Abstract
We construct a countable lattice S isomorphic to a bounded sublattice of the subspace lattice of a vector space with two non-isomorphic maximal Boolean sublattices. We represent one of them as the range of a Banaschewski function and we prove that this is not the case of the other.
Hereby we solve a problem of F. Wehrung.
We study coordinatizability of the lattice S. We prove that although it does not contain a 3-frame, the lattice S is coordinatizable.
We show that the two maximal Boolean sublattices correspond to maximal Abelian regular subalgebras of the coordinatizating ring.