Israel Journal of Mathematics

, Volume 193, Issue 1, pp 459–505

Topological and algebraic structures on the ring of Fermat reals



The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring by using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non-standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensors, Analysis and Applications. second edition, Springer-Verlag, Berlin, 1988.CrossRefGoogle Scholar
  2. [2]
    W. Bertram, Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings, American Mathematical Society, Providence, RI, 2008.Google Scholar
  3. [3]
    M. Berz, G. Hoffstatter, W. Wan, K. Shamseddine and K. Makino, COSY INFINITY and its Applications to Nonlinear Dynamics, in Chapter Computational Differentiation: Techniques, Applications, and Tools, SIAM, Philadelphia, Penn, 1966, pp. 363–367.Google Scholar
  4. [4]
    A. Bigard, K. Keimel and S. Wolfenstein, Groupes et anneaux reticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin, 1977.MATHGoogle Scholar
  5. [5]
    J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland Mathematics Studies 84, North-Holland, Amsterdam, 1984.MATHGoogle Scholar
  6. [6]
    J. F. Colombeau, Multiplication of Distributions, Springer, Berlin, 1992.MATHGoogle Scholar
  7. [7]
    J. H. Conway, On Numbers and Games, London Mathematical Society Monographs, No. 6, Academic Press, London & New York, 1976.MATHGoogle Scholar
  8. [8]
    P. Ehrlich, An alternative construction of Conway’s ordered field No. Algebra Universalis 25 (1988), 7–16.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    P. Giordano, Fermat reals: Nilpotent infinitesimals and infinite dimensional spaces, arXiv:0907.1872, July 2009.Google Scholar
  10. [10]
    P. Giordano, Fermat-Reyes method in the ring of Fermat reals, Advances in Mathematics 228 (2011), 862–893. DOI: 10.1016/j.aim.2011.06.008.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    P. Giordano, Infinitesimals without logic, Russian Journal of Mathematical Physics 17 (2010), 159–191.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    P. Giordano, Order relation and geometrical representation of Fermat reals, American Mathematical Journal, Mathematical Proceedings of the Cambridge Philosophical Society, submitted.Google Scholar
  13. [13]
    P. Giordano, The ring of Fermat reals, Advances in Mathematics 225 (2010), 2050–2075. DOI: 10.1016/j.aim.2010.04.010.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    P. Iglesias-Zemmour, Diffeology,, July 9 2012.
  15. [15]
    A. Kock, Synthetic Differential Geometry, Volume 51 London Mathematical Society Lecture Note Series, Cambridge University Press, 1981.Google Scholar
  16. [16]
    I. Kolár, P.W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1993.MATHCrossRefGoogle Scholar
  17. [17]
    A. Kriegl and P.W. Michor, Product preserving functors of infinite dimensional manifolds, Archivum Mathematicum (Brno) 32 (1996), 289–306.MathSciNetMATHGoogle Scholar
  18. [18]
    A. Kriegl and P.W. Michor, The Convenient Settings of Global Analysis, Mathematical Surveys and Monographs 53, American Mathematical Society, Providence, RI, 1997.Google Scholar
  19. [19]
    R. Lavendhomme, Basic Concepts of Synthetic Differential Geometry, Kluwer Academic Publishers, Dordrecht, 1996.MATHCrossRefGoogle Scholar
  20. [20]
    T. Levi-Civita, Sugli infiniti ed infinitesimi attuali quali elementi analitici, Atti del Regio Istituto Veneto di Scienze, Lettere ed Arti VII (1893), 1765–1815.Google Scholar
  21. [21]
    I. Moerdijk and G.E. Reyes, Models for Smooth Infinitesimal Analysis, Springer, Berlin, 1991.MATHCrossRefGoogle Scholar
  22. [22]
    M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics Series 259, Longman Scientific & Technical, Harlow 1992.MATHGoogle Scholar
  23. [23]
    Z. M. Odibat and N. T. Shawagfeh, Generalized Taylor’s formula, Applied Mathematics and Computation 186 (2007), 286–293.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    A. Robinson, Non-standard Analysis, Princeton University Press, 1966.Google Scholar
  25. [25]
    K. Shamseddine, New Elements of Analysis on the Levi-Civita Field, PhD thesis, Michigan State University, East Lansing, Michigan, 1999.Google Scholar
  26. [26]
    K. Shamseddine and M. Berz, Intermediate value theorem for analytic functions on a Levi-Civita field, The Bulletin of the Belgian Mathematical Society Simon Stevin 14 (2007), 1001–1015.MathSciNetMATHGoogle Scholar
  27. [27]
    H. Vernaeve, Ideals in the ring of Colombeau generalized numbers, Communications in Algebra 38 (2010), 2199–2228.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2012

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria

Personalised recommendations