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Algebra universalis

, 79:15 | Cite as

On Boolean ranges of Banaschewski functions

  • Samuel Mokriš
  • Pavel Růžička
Article
  • 33 Downloads

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.

Keywords

Lattice Complemented Modular Boolean Schmidt’s construction Banaschewski function Closure operator Adjunction 

Mathematics Subject Classification

06A15 06C20 06D75 

Notes

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Algebra, Faculty of mathematics and PhysicsCharles University in PraguePragueCzech Republic

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