On the algebraic models of lambda calculus

The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory logic in this regard since it is a rst-order algebraic description of lambda calculus, which allows to keep the lambda notation and hence all the functional intuitions. In this paper we show that the lattice of the subvarieties of lambda abstraction algebras is isomorphic to the lattice of lambda theories of the lambda calculus; for every variety of lambda abstraction algebras there exists exactly one lambda theory whose term algebra generates the variety. For example, the variety generated by the term algebra of the minimal lambda theory is the variety of all lambda abstraction algebras. This result is applied to obtain a generalization of the genericity lemma of nitary lambda calculus to the in nitary lambda calculus. Another result of the paper is an algebraic proof of consistency of the in nitary lambda calculus. Finally, some algebraic constructions by Krivine are generalized to lambda abstraction algebras. c © 2000 Elsevier Science B.V. All rights reserved.