# Heyting Algebras
[](http://github.com/coot)
[](https://travis-ci.org/coot/heyting-algebras)
This package contains type classes and instances for many Heyting algebras
which are in the Haskell eco-system. It is build on top of
[lattices](https://hackage.haskell.org/package/lattices) and
[free-algebras](https://hackage.haskell.org/package/free-algebras) (to provide
combinators for free Heyting algebras). The package also defines a type class
for Boolean algebras and comes with a handful of instances.
A very good introduction to Heyting algebras can be found at
[ncatlab](https://ncatlab.org/nlab/show/Heyting%2Balgebra). Heyting algebras
are the crux of [intuitionistic
logic](https://en.wikipedia.org/wiki/Intuitionistic_logic), which drops the
axiom of exluded middle. From categorical point of view, Heyting algebras are
posets (categories with at most one arrow between any objects), which are also
Cartesian closed (and finitely (co-)complete). Note that this makes any
Heyting algebra a simply typed lambda calculus; hence one more incentive to
learn how to use them.
The most important operation is implication `(==>) :: HeytingAlgebra a => a ->
a -> a`; since every Boolean algebra is a Heyting algebra via `a ==>
b = not a \/ b` (using the lattice notation for `or`). It is very handy in
expression conditional logic.
Some basic examples of Heyting algebras:
* `Bool` is a Boolean algebra
* `(Ord a, Bounded a) => a`; the implication is defined as: if `a ≤ b` then `a
⇒ b = maxBound`; otherwise `a ⇒ b = b`; e.g. integers with both `±∞` (it can
be represented by `Levitated Int`. This type is not a Boolean algebra.
* The power set is a Boolean algebra, in Haskell it can be represented by `Set
a` (one might need to require `a` to be finite though, otherwise `not (not
empty)` might be `undefined` rather than `empty`).
* More generally every type `(Ord k, Finite k, HeytingAlgebra v) => Map k a` is
a Heyting algebra (though in general not a Boolean one).