Abstract
We construct a countable lattice $${\varvec{\mathcal {S}}}$$ isomorphic to a bounded sublattice of the subspace lattice of a vector space with two non-iso-morphic 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 $${\varvec{\mathcal {S}}}$$ . We prove that although it does not contain a 3-frame, the lattice $${\varvec{\mathcal {S}}}$$ is coordinatizable. We show that the two maximal Boolean sublattices correspond to maximal Abelian regular subalgebras of the coordinatizating ring.
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