Easiness in graph models

We generalize Baeten and Boerboom’s method of forcing, and apply this to show that there is a fixed sequence (uk)k∈ω of closed (untyped) λ-terms satisfying the following properties: a) For any countable sequence (gk)k∈ω of continuous functions (of arbitrary arity) on the power set of an arbitrary countable set, there is a graph model such that (λx.xx)(λx.xx)uk represents gk in the model. b) For any countable sequence (tk)k∈ω of closed λ -terms there is a graph model that satisfies (λx.xx)(λx.xx)uk = tk for all k. We apply these two results to show the existence of 1. a finitely axiomatized λ-theory L such that the interval lattice constituted by the λ-theories extending L is distributive; 2. a continuum of pairwise inconsistent graph theories (= λ-theories that can be realized as theories of graph models); 3. a congruence distributive variety of combinatory algebras (lambda abstraction algebras, respectively).