Easiness in graph models
Résumé
We generalize Baeten and Boerbom's method of forcing, and apply it to show that there is a fixed sequence (uk)k in N, of closed untyped λ-terms satisfying the following properties: * For any countable sequence (gk)k in N, of continuous functions (of arbitrary arity) on the power set P(D) of an arbitrary countable set D, there is a graph model (D,p) such that (λx.xx)(λx.xx)uk represents gk in the model. * For any countable sequence (tk)k in N, of closed λ-terms there is a graph model that satisfies (λx.xx)(λx.xx)uk=tk for all k. These two results bring information on the landscape of graph theories (= λ-theories that can be realized as theories of graph models), and more generally on the lattice of λ-theories (ordered by inclusion).