Intersection Types and Lambda Theories
  We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of lambda theories. Relying on the notion of easy intersection type theory we successfully build a filter model in which the interpretation of an arbitrary simple easy term is any filter which can be described in an uniform way by a predicate. This allows us to prove the consistency of a well-know lambda theory: this consistency has interesting consequences on the algebraic structure of the lattice of lambda theories. 
M.Dezani-Ciancaglini and S.Lusin