A complete Heyting algebra is a Heyting algebra which is also a complete lattice; that is, it is a poset with arbitrary limits and colimits, that is also cartesian closed.
By the adjoint functor theorem, one can demonstrate that every frame is a complete Heyting algebra, and vice versa, so far as the underlying poset goes.
However, morphisms of frames needn’t preserve exponential objects or infinitary meets, as would most naturally be required of morphisms of complete Heyting algebras. Also, when considering large lattices which are only small-complete, then frames and complete Heyting algebras are different objects. This becomes important in predicative constructive mathematics, where frames are not provably small, but only large, locally small, and small-complete, and so one can’t show that every frame is a complete Heyting algebra.
However, it is still the case in predicative constructive mathematics that any frame with a small base satisfies the adjoint functor theorem and is a complete Heyting algebra, since having a small base is the posetal verison of the solution set condition.
The initial small-frame, which is the category of small subsingletons, is a complete Heyting algebra, since it is locally small and small-complete, and its base is given by the boolean domain, which is always small.
The frame of opens of the locale of small real numbers is a complete Heyting algebra, since it locally small and small-complete, and its base is the set of pairs of rational numbers such that , which is always small.
For that frames are not complete Heyting algebras in predicative constructive mathematics, see:
See also at Heyting algebra.
Last revised on May 29, 2026 at 22:10:19. See the history of this page for a list of all contributions to it.