A representation theorem for lambda abstraction algebras

Abstract

The concept of a lambda abstraction algebra (LAA) is designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. Like cylindric and polyadic algebras LAA's can be defined by true identities and thus form a variety in the sense of universal algebra. They provide a distinctly algebraic alternative to the highly combinatorial lambda calculus. A characteristic feature of LAA's is the algebraic reformulation of (β)-conversion as the definition of abstract substitution. The equational axioms of LAA's reflect (α)-conversion and Curry's recursive axiomatization of substitution in the lambda calculus. Functional LAA's arise from environment models or lambda models, the natural models of the lambda calculus. The main result of the paper is a stronger version of the functional representation theorem for locally finite LAA's, the algebraic analogue of the completeness theorem of lambda calculus.