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Invariance of interpretation by beta-conversion is one of the minimal requirements for any standard model for the gimel-calculus. With the intersection-type systems being a general framework for the study of semantic domains for the gimel-calculus, the present paper provides a (syntactic) characterisation of the above mentioned requirement in terms of characterisation results for intersection-type assignment systems. Instead of considering conversion as a whole, reduction and expansion will be considered separately. Not only for usual computational rules like beta, eta, but also for a number of relevant restrictions of those. Characterisations will be also provided for (intersection) filter structures that are indeed gimel-models. (c) 2006 Elsevier B.V. All rights reserved

We use intersection types as a tool for obtaining lambda-models. Relying on the notion of easy intersection type theory, we successfully build a lambda-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 lambda-theory where the lambda-term (lambdax.xx)(lambdax.xx) is forced to behave as the join operator. This result has interesting consequences on the algebraic structure of the lattice of lambda-theories. The second result is that for any simple easy term, there is a lambda-model, where the interpretation of the term is the minimal fixed point operator. (C) 2004 Elsevier B.V. All rights reserved

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Abstract. Filter models and (solutions of) recursive domain equations are two different ways of constructing lambda models. Many partial results have been shown about the equivalence between these two constructions (in some specific cases). This paper deepens the connection by showing that the equivalence can be shown in a general framework. We will introduce the class of disciplined intersection type theories and its four subclasses: natural split, lazy split, natural equated and lazy equated. We will prove that each class corresponds to a different recursive domain equation. For this result, we are extracting the essence of the specific proofs for the particular cases of intersection type theories and making one general construction that encompasses all of them. This general approach puts together all these results which may appear scattered and sometimes with incomplete proofs in the literature.