Substructure lattices of models of arithmetic

We completely characterize those distributive lattices which can be obtained as elementary substructure lattices of models of Peano arithmetic. Stated concisely: every plausible distributive lattice occurs abundantly. Our proof employs the notion of a strongly definable type in many variables. With slight modifications the method also yields a characterization of those distributive lattices which can be obtained uniformly by Gaifman's methods of definable and end extensional 1-types. As a special case this gives another proof of two conjectures involving finite distributive lattices and models of arithmetic posed by Gaifman and initially proved by Schmerl. We also show that every minimal type (in the sense of Gaifman) satisfies a strong partition property which we will call being “uniformly Ramsey”.