Bounded lattices (that is lattices that are both lower bounded and upper bounded) form a large class of lattices that include all distributive lattices, many nondistributive finite lattices such as the pentagon lattice N5, and all lattices in any variety generated by a finite bounded lattice. Extending a theorem of Paris for distributive lattices, we prove that if L is an ℵ0-algebraic bounded lattice, then every countable nonstandard model ℳ of Peano Arithmetic has a cofinal elementary extension 𝒩 such that the interstructure lattice Lt(𝒩/ℳ) is isomorphic to L.
"Infinite substructure lattices of models of Peano Arithmetic." J. Symbolic Logic 75 (4) 1366 - 1382, December 2010. https://doi.org/10.2178/jsl/1286198152