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Positivity

, Volume 23, Issue 4, pp 811–827 | Cite as

On the topological mass lattice groups

  • M. PourgholamhosseinEmail author
  • M. A. Ranjbar
Article
  • 15 Downloads

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.

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 

Notes

References

  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics, 2nd edn. American Mathematics Society, Providence (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematics Society, Providence (1967)zbMATHGoogle Scholar
  3. 3.
    Boccuto, A., Dimitriou, X.: Convergence Theorems for Lattice Group-Valued Measures. Bentham Science Publishers Ltd, Sharjah (2015)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cernak, S., Lihova, J.: Convergence with a regulator in lattice ordered groups. Tatra Mt. Math. Publ. 39, 35–45 (2005)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cernak, S.: Convergence with a fixed regulator in Archimedean lattice ordered groups. Math. Slovaca 56(2), 167–180 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Foulis, D.J.: Removing the torsion from a unital group. Rep. Math. Phys. 52(2), 187–203 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Foulis, D.J.: Compressions on partially ordered abelian groups. Proc. Am. Math. Soc. 132(12), 3581–3587 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Foulis, D.J., Pulmannova, S.: Monotone \(\sigma \)-complete RC-groups. J. Lond. Math. Soc. (2) 73, 304–324 (2006)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hungerford, T.W.: Algebra. Springer, New York (1974)zbMATHGoogle Scholar
  12. 12.
    Knapp, A.W.: Basic Real Analysis. Birkhauser, Boston (2005)zbMATHGoogle Scholar
  13. 13.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces I. North-Holland, Amsterdam (1971)zbMATHGoogle Scholar
  14. 14.
    Pulmannova, S.: Effect algebras with the Riesz decomposition property and AF C*algebras. Found. Phys. 29, 1389–1401 (1999)MathSciNetCrossRefGoogle 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)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesUniversity of QomQomIran

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