Advertisement

Fields, Meadows and Abstract Data Types

  • Jan Bergstra
  • Yoram Hirshfeld
  • John Tucker
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4800)

Abstract

Fields and division rings are not algebras in the sense of “Universal Algebra”, as inverse is not a total function. Mending the inverse by any definition of 0− 1 will not suffice to axiomatize the axiom of inverse x − 1·x = 1, by an equation. In particular the theory of fields cannot be used for specifying the abstract data type of the rational numbers.

We define equational theories of Meadows and of Skew Meadows, and we prove that these theories axiomatize the equational properties of fields and of division rings, respectively, with 0− 1= 0 . Meadows are then used in the theory of Von Neumann regular ring rings to characterize strongly regular rings as those that support an inverse operation that turns it into a skew meadow. To conclude, we present in this framework the specification of the abstract type of the rational numbers, as developed by the first and third authors in [2].

Keywords

Rational Number Inverse Function Commutative Ring Equational Theory Division Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bergman, G.M.: An Invitation to General Algebra and Universal Constructions. Henry Helson (1998)Google Scholar
  2. 2.
    Bergstra, J.A., Tucker, J.V.: The rational numbers as an abstract data type. J. ACM 54(2), Article 7 (April 2007)Google Scholar
  3. 3.
    Burris, S.N., Sankappanavar, H.P.: A Course in Universal Algebra, free online edition. Springer, Heidelberg (1981)Google Scholar
  4. 4.
    Cohn, P.M.: Universal Algebra. D. Reidel Publishing, Dordrecht (1981)zbMATHGoogle Scholar
  5. 5.
    Gougen, J.A., Thacher, J.W., Wagner, E.G.: An initial algebra approach to the specification, correctness and implementation of abstract data types. In: Yeh, R.T. (ed.) Current Trends in Programing Methodology, VI, Data Structuring, pp. 80–149. Prentice Hall, Englewood Cliffs (1978)Google Scholar
  6. 6.
    Goodearl, K.R.: Von Neumann regular rings. Pitman, London, San Francisco, Melburne (1979)Google Scholar
  7. 7.
    Graetzer, G.: Universal Algebra. Hobby, David, and Ralph McKenzie (1988)Google Scholar
  8. 8.
    Loeckx, J., Ehrich, H.D., Wolf, M.: Specification of Abstract Data Types. and Teubner. Wiley and Teubner, Chichester (1996)zbMATHGoogle Scholar
  9. 9.
    Loeckx, J., Ehrich, H.-D., Wolf, M.: Algebraic specification of abstract data types. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, Oxford University Press, Oxford (2000)Google Scholar
  10. 10.
    Meinke, K., Tucker, J.V.: Universal Algebra. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of logic in computer science I, Mathematical structures, pp. 189–411 (1992)Google Scholar
  11. 11.
    Moore, E.H.: On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society 26, 394–395 (1920)Google Scholar
  12. 12.
    Penrose, R.: A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society 51, 406–413 (1955)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    von Neumann, J.: Continuous geometries. Princeton University Press, Princeton (1960)Google Scholar
  14. 14.
    Maclagan-Wedderburn, J.H.: A theorem on finite algebras. Transactions of the American Mathematical Society 6, 349–352 (1905)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Wikipedia, Universal Algebra, http://en.wikipedia.org/wiki/Universal_algebra
  16. 16.
    Wirsing, M.: Algebraic Specification. In: Leeuwen, J.v. (ed.) Handbook of Theoretical Computer Science. Formal Models and Sematics, vol. B, Elsevier and MIT Press (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jan Bergstra
    • 1
  • Yoram Hirshfeld
    • 2
  • John Tucker
    • 3
  1. 1.University of Amsterdam 
  2. 2.Tel Aviv University 
  3. 3.University of Wales, Swansea 

Personalised recommendations