The Lattice Problem for Models of $\mathsf{PA}$

TL;DR

This paper surveys the longstanding problem of characterizing which lattices can be realized as elementary submodel lattices within models of Peano Arithmetic, highlighting key results, techniques, and lesser-known findings.

Contribution

It provides a comprehensive survey of the lattice problem for models of PA, including detailed analysis of special cases and the development of construction techniques.

Findings

Identification of key classes of representable lattices

Development of Schmerl's construction technique

Discussion of results for countable recursively saturated models

Abstract

The lattice problem for models of Peano Arithmetic ($\mathsf{PA}$) is to determine which lattices can be represented as lattices of elementary submodels of a model of $\mathsf{PA}$, or, in greater generality, for a given model $\mathcal{M}$, which lattices can be represented as interstructure lattices of elementary submodels $\mathcal{K}$ of an elementary extension $\mathcal{N}$ such that $\mathcal{M} \preccurlyeq \mathcal{K} \preccurlyeq \mathcal{N}$. The problem has been studied for the last 60 years and the results and their proofs show an interesting interplay between the model theory of PA, Ramsey style combinatorics, lattice representation theory, and elementary number theory. We present a survey of the most important results together with a detailed analysis of some special cases to explain and motivate a technique developed by James Schmerl for constructing elementary extensions with prescribed interstructure lattices. The last section is devoted to a discussion of lesser-known results about lattices of elementary submodels of countable recursively saturated models of PA.