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On Boolean ranges of Banaschewski functions

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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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References

  1. Banaschewski, B.: Totalgeordnete moduln. Arch. Math. 7, 430–440 (1957). (German)

    Article  MathSciNet  MATH  Google Scholar 

  2. Burris, S., Sankappanavar, H.B.: A Course in Universal Algebra. Graduate Texts in Mathematics, vol. 78. Springer, New-York (1981)

    MATH  Google Scholar 

  3. Engelking, R.: General Topology. Heldermann, Berlin (1989)

    MATH  Google Scholar 

  4. Goodearl, K.R.: Von Neumann Regular Rings, 2nd edn. Krieger Pub. Co., Malabar, FL (1991)

    MATH  Google Scholar 

  5. Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  6. Grätzer, G., Schmidt, E.T.: A lattice construction and congruence-preserving extensions. Acta Math. Hung. 66, 275–288 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Grätzer, G., Schmidt, E.T.: On the independence theorem of related structures for modular (arguesian) lattices. Stud.Sci. Math. Hungar. 40, 1–12 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Grätzer, G., Wehrung, F.: Flat semilattices. Colloq. Math. 79, 185–191 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grätzer, G., Wehrung, F.: The \(M_3 [D]\) construction and \(n\)-modularity. Algebra Univers. 41, 87–114 (1999)

    Article  MATH  Google Scholar 

  10. Grätzer, G., Wehrung, F.: A new lattice construction: the box product. J. Algebra 221, 5893–5919 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grätzer, G., Wehrung, F.: Proper congruence-preserving extension of lattices. Acta Math. Hungar. 85, 169–179 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grätzer, G., Wehrung, F.: Tensor product and transferability of semilattices. Can. J. Math. 51, 792–815 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Grätzer, G., Wehrung, F.: Tensor product and semilattices with zero, revisited. J. Pure Appl. Algebra 147, 273–301 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grätzer, G., Wehrung, F.: A survey of tensor product and related structures in two lectures. Algebra Univers. 45, 117–143 (2001)

    MATH  Google Scholar 

  15. Jónsson, B.: Representations of complemented modular lattices. Trans. Am. Math. Soc. 60, 64–94 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  16. Murphy, G.J.: \(C^\ast \)-Algebras and Operator Theory. Academic Press. Inc, London (1990)

    MATH  Google Scholar 

  17. Ore, Ø.: Galois connexions. Trans. Am. Math. Soc. 55, 493–513 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schmidt, E.T.: Zur Charakterisierung der Kongruenzverbände der Verbände. Mat. Časopis Sloven. Akad. Vied. 18, 3–20 (1968). (German)

    MATH  Google Scholar 

  19. Schmidt, E.T.: Every finite distributive lattice is the congruence lattice of a modular lattice. Algebra Univers. 4, 49–57 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wehrung, F.: Coordinatization of lattices by regular rings without unit and Banaschewski functions. Algebra Univers. 64, 49–67 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wehrung, F.: A non-coordinatizable sectionally complemented modular lattice with a large Jónsson four-frame. Adv. Appl. Math. 47, 173–193 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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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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Authors and Affiliations

  1. Department of Algebra, Faculty of mathematics and Physics, Charles University in Prague, Sokolovská 83, 186 75, Prague, Czech Republic

    Samuel Mokriš & Pavel Růžička

Authors
  1. Samuel Mokriš
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  2. Pavel Růžička
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Correspondence to Pavel Růžička.

Additional information

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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