# A maximal Boolean sublattice that is not the range of a Banaschewski function

## TL;DR

The paper constructs a specific countable bounded sublattice of a vector space's subspace lattice, demonstrating that not all maximal Boolean sublattices are ranges of Banaschewski functions, thus solving an open problem.

## Contribution

It provides a counterexample of a maximal Boolean sublattice not arising from a Banaschewski function, addressing a question posed by Wehrung.

## Key findings

- Constructed a countable bounded sublattice with two non-isomorphic maximal Boolean sublattices.
- One sublattice is the range of a Banaschewski function, the other is not.
- Resolved an open problem in lattice theory regarding Banaschewski functions.

## Abstract

We construct a countable bounded sublattice of the lattice of all subspaces of a vector space with two non-isomorphic maximal Boolean sublattice. We represent one of them as the range of a Banschewski function and we prove that this is not the case of the other. Hereby we solve a problem of F. Wehrung.