Data.Semilattice.Meet

Join semilattices, related to Lower and Upper.

Documentation

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

Meet a => Meet (GreaterThan a)

Defined in Data.Semilattice.Meet

Methods

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

Meet a => Meet (Meeting a)

Defined in Data.Semilattice.Meet

Methods

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

Join a => Meet (Tumble a)

Defined in Data.Semilattice.Tumble

Methods

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

(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

(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

newtype Meeting a

A Semigroup for any Meet semilattice.

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

Constructors

Meeting

Fields

• getMeeting :: a

Instances

Functor Meeting

Defined in Data.Semilattice.Meet

Methods

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

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

Foldable Meeting

Defined in Data.Semilattice.Meet

Methods

fold :: Monoid m => Meeting m -> m

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

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

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

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

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

foldr1 :: (a -> a -> a) -> Meeting a -> a

foldl1 :: (a -> a -> a) -> Meeting a -> a

toList :: Meeting a -> [a]

null :: Meeting a -> Bool

length :: Meeting a -> Int

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

maximum :: Ord a => Meeting a -> a

minimum :: Ord a => Meeting a -> a

sum :: Num a => Meeting a -> a

product :: Num a => Meeting a -> a

Traversable Meeting

Defined in Data.Semilattice.Meet

Methods

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

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

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

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

Bounded a => Bounded (Meeting a)

Defined in Data.Semilattice.Meet

Methods

minBound :: Meeting a

maxBound :: Meeting a

Enum a => Enum (Meeting a)

Defined in Data.Semilattice.Meet

Methods

succ :: Meeting a -> Meeting a

pred :: Meeting a -> Meeting a

toEnum :: Int -> Meeting a

fromEnum :: Meeting a -> Int

enumFrom :: Meeting a -> [Meeting a]

enumFromThen :: Meeting a -> Meeting a -> [Meeting a]

enumFromTo :: Meeting a -> Meeting a -> [Meeting a]

enumFromThenTo :: Meeting a -> Meeting a -> Meeting a -> [Meeting a]

Eq a => Eq (Meeting a)

Defined in Data.Semilattice.Meet

Methods

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

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

Num a => Num (Meeting a)

Defined in Data.Semilattice.Meet

Methods

(+) :: Meeting a -> Meeting a -> Meeting a

(-) :: Meeting a -> Meeting a -> Meeting a

(*) :: Meeting a -> Meeting a -> Meeting a

negate :: Meeting a -> Meeting a

abs :: Meeting a -> Meeting a

signum :: Meeting a -> Meeting a

fromInteger :: Integer -> Meeting a

Ord a => Ord (Meeting a)

Defined in Data.Semilattice.Meet

Methods

compare :: Meeting a -> Meeting a -> Ordering

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

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

(>) :: Meeting a -> Meeting a -> Bool

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

max :: Meeting a -> Meeting a -> Meeting a

min :: Meeting a -> Meeting a -> Meeting a

Read a => Read (Meeting a)

Defined in Data.Semilattice.Meet

Methods

readsPrec :: Int -> ReadS (Meeting a)

readList :: ReadS [Meeting a]

readPrec :: ReadPrec (Meeting a)

readListPrec :: ReadPrec [Meeting a]

Show a => Show (Meeting a)

Defined in Data.Semilattice.Meet

Methods

showsPrec :: Int -> Meeting a -> ShowS

show :: Meeting a -> String

showList :: [Meeting a] -> ShowS

Meet a => Semigroup (Meeting a)

Meeting <> is associative.

Meeting a <> (Meeting b <> Meeting c) == (Meeting a <> Meeting b) <> Meeting (c :: IntSet)

Defined in Data.Semilattice.Meet

Methods

(<>) :: Meeting a -> Meeting a -> Meeting a

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

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

(Upper a, Meet a) => Monoid (Meeting a)

Meeting mempty is the left- and right-identity.

let (l, r) = (mappend mempty (Meeting x), mappend (Meeting x) mempty) in l == Meeting x && r == Meeting (x :: Bool)

Defined in Data.Semilattice.Meet

Methods

mempty :: Meeting a

mappend :: Meeting a -> Meeting a -> Meeting a

mconcat :: [Meeting a] -> Meeting a

Upper a => Upper (Meeting a)

Defined in Data.Semilattice.Meet

Methods

upperBound :: Meeting a

Meet a => Meet (Meeting a)

Defined in Data.Semilattice.Meet

Methods

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

newtype GreaterThan a

Meet semilattices give rise to a partial Ordering.

Constructors

GreaterThan

Fields

• getGreaterThan :: a

Instances

Functor GreaterThan

Defined in Data.Semilattice.Meet

Methods

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

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

Foldable GreaterThan

Defined in Data.Semilattice.Meet

Methods

fold :: Monoid m => GreaterThan m -> m

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

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

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

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

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

foldr1 :: (a -> a -> a) -> GreaterThan a -> a

foldl1 :: (a -> a -> a) -> GreaterThan a -> a

toList :: GreaterThan a -> [a]

null :: GreaterThan a -> Bool

length :: GreaterThan a -> Int

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

maximum :: Ord a => GreaterThan a -> a

minimum :: Ord a => GreaterThan a -> a

sum :: Num a => GreaterThan a -> a

product :: Num a => GreaterThan a -> a

Traversable GreaterThan

Defined in Data.Semilattice.Meet

Methods

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

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

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

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

Enum a => Enum (GreaterThan a)

Defined in Data.Semilattice.Meet

Methods

succ :: GreaterThan a -> GreaterThan a

pred :: GreaterThan a -> GreaterThan a

toEnum :: Int -> GreaterThan a

fromEnum :: GreaterThan a -> Int

enumFrom :: GreaterThan a -> [GreaterThan a]

enumFromThen :: GreaterThan a -> GreaterThan a -> [GreaterThan a]

enumFromTo :: GreaterThan a -> GreaterThan a -> [GreaterThan a]

enumFromThenTo :: GreaterThan a -> GreaterThan a -> GreaterThan a -> [GreaterThan a]

Eq a => Eq (GreaterThan a)

Defined in Data.Semilattice.Meet

Methods

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

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

Num a => Num (GreaterThan a)

Defined in Data.Semilattice.Meet

Methods

(+) :: GreaterThan a -> GreaterThan a -> GreaterThan a

(-) :: GreaterThan a -> GreaterThan a -> GreaterThan a

(*) :: GreaterThan a -> GreaterThan a -> GreaterThan a

negate :: GreaterThan a -> GreaterThan a

abs :: GreaterThan a -> GreaterThan a

signum :: GreaterThan a -> GreaterThan a

fromInteger :: Integer -> GreaterThan a

(Eq a, Meet a) => Ord (GreaterThan a)

NB: This is not in general a total ordering.

Defined in Data.Semilattice.Meet

Methods

compare :: GreaterThan a -> GreaterThan a -> Ordering

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

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

(>) :: GreaterThan a -> GreaterThan a -> Bool

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

max :: GreaterThan a -> GreaterThan a -> GreaterThan a

min :: GreaterThan a -> GreaterThan a -> GreaterThan a

Read a => Read (GreaterThan a)

Defined in Data.Semilattice.Meet

Methods

readsPrec :: Int -> ReadS (GreaterThan a)

readList :: ReadS [GreaterThan a]

readPrec :: ReadPrec (GreaterThan a)

readListPrec :: ReadPrec [GreaterThan a]

Show a => Show (GreaterThan a)

Defined in Data.Semilattice.Meet

Methods

showsPrec :: Int -> GreaterThan a -> ShowS

show :: GreaterThan a -> String

showList :: [GreaterThan a] -> ShowS

Meet a => Meet (GreaterThan a)

Defined in Data.Semilattice.Meet

Methods

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