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 use intersection types as a tool for obtaining λ-models. Relying on the notion of easy intersection type theory we successfully build a λ-model in which the interpretation of an arbitrary simple easy term is any filter which can be described by a continuous predicate. This allows us to prove two results. The first gives a proof of consistency of the λ-theory where the λ-term (λx.xx)(λx.xx) is forced to behave as the join operator. This result has interesting consequences on the algebraic structure of the lattice of λ-theories. The second result is that for any simple easy term there is a λ-model where the interpretation of the term is the minimal fixed point operator.

We generalize Baeten and Boerboom’s method of forcing, and apply this to show that there is a fixed sequence (uk)k∈ω of closed (untyped) λ-terms satisfying the following properties: a) For any countable sequence (gk)k∈ω of continuous functions (of arbitrary arity) on the power set of an arbitrary countable set, there is a graph model such that (λx.xx)(λx.xx)uk represents gk in the model. b) For any countable sequence (tk)k∈ω of closed λ-terms there is a graph model that satisfies (λx.xx)(λx.xx)uk = tk for all k. We apply these two results to show the existence of 1. a finitely axiomatized λ-theory L such that the interval lattice constituted by the λ-theories extending L is distributive; 2. a continuum of pairwise inconsistent graph theories ( = λ-theories that can be realized as theories of graph models); 3. a congruence distributive variety of combinatory algebras (lambda

We illustrate the use of intersection types as a semantic tool for showing properties of the lattice of l-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 recursive predicate. This allows us to prove the consistency of a well-know l-theory: this consistency has interesting consequences on the algebraic structure of the lattice of l-theories.

We show how to use intersection types for building models of a #-calculus enriched with recursive terms, whose intended meaning is of minimal fixed points. As a by-product we prove an interesting consistency result.

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The invariance of the meaning of a #-term by reduction/expansion w.r.t. the considered computational rules is one of the minimal requirements one expects to hold for a #-model. Being the intersection type systems a general framework for the study of semantic domains for the Lambda-calculus, the present paper provides a characterisation of "meaning invariance" in terms of characterisation results for intersection type systems enabling typing invariance of terms w.r.t. various notions of reduction/expansion, like #, # and a number of relevant restrictions of theirs. 1.

Abstract. A closed λ-term M is easy if, for any other closed term N, the lambda theory generated by M = N is consistent, while it is simple easy if, given an arbitrary intersection type τ, one can find a suitable pre-order on types which allows to derive τ for M. Simple easiness implies easiness. The question whether easiness implies simple easiness constitutes Problem 19 in the TLCA list of open problems. In this paper we negatively answer the question providing a nonempty co-r.e. (complement of a recursively enumerable) set of easy, but non simple easy, λ-terms. Key words: Lambda calculus, easy terms, simple easy terms, filter models 1