Type Preorders

Various type structures, called Intersection Type Structures, have been introduced in the literature in order to define models of λ-calculus and simultaneously to reason in a finitary way about λ-terms. All these systems are only employed as meet-semilattices generated by preorders built on prime types. For this reason these structures are linguistically redundant. Starting from this observation we introduce the category of Type Preorders, which arises naturally when we eliminate redundant types from Intersection Type Structures. We give a Stoneduality type result for Type Preorders, showing that they are equivalent to the category of prime-algebraic complete lattices and Scott continuous functions. Thus we clarify the domain-theoretic description of Intersection Type Structures, which often appears opaque. As an application we give the domain-theoretic reading of the Intersection Union Type Structure.