Return a CommutativeRig using join and meet.
Return a CommutativeRig using join and meet. Note this must obey the commutative rig laws since meet(a, one) = a, and meet and join are associative, commutative and distributive.
This is the lattice with meet and join swapped
This is the lattice with meet and join swapped
Heyting algebras are bounded lattices that are also equipped with an additional binary operation
imp
(for implication, also written as →).Implication obeys the following laws:
In heyting algebras,
and
is equivalent tomeet
andor
is equivalent tojoin
; both methods are available.Heyting algebra also define
complement
operation (sometimes written as ¬a). The complement ofa
is equivalent to(a → 0)
, and the following laws hold:However, in Heyting algebras this operation is only a pseudo-complement, since Heyting algebras do not necessarily provide the law of the excluded middle. This means that there is no guarantee that (a ∨ ¬a) = 1.
Heyting algebras model intuitionistic logic. For a model of classical logic, see the boolean algebra type class implemented as
Bool
.