# Difference between revisions of "Lattices of elementary substructures"

For $M\models PA$ let ${\rm{Lt}(M)}=(\{K: K\prec M\},\prec)$ and for $M\prec N$, let ${\rm Lt}(N/M)=(\{K: M\prec K\prec M\},\prec)$.

In general, the lattice problem is: Which lattices can be represented as ${\rm Lt}(N/M)=(\{K: M\prec K\prec M\},\prec)$, for some $M\prec N$?

There is a vast literature on the problem and many special cases remain open. Here are basic references: [1, 2, 3, 4, 5, 6]. Chapter 4 of  is devoted to the lattice problem.

## Finite lattices

Is every finite lattice lattice a substructure lattice of a model of $PA$?

By a result of Schmerl the answer is positive for all ${\bf M}_n$, where $n=q+1$ or $n=q+2$ and $q$ is a power of a prime. ${\bf M}_n$ is the lattice with a top element, bottom element, and $n$ incomparable elements in between. The simplest lattice for which the problem is open is ${\bf M}_{16}$.

Every countable $M\models PA$ has an elementary end extension $N$ such that ${\rm Lt}(N/M)$ is isomorphic to the pentagon lattice ${\bf N}_5$ , but no $M\models PA$ at all has an elementary end extension such that ${\rm Lt}(N/M) \cong {\bf M}_3$ (.

Schmerl has asked: What finite lattices $L$ are such that every $M\models PA$ has an elementary end extension $N$ such that ${\rm}Lt(N/M) \cong L$? What finite lattices $L$ are such that every countable $M\models PA$ has an elementary end extension $N$ such that ${\rm Lt}(N/M) \cong L$?

## First-order theory of ${\rm Lt}(N/M)$.

Suppose $M_1\prec_{cof} N_1$, $M_2\prec_{cof} N_2$, and $(N_1,M_1)\equiv (N_2,M_2)$. Is ${\rm Lt}(N_1/M_1)$ elementarily equivalent to ${\rm Lt}(N_2/M_2)$? The problem is motivated by a result from  showing that the answer is positive for lattices of finitely generated interstructures.