Skip to main content
Log in

Topological and algebraic structures on the ring of Fermat reals

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensors, Analysis and Applications. second edition, Springer-Verlag, Berlin, 1988.

    Book  Google Scholar 

  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. 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. A. Bigard, K. Keimel and S. Wolfenstein, Groupes et anneaux reticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin, 1977.

    MATH  Google Scholar 

  5. J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, North-Holland Mathematics Studies 84, North-Holland, Amsterdam, 1984.

    MATH  Google Scholar 

  6. J. F. Colombeau, Multiplication of Distributions, Springer, Berlin, 1992.

    MATH  Google Scholar 

  7. J. H. Conway, On Numbers and Games, London Mathematical Society Monographs, No. 6, Academic Press, London & New York, 1976.

    MATH  Google Scholar 

  8. P. Ehrlich, An alternative construction of Conway’s ordered field No. Algebra Universalis 25 (1988), 7–16.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Giordano, Fermat reals: Nilpotent infinitesimals and infinite dimensional spaces, arXiv:0907.1872, July 2009.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Giordano, Infinitesimals without logic, Russian Journal of Mathematical Physics 17 (2010), 159–191.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Giordano, Order relation and geometrical representation of Fermat reals, American Mathematical Journal, Mathematical Proceedings of the Cambridge Philosophical Society, submitted.

  13. P. Giordano, The ring of Fermat reals, Advances in Mathematics 225 (2010), 2050–2075. DOI: 10.1016/j.aim.2010.04.010.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Iglesias-Zemmour, Diffeology, http://math.huji.ac.il/~piz/documents/Diffeology.pdf, July 9 2012.

  15. A. Kock, Synthetic Differential Geometry, Volume 51 London Mathematical Society Lecture Note Series, Cambridge University Press, 1981.

  16. I. Kolár, P.W. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, Heidelberg, New York, 1993.

    Book  MATH  Google Scholar 

  17. A. Kriegl and P.W. Michor, Product preserving functors of infinite dimensional manifolds, Archivum Mathematicum (Brno) 32 (1996), 289–306.

    MathSciNet  MATH  Google Scholar 

  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. R. Lavendhomme, Basic Concepts of Synthetic Differential Geometry, Kluwer Academic Publishers, Dordrecht, 1996.

    Book  MATH  Google Scholar 

  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. I. Moerdijk and G.E. Reyes, Models for Smooth Infinitesimal Analysis, Springer, Berlin, 1991.

    Book  MATH  Google Scholar 

  22. M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics Series 259, Longman Scientific & Technical, Harlow 1992.

    MATH  Google Scholar 

  23. Z. M. Odibat and N. T. Shawagfeh, Generalized Taylor’s formula, Applied Mathematics and Computation 186 (2007), 286–293.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Robinson, Non-standard Analysis, Princeton University Press, 1966.

  25. K. Shamseddine, New Elements of Analysis on the Levi-Civita Field, PhD thesis, Michigan State University, East Lansing, Michigan, 1999.

    Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  27. H. Vernaeve, Ideals in the ring of Colombeau generalized numbers, Communications in Algebra 38 (2010), 2199–2228.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Paolo Giordano.

Additional information

Supported by an L. Meitner FWF (Austria) grant (M1247-N13).

Supported by FWF research grants Y237 and P20525.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Giordano, P., Kunzinger, M. Topological and algebraic structures on the ring of Fermat reals. Isr. J. Math. 193, 459–505 (2013). https://doi.org/10.1007/s11856-012-0079-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-012-0079-z

Keywords

Navigation