Inverse Limit Models as Filter Models

Natural intersection type preorders are the type structures which agree with the plain intuition of intersection type constructor as set-theoretic intersection operation and arrow type constructor as set-theoretic function space constructor. In this paper we study the relation between natural intersection type preorders and natural λ-structures, i.e. ω-algebraic lattices D with Galois connections given by F : D → [D → D] and G : [D → D] → D. We prove on one hand that natural intersection type preorders induces natural λ-structures, on the other hand that natural λ-structures admits presentations through intersection type preorders. Moreover we give a concise presentations of classical D∞ λ-models of untyped λ-calculus through suitable natural intersection type preorders and prove that filter λ-models induced by them are isomorphic to D∞.