Hashable k, Join a) => Join (HashMap k a)

HashMap union with Joinable 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))

Methods

(\/) :: HashMap k a -> HashMap k a -> HashMap k a

newtype Joining a

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

Functor Joining

Methods

fmap :: (a -> b) -> Joining a -> Joining b

(<$) :: a -> Joining b -> Joining a

Foldable Joining

Methods

fold :: Monoid m => Joining m -> 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

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)

Bounded a => Bounded (Joining a)

Methods

minBound :: Joining a

maxBound :: Joining a

Enum a => Enum (Joining a)

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]

Eq a => Eq (Joining a)

Methods

(==) :: Joining a -> Joining a -> Bool

(/=) :: Joining a -> Joining a -> Bool

Num a => Num (Joining a)

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

Ord a => Ord (Joining a)

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

Read a => Read (Joining a)

Methods

readsPrec :: Int -> ReadS (Joining a)

readList :: ReadS [Joining a]

readPrec :: ReadPrec (Joining a)

readListPrec :: ReadPrec [Joining a]

Show a => Show (Joining a)

Methods

showsPrec :: Int -> Joining a -> ShowS

show :: Joining a -> String

showList :: [Joining a] -> ShowS

Join a => Semigroup (Joining a)

Joining <> is associative.

\ a b c -> Joining a <> (Joining b <> Joining c) == (Joining a <> Joining b) <> Joining (c :: IntSet)

Methods

(<>) :: Joining a -> Joining a -> Joining a

sconcat :: NonEmpty (Joining a) -> Joining a

stimes :: Integral b => b -> Joining a -> Joining a

(Lower a, Join a) => Monoid (Joining a)

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)

Methods

mempty :: Joining a

mappend :: Joining a -> Joining a -> Joining a

mconcat :: [Joining a] -> Joining a

Lower a => Lower (Joining a)

Methods

lowerBound :: Joining a

Join a => Join (Joining a)

Methods

(\/) :: Joining a -> Joining a -> Joining a

newtype LessThan a

Join semilattices give rise to a partial Ordering.

Constructors

LessThan

Fields

• getLessThan :: a

Instances

Functor LessThan

Methods

fmap :: (a -> b) -> LessThan a -> LessThan b

(<$) :: a -> LessThan b -> LessThan a

Foldable LessThan

Methods

fold :: Monoid m => LessThan m -> 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

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)

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]

Eq a => Eq (LessThan a)

Methods

(==) :: LessThan a -> LessThan a -> Bool

(/=) :: LessThan a -> LessThan a -> Bool

Num a => Num (LessThan a)

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

(Eq a, Join a) => Ord (LessThan a)

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

Read a => Read (LessThan a)

Methods

readsPrec :: Int -> ReadS (LessThan a)

readList :: ReadS [LessThan a]

readPrec :: ReadPrec (LessThan a)

readListPrec :: ReadPrec [LessThan a]

Show a => Show (LessThan a)

Methods

showsPrec :: Int -> LessThan a -> ShowS

show :: LessThan a -> String

showList :: [LessThan a] -> ShowS

Join a => Join (LessThan a)

Methods

(\/) :: LessThan a -> LessThan a -> LessThan a