# Blog Archives

# Topic Archive: Lattices of elementary substructures

# Finite Distributive Lattices II

I will continue the proof (in TSOMOPA 4.3) that if D is a finite distributive lattice, there is a model M such that Lt(M) is isomorphic to D.

# The lattice problem

Ordered by inclusion, the set of elementary substructures of a model of PA is a lattice.

It is an ℵ_{1}-compactly generated lattice, meaning that every element of the lattice is a supremum of its compact elements, and for every compact element, there are at most countably many compact elements below it. The lattice problem for models of PA is to determine which lattices can be represented as lattices of elementary substructures of a model of PA. The condition above puts a restriction on the types of lattices that can be represented this way. As far as we know, this can be the only restriction. Much work on the problem has been done in the 1970s by Gaifman, Knight, Mills, Paris, Schmerl, and Wilkie, and has been continued by Schmerl. It involves some specialized knowledge of models of PA, highly nontrivial lattice representation theory, combinatorics, and sometimes number theory. There are many positive results, but we still do not know if there is a finite lattice which cannot be represented as a substructure lattice of a model of PA. My talk will be an introduction to the lattice problem. I’ll introduce basic definitions and I’ll prove a couple of introductory negative results.

# Boolean algebras of elementary substructures II

This is a continuation of the talk from last week. I will show how to use minimal types to construct elementary end extensions with large interstructure lattices.

# Boolean algebras of elementary substructures

In his 1976 paper Haim Gaifman proved that for every set I, every model M of PA has an elementary end extension N such that Lt(N/M) is isomorphic to P(I). I will present a proof.

# Introduction to Lattices and Substructure Lattices II

We will continue an introduction to Substructure Lattices, a theme for this semester’s seminar. It will still be completely elementary.

# Introduction to Lattices and Substructure Lattices

This talk with be completely elementary. We will provide an introduction to Substructure Lattices, a theme for this semester’s seminar. Given a model M of Peano Arithmetic, its Substructure Lattice is the lattice of elementary substructures of M. We will discuss the basics of lattice theory relevant to understanding this topic and present some of the big questions in this area.