Semilattice

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)

Defined in Data.Semilattice.Join

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)

Defined in Data.Semilattice.Join

Methods

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

Join ()

Defined in Data.Semilattice.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)

Defined in Data.Semilattice.Join

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)

Defined in Data.Semilattice.Join

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))

Defined in Data.Semilattice.Join

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)

Defined in Data.Semilattice.Join

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)

Defined in Data.Semilattice.Join

Methods

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

Defined in Data.Semilattice.Order

Methods

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

Join a => Join (Joining a)

Defined in Data.Semilattice.Join

Methods

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

Join a => Join (LessThan a)

Defined in Data.Semilattice.Join

Methods

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

Defined in Data.Semilattice.Join

Methods

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

(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))

Defined in Data.Semilattice.Join

Methods

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

(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))

Defined in Data.Semilattice.Join

Methods

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

class Meet s where

A meet semilattice is an idempotent commutative semigroup.

Minimal complete definition

(/\)

Methods

(/\) :: s -> s -> s infixr 7

The meet operation.

Laws:

Idempotence:

x /\ x = x

Associativity:

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

Commutativity:

a /\ b = b /\ a

Additionally, if s has an Upper bound, then upperBound must be its identity:

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

If s has a Lower bound, then lowerBound must be its absorbing element:

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

Instances

Meet Bool

Boolean conjunction forms a semilattice.

Idempotence:

x /\ x == (x :: Bool)

Associativity:

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

Commutativity:

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

Identity:

upperBound /\ a == (a :: Bool)

Absorption:

lowerBound /\ a == (lowerBound :: Bool)

Defined in Data.Semilattice.Meet

Methods

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

Meet 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:

upperBound /\ a == (a :: Ordering)

Absorption:

lowerBound /\ a == (lowerBound :: Ordering)

Defined in Data.Semilattice.Meet

Methods

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

Meet ()

Defined in Data.Semilattice.Meet

Methods

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

Meet IntSet

IntSet intersection forms a semilattice.

Idempotence:

x /\ x == (x :: IntSet)

Associativity:

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

Commutativity:

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

Absorption:

lowerBound /\ a == (lowerBound :: IntSet)

Defined in Data.Semilattice.Meet

Methods

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

Ord a => Meet (Min a)

The greatest lowerBound bound gives rise to a meet semilattice.

Idempotence:

x /\ x == (x :: Min Int)

Associativity:

a /\ (b /\ c) == (a /\ b) /\ (c :: Min Int)

Commutativity:

a /\ b == b /\ (a :: Min Int)

Identity:

upperBound /\ a == (a :: Min Int)

Absorption:

lowerBound /\ a == (lowerBound :: Min Int)

Defined in Data.Semilattice.Meet

Methods

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

Meet a => Meet (IntMap a)

IntMap union with Meetable 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))

Absorption:

lowerBound /\ a == (lowerBound :: IntMap (Set Char))

Defined in Data.Semilattice.Meet

Methods

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

Ord a => Meet (Set a)

Set intersection 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)

Absorption:

lowerBound /\ a == (lowerBound :: Set Char)

Defined in Data.Semilattice.Meet

Methods

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

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

HashSet intersection 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)

Absorption:

lowerBound /\ a == (lowerBound :: HashSet Char)

Defined in Data.Semilattice.Meet

Methods

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

Ord a => Meet (Order a)

Total Orderings give rise to a meet 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:

upperBound /\ Order a == Order (a :: Int)

Absorption:

lowerBound /\ Order a == (lowerBound :: Order Int)

Distributivity:

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

Defined in Data.Semilattice.Order

Methods

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

Meet a => Meet (Meeting a)

Defined in Data.Semilattice.Meet

Methods

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

Meet a => Meet (GreaterThan a)

Defined in Data.Semilattice.Meet

Methods

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

Meet b => Meet (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) -> upperBound /\ a ~= (a :: Int -> Bool)

Absorption:

\ (Fn a) -> lowerBound /\ a ~= (lowerBound :: Int -> Bool)

Defined in Data.Semilattice.Meet

Methods

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

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

HashMap union with Meetable 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))

Absorption:

lowerBound /\ a == (lowerBound :: HashMap Char (Set Char))

Defined in Data.Semilattice.Meet

Methods

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

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

Map union with Meetable 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))

Absorption:

lowerBound /\ a == (lowerBound :: Map Char (Set Char))

Defined in Data.Semilattice.Meet

Methods

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

newtype Order a

A Join- and Meet-semilattice for any total Ordering.

Constructors

Order

Fields

• getOrder :: a

Instances

Functor Order

Defined in Data.Semilattice.Order

Methods

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

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

Foldable Order

Defined in Data.Semilattice.Order

Methods

fold :: Monoid m => Order m -> m

foldMap :: Monoid m => (a -> m) -> Order a -> m

foldr :: (a -> b -> b) -> b -> Order a -> b

foldr' :: (a -> b -> b) -> b -> Order a -> b

foldl :: (b -> a -> b) -> b -> Order a -> b

foldl' :: (b -> a -> b) -> b -> Order a -> b

foldr1 :: (a -> a -> a) -> Order a -> a

foldl1 :: (a -> a -> a) -> Order a -> a

toList :: Order a -> [a]

null :: Order a -> Bool

length :: Order a -> Int

elem :: Eq a => a -> Order a -> Bool

maximum :: Ord a => Order a -> a

minimum :: Ord a => Order a -> a

sum :: Num a => Order a -> a

product :: Num a => Order a -> a

Traversable Order

Defined in Data.Semilattice.Order

Methods

traverse :: Applicative f => (a -> f b) -> Order a -> f (Order b)

sequenceA :: Applicative f => Order (f a) -> f (Order a)

mapM :: Monad m => (a -> m b) -> Order a -> m (Order b)

sequence :: Monad m => Order (m a) -> m (Order a)

Bounded a => Bounded (Order a)

Defined in Data.Semilattice.Order

Methods

minBound :: Order a

maxBound :: Order a

Enum a => Enum (Order a)

Defined in Data.Semilattice.Order

Methods

succ :: Order a -> Order a

pred :: Order a -> Order a

toEnum :: Int -> Order a

fromEnum :: Order a -> Int

enumFrom :: Order a -> [Order a]

enumFromThen :: Order a -> Order a -> [Order a]

enumFromTo :: Order a -> Order a -> [Order a]

enumFromThenTo :: Order a -> Order a -> Order a -> [Order a]

Eq a => Eq (Order a)

Defined in Data.Semilattice.Order

Methods

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

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

Num a => Num (Order a)

Defined in Data.Semilattice.Order

Methods

(+) :: Order a -> Order a -> Order a

(-) :: Order a -> Order a -> Order a

(*) :: Order a -> Order a -> Order a

negate :: Order a -> Order a

abs :: Order a -> Order a

signum :: Order a -> Order a

fromInteger :: Integer -> Order a

Ord a => Ord (Order a)

Defined in Data.Semilattice.Order

Methods

compare :: Order a -> Order a -> Ordering

(<) :: Order a -> Order a -> Bool

(<=) :: Order a -> Order a -> Bool

(>) :: Order a -> Order a -> Bool

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

max :: Order a -> Order a -> Order a

min :: Order a -> Order a -> Order a

Read a => Read (Order a)

Defined in Data.Semilattice.Order

Methods

readsPrec :: Int -> ReadS (Order a)

readList :: ReadS [Order a]

readPrec :: ReadPrec (Order a)

readListPrec :: ReadPrec [Order a]

Show a => Show (Order a)

Defined in Data.Semilattice.Order

Methods

showsPrec :: Int -> Order a -> ShowS

show :: Order a -> String

showList :: [Order a] -> ShowS

Ord a => Meet (Order a)

Total Orderings give rise to a meet 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:

upperBound /\ Order a == Order (a :: Int)

Absorption:

lowerBound /\ Order a == (lowerBound :: Order Int)

Distributivity:

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

Defined in Data.Semilattice.Order

Methods

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

Upper a => Upper (Order a)

Defined in Data.Semilattice.Order

Methods

upperBound :: Order 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)

Defined in Data.Semilattice.Order

Methods

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

Lower a => Lower (Order a)

Defined in Data.Semilattice.Order

Methods

lowerBound :: Order a