  We characterize those intersection-type theories which yield complete
intersection-type assignment systems for lambda-calculi, with respect to the
three canonical set-theoretical semantics for intersection-types: the inference
semantics, the simple semantics and the F-semantics. These semantics arise by
taking as interpretation of types subsets of applicative structures, as
interpretation of the intersection constructor set-theoretic inclusion, and by
taking the interpretation of the arrow constructor a' la Scott, with respect to
either any possible functionality set, or the largest one, or the least one.
  These results strengthen and generalize significantly all earlier results in
the literature, to our knowledge, in at least three respects. First of all the
inference semantics had not been considered before. Secondly, the
characterizations are all given just in terms of simple closure conditions on
the preorder relation on the types, rather than on the typing judgments
themselves. The task of checking the condition is made therefore considerably
more tractable. Lastly, we do not restrict attention just to lambda-models, but
to arbitrary applicative structures which admit an interpretation function.
Thus we allow also for the treatment of models of restricted lambda-calculi.
Nevertheless the characterizations we give can be tailored just to the case of
lambda-models.
