Abstract Many familiar models of the untyped lambda calculus are constructed by order theoretic methods. This paper pro-vides some basic new facts about ordered models of the lambda calculus. We show that in any partially ordered model that is complete for the theory of fi- or fij-conversion, the partial order is trivial on term denotations. Equivalently, theopen and closed term algebras of the untyped lambda calculus cannot be non-trivially partially ordered. Our second result is a syntactical characterization, in terms of so-called generalized Mal'cev operators, of those lambda theorieswhich cannot be induced by any non-trivially partially ordered model. We also consider a notion of finite models for the untyped lambda calculus, or more precisely, finite models of reduction. We demonstrate how such models can beused as practical tools for giving finitary proofs of term inequalities. 1 Introduction Perhaps the most important contribution in the area of mathematical programming semantics was the discovery, byD. Scott in the late 1960's, that models for the untyped lambda calculus could be obtained by a combination of ordertheoretic and topological methods. A long tradition of research in domain theory ensued, and Scott's methods havebeen successfully applied to many aspects of programming semantics.

Abstract. If V is a variety of algebras, let L(V) denote the prevariety of all lattices embeddable in congruence lattices of algebras in V. We give some criteria for the first-order axiomatizability or nonaxiomatizability of L(V). One corollary to our results is a nonconstructive proof that every congruence n-permutable variety satisfies a nontrivial congruence identity. 1.