complete Heyting algebra

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, which 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 neednâ€™t preserve exponentials 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.

Revised on February 24, 2010 03:35:18
by Toby Bartels
(75.88.94.75)