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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Acknowledgements
We thank the anonymous referee for their valuable comments that led to remarkable improvements of the paper. Following their suggestions we simplified Section 3 and extended the paper by Sections 7–9.
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Dedicated to Jára Cimrman on the occasion of his 50th birthday.
Presented by F. Wehrung.
The first author was partially supported by the project SVV-2015-260227 of Charles University in Prague. The second author was partially supported by the Grant Agency of the Czech Republic under the Grant no. GACR 14-15479S.
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Mokriš, S., Růžička, P. On Boolean ranges of Banaschewski functions. Algebra Univers. 79, 15 (2018). https://doi.org/10.1007/s00012-018-0489-9
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DOI: https://doi.org/10.1007/s00012-018-0489-9