Lambda theories are equational extensions of the untyped lambda calculus that are closed under derivation. The set of lambda theories is naturally equipped with a structure of complete lattice, where the meet of a family of lambda theories is their intersection, and the join is the least lambda theory containing their union. In this paper we study the structure of the lattice of lambda theories by universal algebraic methods. We show that nontrivial quasi-identities in the language of lattices hold in the lattice of lambda theories, while every nontrivial lattice identity fails in the lattice of lambda theories if the language of lambda calculus is enriched by a suitable finite number of constants. We also show that there exists a sublattice of the lattice of lambda theories which satisfies: (i) a restricted form of distributivity, called meet semidistributivity; and (ii) a nontrivial identity in the language of lattices enriched by the relative product of binary relations.
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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