On the topological mass lattice groups

A Correction to this article is available

This article has been updated

Abstract

Let G to be a torsion free abelian group. In this paper we introduce the following concepts:

  1. (1)

    Algebraic line, algebraic line segment and thin convex subsets of G.

  2. (2)

    Absorbing topological group, that is a generalization of topological vector space.

  3. (3)

    A special subset of a mass lattice group (G, \(\le \)) called a link in G which we can construct a locally solid topology on G by it.

Some interesting results about unital lattice groups and Riesz spaces with the chief link topology on them have been presented.

This is a preview of subscription content, access via your institution.

Change history

  • 08 May 2020

    (i) In the first assertion of the Lemma 1 in [2], it is necessary that .

References

  1. 1.

    Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics, 2nd edn. American Mathematics Society, Providence (2003)

    Google Scholar 

  2. 2.

    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematics Society, Providence (1967)

    Google Scholar 

  3. 3.

    Boccuto, A., Dimitriou, X.: Convergence Theorems for Lattice Group-Valued Measures. Bentham Science Publishers Ltd, Sharjah (2015)

    Google Scholar 

  4. 4.

    Cernak, S., Lihova, J.: Convergence with a regulator in lattice ordered groups. Tatra Mt. Math. Publ. 39, 35–45 (2005)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Cernak, S.: Convergence with a fixed regulator in Archimedean lattice ordered groups. Math. Slovaca 56(2), 167–180 (2006)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Foulis, D.J.: Removing the torsion from a unital group. Rep. Math. Phys. 52(2), 187–203 (2003)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Foulis, D.J.: Compressions on partially ordered abelian groups. Proc. Am. Math. Soc. 132(12), 3581–3587 (2004)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Foulis, D.J., Pulmannova, S.: Monotone \(\sigma \)-complete RC-groups. J. Lond. Math. Soc. (2) 73, 304–324 (2006)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. AMS Mathematical Surveys and Monographs, No. 20. American Mathematical Society, Providence (1986)

    Google Scholar 

  10. 10.

    Gusic, I.: A topology on lattice-ordered groups. Proc. Am. Math. Soc. 126(9), 2593–2597 (1998)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Hungerford, T.W.: Algebra. Springer, New York (1974)

    Google Scholar 

  12. 12.

    Knapp, A.W.: Basic Real Analysis. Birkhauser, Boston (2005)

    Google Scholar 

  13. 13.

    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)

    Google Scholar 

  14. 14.

    Pulmannova, S.: Effect algebras with the Riesz decomposition property and AF C*algebras. Found. Phys. 29, 1389–1401 (1999)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill, New York (1991)

    Google Scholar 

  16. 16.

    Vulikh, B.Z.: Introduction to the Theory of Partially Ordered Spaces. Wolters-Noordhoff Scientific Publications LTD, Groningen (1967) (translated from Russian by L.F. Boron)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to M. Pourgholamhossein.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pourgholamhossein, M., Ranjbar, M.A. On the topological mass lattice groups. Positivity 23, 811–827 (2019). https://doi.org/10.1007/s11117-018-0639-5

Download citation

Keywords

  • Lattice group
  • Mass group
  • Locally solid topology
  • Retraction
  • Link
  • Absorbing topological group

Mathematics Subject Classification

  • Primary: 22A10
  • 06F20
  • 46A40
  • 57N17
  • Secondary: 54H12
  • 22A26
  • 54F05