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Archiv der Mathematik

, Volume 111, Issue 2, pp 177–186 | Cite as

Approximate homomorphisms on lattices

  • Roman Badora
  • Tomasz KochanekEmail author
  • Barbara Przebieracz
Article
  • 51 Downloads

Abstract

We prove two results concerning an Ulam-type stability problem for homomorphisms between lattices. One of them involves estimates by quite general error functions; the other deals with approximate (join) homomorphisms in terms of certain systems of lattice neighborhoods. As a corollary, we obtain a stability result for approximately monotone functions.

Keywords

Ulam stability Distributive lattice Lattice homomorphism 

Mathematics Subject Classification

06B23 06D99 39B82 

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Notes

Acknowledgements

We thank the referee for his/her valuable remarks and the suggestion of extending Corollary 10 to the case of general linearly ordered sets.

References

  1. 1.
    I. Farah, Approximate homomorphisms, Combinatorica 18 (1998), 335–348.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    W. Förg-Rob, K. Nikodem, and Zs. Páles, Separation by monotonic functions, Math. Pannon. 7 (1996), 191–196.MathSciNetzbMATHGoogle Scholar
  3. 3.
    G. Grätzer, General Lattice Theory, Academic Press, New York–San Francisco, 1978.CrossRefzbMATHGoogle Scholar
  4. 4.
    P.M. Gruber, Stability of isometries, Trans. Amer. Math. Soc. 245 (1978), 263–277.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    D.H. Hyers, G. Isac, and Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Basel, 1998.CrossRefzbMATHGoogle Scholar
  6. 6.
    N.J. Kalton and J.W. Roberts, Uniformly exhaustive submeasures and nearly additive set functions, Trans. Amer. Math. Soc. 278 (1983), 803–816.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    N. Kalton, Quasi-Banach spaces, In: Handbook of the Geometry of Banach Spaces, Vol. 2, 1099–1130, North-Holland, Amsterdam, 2003.Google Scholar
  8. 8.
    T. Kochanek, Stability of vector measures and twisted sums of Banach spaces, J. Funct. Anal. 264 (2013), 2416–2456.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    W. Kubiś, A sandwich theorem for convexity preserving maps, Tatra Mt. Math. Publ. 24 (2002), 125–131.MathSciNetzbMATHGoogle Scholar
  10. 10.
    S.M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York–London, 1960.Google Scholar
  11. 11.
    M. van de Vel, Theory of convex structures, North-Holland, Amsterdam, 1993.zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Roman Badora
    • 1
  • Tomasz Kochanek
    • 2
    • 3
    Email author
  • Barbara Przebieracz
    • 1
  1. 1.Institute of MathematicsUniversity of SilesiaKatowicePoland
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  3. 3.Institute of MathematicsUniversity of WarsawWarsawPoland

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