On the strong Freese-Nation property

TL;DR

This paper demonstrates the existence of a boolean algebra with the Freese-Nation property that does not have the stronger version, answering a previously open question and providing new characterizations of these properties.

Contribution

It constructs a boolean algebra with FN but not SFN and introduces new characterizations of these properties using sequences of elementary submodels.

Findings

Existence of boolean algebra with FN but not SFN

New characterizations of FN and SFN

Answer to an open question by Heindorf and Shapiro

Abstract

We show that there is a boolean algebra that has the Freese-Nation property (FN) but not the strong Freese-Nation property (SFN), thus answering a question of Heindorf and Shapiro. Along the way, we produce some new characterizations of the FN and SFN in terms of sequences of elementary submodels.