 
Summary: A Complete Characterization of Complete
IntersectionType Preorders
M. DEZANICIANCAGLINI
Universita di Torino
and
F. HONSELL and F. ALESSI
Universita di Udine
We characterize those type preorders which yield complete intersectiontype assignment systems
for calculi, with respect to the three canonical settheoretical semantics for intersectiontypes:
the inference semantics, the simple semantics and the Fsemantics. These semantics arise by taking
as interpretation of types subsets of applicative structures, as interpretation of the preorder rela
tion, , settheoretic inclusion, as interpretation of the intersection constructor, \, settheoretic
intersection, 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 signicantly 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 models, but to arbitrary applicative structures which admit an interpretation
