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Journal of Logic and Computation 2004 14(3):373-394; doi:10.1093/logcom/14.3.373
© 2004 by Oxford University Press
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Original Article

The Lattice of Lambda Theories

Stefania Lusin and Antonino Salibra

Dipartimento di Informatica, Università Ca'Foscari di Venezia, Via Torino 155, 30172 Venezia, Italy. E-mail: slusin{at}dsi.unive.it, salibra{at}dsi.unive.it

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.

Keywords: Lambda calculus, lattice of lambda theories, lattice identities, commutator, lambda abstraction algebras.

Received 4 September 2002.


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