An abstract notion of application

Abstract

Many concrete notions of function application, suitable for interpreting typed lambda calculi with recursive types, have been introduced in the literature. These arise in different fields such as set theory, multiset theory, type theory and functor theory and are apparently unrelated. In this paper we introduce the general concept of applicative exponential structure and show that it subsumes all these notions. Our approach is based on a generalization of the notion of intersection type. We construe all these structures in a finitary way, so as to be able to utilize uniformly a general form of type assignment system for defining the interpretation function. Applicative exponential structures are just combinatory algebras, in general. Our approach suggests a wide variety of entirely new concrete notions of function application: e.g. in connection with boolean sets. Applicative exponential structures can be used for modeling various forms of non-deterministic operators.