  We study the question of extending the BCD intersection type system with
additional type constructors. On the typing side, we focus on adding the usual
rules for product types. On the subtyping side, we consider a generic way of
defining a subtyping relation on families of types which include intersection
types. We find back the BCD subtyping relation by considering the particular
case where the type constructors are intersection, omega and arrow. We obtain
an extension of BCD subtyping to product types as another instance. We show how
the preservation of typing by both reduction and expansion is satisfied in all
the considered cases. Our approach takes benefits from a "subformula property"
of the proposed presentation of the subtyping relation.
