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A sandwich theorem for monotone additive functions

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Abstract

The followingsandwich problem is considered: Given two real-valued functionsg:X→ℝ,h:Y→ℝ defined on subsetsX, Y ⊆ S of a preordered abelian monoids (S, +, 0, ≺), does there exist a real-valued ≺ additive functionf:S→ℝ such thatg≤f|X andf|Y≤h? We study several necessary conditions for the existence of such a functionf, some of which are also sufficient.

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References

  1. Anger, B., and J. Lembcke,Hahn-Banach type theorems for hypolinear functionals, Math. Ann.209 (1974), 127–151.

    Article  MATH  MathSciNet  Google Scholar 

  2. Aumann, G.,Über die Erweiterung von additiven monotonen Funktionen auf regulär geordneten Halbgruppen, Arch. Math.8 (1957), 422–427.

    Article  MathSciNet  Google Scholar 

  3. Cohn, P.M.,Rank functions on rings, J. Algebra133 (1990), 373–385.

    Article  MATH  MathSciNet  Google Scholar 

  4. Dinges, H.,Decompositon in ordered semigroups, J. Funct. Anal.5 (1970), 436–483.

    Article  MATH  MathSciNet  Google Scholar 

  5. Fuchssteiner, B.,Sandwich theorems and lattice semigroups, J. Funct. Anal.16 (1974), 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  6. Fuchssteiner, B. and W. Lusky, “Convex Cones”, Mathematics Studies 56, North Holland, Amsterdam 1981.

    MATH  Google Scholar 

  7. Kindler, J.,Sandwich theorems for set functions, J. Math. Anal. Appl.133 (1988), 529–542.

    Article  MATH  MathSciNet  Google Scholar 

  8. Laczkovich, M.,Invariant signed measures and the cancellation law, Proc. Amer. Math. Soc.111 (1991), 421–431.

    Article  MATH  MathSciNet  Google Scholar 

  9. Páles, Z.,A Hahn-Banach theorem for separation of semigroups and its applications, Aequationes Math.37 (1989), 141–161.

    Article  MathSciNet  Google Scholar 

  10. Plappert, P.,A sandwich theorem for precharges, J. Math. Anal. Appl.156 (1991), 395–409.

    Article  MATH  MathSciNet  Google Scholar 

  11. Plappert, P.,Sandwichsätze für Funktionen auf abelschen Monoiden und für Mengenfunktionen, Dissertation, Fachbereich Mathematik, Technische Hochschule Darmstadt 1994.

  12. Popov, V. A.,Additive and semiadditive functions on Boolean algebras, Siberian Math. J.17 (1976), 258–264.

    Article  MATH  Google Scholar 

  13. Tarski, A.,Algebraische Fassung des Maßproblems, Fundam. Math.31 (1938), 47–66.

    MATH  Google Scholar 

  14. Topsøe, F.,The naive approach to the Hahn-Banach theorem, Commentat. math., Spec. Vol. I, dedic. L. Orlicz (1978), 315–330.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Hochschule Darmstadt, Schloßgartenstr. 7, D-64289, Darmstadt, Germany

    Peter Plappert

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  1. Peter Plappert
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Communicated by Karl H. Hofmann

The results of this paper formed part of the author's doctoral dissertation [11] at Technische Hochschule Darmstadt. The author would like to thank his supervisor, Professor Dr. J. Kindler, for his help and guidance and Professor Dr. K. H. Hofmann and Professor Dr. K. Keimel for useful suggestions and comments.

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Plappert, P. A sandwich theorem for monotone additive functions. Semigroup Forum 51, 347–355 (1995). https://doi.org/10.1007/BF02573643

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  • DOI: https://doi.org/10.1007/BF02573643

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