On the topological mass lattice groups

A Correction to this article was published on 08 May 2020

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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.

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  • 08 May 2020

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


  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)

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Correspondence to M. Pourgholamhossein.

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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

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

Mathematics Subject Classification

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