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On Values of Elements in Partially Ordered Groups


We consider properties of convex directed subgroups for the interpolation groups in which each element is a quotient of two almost orthogonal elements. A series of results on values for almost orthogonal elements in those groups is obtained. We investigate characteristics of lexicographic extensions for partially ordered groups in which each element is a quotient of two almost orthogonal elements.

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  1. 1.

    P. Conrad, “The lattice of all convex l-subgroups of lattice-ordered group,” Czech. Math. J., 15, 101–123 (1965).

    MathSciNet  Google Scholar 

  2. 2.

    P. Conrad, “Representation of partially ordered Abelian groups as groups of real valued functions,” Acta Math., 116, 199–221 (1966).

    MATH  MathSciNet  Google Scholar 

  3. 3.

    L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford; Addison-Wesley, Reading (1963).

  4. 4.

    L. Fuchs, “Riesz groups,” Ann. Scu. Norm. Sup. Pisa, 19, No. 3, 1–34 (1965).

    MATH  Google Scholar 

  5. 5.

    A. M. W. Glass, “Polars and their application in directed interpolation groups,” Trans. Am. Math. Soc., 166, 1–25 (1972).

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    V. M. Kopytov, Lattice-Ordered Groups [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  7. 7.

    S. Lang, Algebra, Addison-Wesley, Reading (1965).

  8. 8.

    E. E. Shirshova, “Pseudo-lattice ordered groups,” in: Groups and Modules; Game Theory [in Russian], Mosk. Obl. Ped. Inst., Moscow (1973), pp. 10–18.

  9. 9.

    E. E. Shirshova, “Associated subgroups of pseudo-lattice ordered groups,” in: Algebraic Systems [in Russian], Ivanov. Gos. Univ., Ivanovo (1991), pp. 78–85.

  10. 10.

    E. E. Shirshova, “Lexicographic extensions and pl-groups,” Fundam. Prikl. Mat., 1, No. 4, 1133–1138 (1995).

    MATH  MathSciNet  Google Scholar 

  11. 11.

    E. E. Shirshova, “On homomorphisms of pl-group,” Fundam. Prikl. Mat., 3, No. 1, 303–314 (1997).

    MATH  MathSciNet  Google Scholar 

  12. 12.

    E. E. Shirshova, “On groups with the almost orthogonality condition,” Commun. Algebra, 28, No. 10, 4803–4818 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    E. E. Shirshova, “On a generalization of the notion of orthogonality and on the Riesz groups,” Math. Notes, 69, No. 1, 107–115 (2001).

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to E. E. Shirshova.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 3, pp. 199–212, 2013.

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Shirshova, E.E. On Values of Elements in Partially Ordered Groups. J Math Sci 206, 597–607 (2015).

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  • Normal Subgroup
  • Maximal Subgroup
  • Positive Element
  • Directed Group
  • Convex Subgroup