The extensional ordering of the sequential functionals

We investigate the extensional ordering of the sequential functionals of finite types, with a focus on when the sequential functionals of a given type form a directed complete partial ordering, and on when a finite sequential functional will be the nontrivial least upper bound of an infinite chain of sequential functionals. We offer a full characterization for finite functionals of pure types.

► The sequential functionals are introduced via finite sequential procedures.
► New types with incomplete orderings of sequential functionals are discovered.
► Finitary sequential functionals may be nontrivial least upper bounds.
► Finitary sequential functionals may be discontinuous.