For a model ℳ of Peano Arithmetic, let Lt(ℳ) be the lattice of its elementary substructures, and let Lt+(ℳ) be the equivalenced lattice (Lt(ℳ), ≅ℳ), where ≅ℳ is the equivalence relation of isomorphism on Lt(ℳ). It is known that Lt+(ℳ) is always a reasonable equivalenced lattice.
Theorem Let L be a finite distributive lattice and let (L,E) be reasonable. If ℳ0 is a nonstandard prime model of PA, then ℳ0 has a cofinal extension ℳ such that Lt+(ℳ) ≅ (L,E).
A general method for proving such theorems is developed which, hopefully, will be able to be applied to some nondistributive lattices.
"Nondiversity in substructures." J. Symbolic Logic 73 (1) 193 - 211, March 2008. https://doi.org/10.2178/jsl/1208358749