Data.Lattice

class JoinSemilattice a  where

A JoinSemilattice must satisfy the following laws:

• Associativity of join: forall a b c, join a (join b c) == join (join a b) c
• Commutativity of join: forall a b, join a b == join b a
• Idempotency of join: forall a, join a a == a

Join semilattices capture the notion of sets with a "least upper bound".

Members

• join :: a -> a -> a

Instances

• JoinSemilattice Boolean

class MeetSemilattice a  where

A MeetSemilattice must satisfy the following laws:

• Associativity of meet: forall a b c, meet a (meet b c) == meet (meet a b) c
• Commutativity of meet: forall a b, meet a b == meet b a
• Idempotency of meet: forall a, meet a a == a

Meet semilattices capture the notion of sets with a "greatest lower bound".

Members

• meet :: a -> a -> a

Instances

• MeetSemilattice Boolean

class (JoinSemilattice a) <= BoundedJoinSemilattice a  where

A BoundedJoinSemilattice must satisfy the following laws in addition to JoinSemilattice laws:

• Bottom (Unitary Element): forall a, join a bottom == a

  Join semilattices capture the notion of sets with a "least upper bound" equipped with a "bottom" element.

Members

• bottom :: a

Instances

• BoundedJoinSemilattice Boolean

class (MeetSemilattice a) <= BoundedMeetSemilattice a  where

A BoundedMeetSemilattice must satisfy the following laws in addition to MeetSemilattice laws:

• Top (Unitary Element): forall a, meet a top == a

Meet semilattices capture the notion of sets with a "greatest lower bound" equipped with a "top" element.

Members

• top :: a

Instances

• BoundedMeetSemilattice Boolean

class (JoinSemilattice a, MeetSemilattice a) <= Lattice a 

A Lattice must satisfy the following in addition to JoinSemilattice and MeetSemilattice laws:

• Absorbtion laws for meet and join: forall a b, meet a (join a b) == a forall a b, join a (meet a b) == a

Instances

• Lattice Boolean

class (BoundedJoinSemilattice a, BoundedMeetSemilattice a) <= BoundedLattice a 

A BoundedLattice must satisfy the following in addition to BoundedMeetSemilattice and BoundedJoinSemilattice laws:

• Absorbtion laws for meet and join: forall a b, meet a (join a b) == a forall a b, join a (meet a b) == a

Instances

• BoundedLattice Boolean