Roundings and Approximations in Ordered Sets

  • R. Albrecht
Part of the Computing Supplementum book series (COMPUTING, volume 2)


The topological aspect of rounding was introduced by the author in an earlier article [1] and applied to the theory of a rounded algebraic law of composition. For example common rounding in ℝ as well as interval arithmetic on ℝ are covered by this general theory. In the present paper the development of the topological theory is continued by introducing the notions of adherence, co-adherence, and limit of a rounding, by a uniform topologization of the Space of filter- and ideal-basis on an ordered set with a rounding, by giving a general contraction principle, by a comparison of two roundings on the same ordered set, and by extending the theory of consistency and stability of approximations to mapping problems.


Interval Arithmetic Uniform Topologization Classical Topology Topological Aspect Royal Irish Academy 


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  1. [1]
    Albrecht, R.: Grundlagen einer Theorie gerundeter algebraischer Verknüpfungen in topologischen Vereinen. Computing, Suppl. 1, pp. 1–14. Wien-New York: Springer 1977.Google Scholar
  2. [2]
    Albrecht, R., Karrer, G.: Fixpunktsätze in uniformen Räumen. Math. Zeitschr. 74, 387–391 (1960).CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    Kulisch, U.: Grundlagen des Numerischen Rechnens. (Reihe Informatik, Bd. 19.) Zürich: Bibliograph. Institut 1976.MATHGoogle Scholar
  4. [4]
    Nöbeling, G.: Grundlagen der analytischen Topologie. Berlin-Göttingen-Heidelberg: Springer 1954.MATHGoogle Scholar
  5. [5]
    Stummel, F.: Discrete convergence of mappings. Topics in numerical analysis. Proceedings of the Royal Irish Academy Conference on Numerical Analysis 1972, pp. 285–310. London: Academic Press 1973.Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • R. Albrecht
    • 1
  1. 1.Institut für Informatik und Numerische MathematikUniversität InnsbruckInnsbruckAustria

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