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k-metric spaces

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Abstract

In this paper, we give a new generalization of metric spaces called k-metric spaces. Our k-metrics are valued in lattice ordered groups, which allows us to talk about distance in non-abelian lattice ordered groups. We also discuss a class of (not necessarily abelian) lattice ordered groups in which every k-metric induces a topology. Then we show that every k-metric valued in the real numbers is metrizable. In the last section, we characterize intrinsic metrics on lattice ordered rings that are almost f-rings and prove that being an almost f-ring is necessary and sufficient for this characterization. Then we show that if a lattice ordered ring is representable, then every intrinsic metric is a k-metric.

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References

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

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

    MATH  Google Scholar 

  3. Holland W.C.: Intrinsic metrics for lattice-ordered groups. Algebra Universalis 19, 142–150 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Holland W.C., Kopperman R., Pajoohesh H.: Intrinsic generalized metrics. Algebra Universalis 67, 1–18 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Holland W.C., Mekler A.H., Reilly N.R.: Varieties of lattice-ordered groups in which prime powers commute. Algebra Universalis 23, 196–214 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jasem M.: Intrinsic metric preserving maps on partially ordered groups. Algebra Universalis 36, 135–140 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kalman J.A.: Triangle inequality in l-groups. Proc. Amer. Math. Soc. 11, 395 (1960)

    MathSciNet  MATH  Google Scholar 

  8. Kunen, K., Vaughan, J.E.: Handbook of Set-Theoretic Topology. North Holland (1984)

  9. Kopperman R.: All topologies come from generalized metrics. Amer. Math. Monthly 95, 89–97 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kopperman R., Pajoohesh H., Richmond T.: Topologies arising from metrics valued in abelian ℓ-groups. Algebra Universalis 65, 315–330 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nachbin L.: Topology and Order. Van Nostrand, Princeton (1965)

    MATH  Google Scholar 

  12. Steinberg S.: Lattice-ordered Rings and Modules. Springer, New York (2010)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Medgar Evers College, CUNY, Brooklyn, NY, USA

    H. Pajoohesh

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  1. H. Pajoohesh
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Correspondence to H. Pajoohesh.

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Presented by W. McGovern.

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Pajoohesh, H. k-metric spaces. Algebra Univers. 69, 27–43 (2013). https://doi.org/10.1007/s00012-012-0218-8

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  • DOI: https://doi.org/10.1007/s00012-012-0218-8

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