The Minimal Graph Model of Lambda Calculus

A longstanding open problem in lambda-calculus, raised by G.Plotkin, is whether there exists a continuous model of the untyped lambda-calculus whose theory is exactly the beta-theory or the beta-eta-theory. A related question, raised recently by C.Berline, is whether, given a class of lambda-models, there is a minimal theory represented by it. In this paper, we give a positive answer to this latter question for the class of graph models à la Plotkin-Scott-Engeler. In particular, we build a graph model in which the equations satisfied are exactly those satisfied in any graph model.