Skip to main content
SpringerLink
Log in

A theorem and a question about epicomplete archimedean lattice-ordered groups

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

An archimedean ℓ-group is called epicomplete (or universally σ-complete, or sequentially inextensible) if it is divisible, σ-complete and laterally σ-complete. Various characterizations of such G are known in case the G have weak order units. The “theorem” of the title is a characterization of such G which have no weak order unit; it involves the requirement that G have a certain kind of representation. The “question” of the title is whether every epicomplete G has such a representation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, M., Feil, T.: Lattice-Ordered Groups. Reidel (1988)

  2. Ball R., Hager A.: Epicomplete archimedean ℓ-groups and vector lattices. Trans. Amer. Math. Soc. 322, 459–478 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ball R., Hager A.: Epicompletion of archimedean ℓ-groups and vector lattices with weak unit. J. Austral. Math. Soc. 48, 25–56 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Ball , R., Hager, A. Johnson, D., Kizanis, A.: The epicomplete monoreflection of an archimedean lattice-ordered group. Alg. Universalis 54, 417–434 (2005)

  5. Bernau S.: Unique representation of archimedean lattice groups and normal lattice rings. Proc. London Math. Soc. 15, 599–631 (1965)

    Article  MathSciNet  Google Scholar 

  6. Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés. Springer (1977)

  7. Brown L., Nakano H.: A representation theorem for Archimedean linear lattices. Proc. Amer. Math. Soc. 17, 835–837 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  8. Buskes G.: Disjoint sequences and completeness properties. Indag. Math 47, 11–19 (1985)

    MATH  MathSciNet  Google Scholar 

  9. Comfort, W., Negrepontis, S.: Continuous Pseudometrics. Dekker (1975)

  10. Conrad P.: The additive group of an f-ring. Can. J. Math. 26, 1157–1168 (1974)

    MATH  MathSciNet  Google Scholar 

  11. Darnel, M.: Theory of Lattice-Ordered Groups. Dekker (1995)

  12. Dashiell F., Hager A., Henriksen M.: Order-Cauchy completions of rings and vector lattices of continuous functions. Canad. J. Math. 32, 657–685 (1980)

    MATH  MathSciNet  Google Scholar 

  13. Engelking, R.: General Topology. Heldermann (1989)

  14. Fremlin D.: Inextensible Riesz spaces. Math. Proc. Cambridge Phil. Soc. 77, 71–89 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand (1960)

  16. Hager, A., Martinez, J.: α-projectible and laterally α-complete archimedean lattice-ordered groups. Ethiopian J. Sci. 19 (Supplement), 73–84 (1996)

    Google Scholar 

  17. Hager, A., Robertson, L.: Representing and ringifying a Riesz space. Symp. Math. 21, pp.411–431. Academic Press (1977)

    Google Scholar 

  18. Henriksen M., Johnson D.: On the structure of a class of archimedean lattice-ordered algebras. Fund. Math. 50, 73–94 (1961)

    MATH  MathSciNet  Google Scholar 

  19. Holland, C. (curator): The Black Swamp Problem Book. Problem 74 (2003)

  20. Johnson D.: On a representation theory for a class of lattice-ordered rings. Proc. London Math. Soc. 12, 207–226 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  21. Johnson D.: A representation theorem revisited. Alg. Universalis 56, 303–314 (2007)

    Article  MATH  Google Scholar 

  22. Johnson D., Kist J.: Prime ideals in vector lattices. Canad. J. Math. 14, 517–528 (1962)

    MATH  MathSciNet  Google Scholar 

  23. Kizanis A.: Weak units in epicompletions of archimedean lattice-ordered groups. Rocky Mt. J. Math. 30, 1323–1341 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Luxemburg, W., Zaanen, A.: Riesz Spaces, Vol. I. North - Holland (1971)

  25. Maeda, F., Ogasawara, T.: Representation of vector lattices. J. Sci. Hiroshima Univ. A, 17–35 (1942)

  26. Papert D.: A representation theory for lattice-groups. Proc. London Math. Soc. 12(3), 100–120 (1960)

    MathSciNet  Google Scholar 

  27. Sikorski, R.: Boolean Algebras (3rd ed.). Springer-Verlag (1969)

  28. Tzeng, Z.: Extended real-valued functions and the projective resolution of a compact space. Ph. D. Thesis, Wesleyan Univ. (1970)

  29. Veksler A.: A new construction of Dedekind completion of vector lattices and ℓ-groups with division. Siber. Math. J. 10, 890–896 (1969)

    MathSciNet  Google Scholar 

  30. Veksler A.: P-sets in topological spaces. Soviet Math. Socl. Dokl. 11, 953–956 (1970)

    MATH  Google Scholar 

  31. Veksler A., Geiler V.: Order and disjoint completeness of linear partially ordered spaces, Siber. Math. J. 13, 30–35 (1972)

    MATH  MathSciNet  Google Scholar 

  32. Yosida, K.: On the representation of the vector lattice. Proc. Imp. Acad. Tokyo 18, 479–482 (1941-42)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Denver, Denver, CO, 80208, USA

    R. N. Ball

  2. Department of Mathematics, Wesleyan University, Middletown, CT, 06459, USA

    A. W. Hager

  3. 5 W. Oak St., Ramsey, NJ, 07446, USA

    D. G. Johnson

  4. Department of Mathematics, Western New England College, Springfield, MA, 01119, USA

    A. Kizanis

Authors
  1. R. N. Ball
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. W. Hager
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. G. Johnson
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Kizanis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. N. Ball.

Additional information

Presented by J. Martinez.

In memory of Paul F. Conrad

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ball, R.N., Hager, A.W., Johnson, D.G. et al. A theorem and a question about epicomplete archimedean lattice-ordered groups. Algebra Univers. 62, 165–184 (2009). https://doi.org/10.1007/s00012-010-0049-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-010-0049-4

2000 Mathematics Subject Classification

Key words and phrases

Search

Navigation