  This paper obtains a completeness result for inequational reasoning with
applicative terms without variables in a setting where the intended semantic
models are the full structures, the full type hierarchies over preorders for
the base types. The syntax allows for the specification that a given symbol be
interpreted as a monotone function, or an antitone function, or both. There is
a natural set of five rules for inequational reasoning. One can add variables
and also add a substitution rule, but we observe that this logic would be
incomplete for full structures. This is why the completeness result in this
paper pertains to terms without variables. Since the completeness is already
known for the class of general (Henkin) structures, we are interested in full
structures. We present a completeness theorem. Our result is not optimal
because we restrict to base preorders which have a weak completeness property:
every pair of elements has an upper bound and a lower bound. To compensate we
add several rules to the logic. We also present extensions and variations of
our completeness result.
