Invariance of interpretation by #-conversion is one of the minimal requirements for any standard model for the #-calculus. With the intersection type systems being a general framework for the study of semantic domains for the #-calculus, the present paper provides a (syntactic) characterisation of the above mentioned requirement in terms of characterisation results for intersection type assignment systems.

We show how to characterise compositionally a number of evaluation properties of λ-terms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalisation, normalisation, head normalisation, and weak head normalisation. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterisation results are new, to our knowledge, or else they streamline, strengthen, or generalise earlier results in the literature. The completeness parts of the characterisations are proved uniformly for all the properties, using a set-theoretical semantics of intersection types over suitable kinds of stable sets. This technique generalises Krivine's and Mitchell's methods for strong normalisation to other evaluation properties.

We construct two inverse limit λ-models which completely characterise sets of terms with similar computational behaviours: the sets of normalising, head normalising, weak head normalising λ-terms, those corresponding to the persistent versions of these notions, and the sets of closable, closable normalising, and closable head normalising λ-terms. More precisely, for each of these sets of terms there is a corresponding element in at least one of the two models such that a term belongs to the set if and only if its interpretation (in a suitable environment) is greater than or equal to that element. We use the finitary logical description of the models, obtained by defining suitable intersection type assignment systems, to prove this.