What is a Model of the Lamda Calculus? Expanded Version

Abstract

An elementary, purely algebraic definition of model for the untypes lambda calculus is given. This definition is shown to be equivalent to the natural semantic definition based on environments. These definitions of model are consistent with, and yield a completeness theorem for, the standard axioms for lambda convertibility. A simple construction of models for lambda calculus is reviewed. The algebraic formulation clarifies the relation between combinators and lambda terms.