Skip to main content
SpringerLink
Log in

Relational Algebraic Approach to the Real Numbers the Additive Group

  • Conference paper
  • First Online:
Relational and Algebraic Methods in Computer Science (RAMiCS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13896))

  • 220 Accesses

Abstract

In this paper we start the investigation of an object representing the real numbers in categories of relations. Our axiomatization uses the construction of a relation power, i.e., an abstract version of power sets within the category. This allows us to utilize a relation algebraic version of Tarski’s axioms of the real numbers as a first-order definition of a real number object. The current paper focuses on the addition operation of the real number object. It is shown that addition forms a densely and linearly ordered abelian group.

The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada (283267).

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Freyd, P., Scedrov, A.: Categories, Allegories. North-Holland Mathematical Library, Vol. 39, North-Holland, Amsterdam (1990)

    Google Scholar 

  2. Olivier J.P., Serrato D.: Catégories de Dedekind. Morphismes dans les Catégories de Schröder. C.R. Acad. Sci. Paris 290, 939–941 (1980)

    Google Scholar 

  3. Olivier, J.P., Serrato, D.: Squares and Rectangles in Relational Categories - Three Cases: Semilattice, Distributive lattice and Boolean Non-unitary. Fuzzy Sets Syst. 72(2), 167–178 (1995)

    Article  MATH  Google Scholar 

  4. Schmidt, G., Ströhlein, T.: Relations and graphs: discrete mathematics for computer scientists. In: EATCS Monographs on Theoretical Computer Science, Springer, Heidelberg (1993). https://doi.org/10.1007/978-3-642-77968-8

  5. Schmidt, G.: Relational mathematics. In: Encyplopedia of Mathematics and its Applications, vol. 132, Cambridge University Press, Cambridge (2011)

    Google Scholar 

  6. Schmidt, Gunther, Winter, Michael: Closures and their Aumann contacts. In: Relational Topology. LNM, vol. 2208, pp. 113–124. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-74451-3_7

    Chapter  MATH  Google Scholar 

  7. Tarski, A.: Introduction to Logic. Oxford University Press, Oxford (1941)

    MATH  Google Scholar 

  8. Winter, M.: Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdetermismus in Programmiersprachen. Dissertationsverlag NG Kopierladen GmbH, München (1998)

    Google Scholar 

  9. Winter M.: Goguen categories - a categorical approach to \(L\)-fuzzy relations. In: Trends in Logic, vol. 25, Springer, Heidelberg (2007). https://doi.org/10.1007/978-1-4020-6164-6

  10. Winter, M.: Arrow categories. Fuzzy Sets Syst. 160, 2893–2909 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Winter, M.: Fixed point operators in Heyting categories. Part I - internal fixed point theorem and calculus. J. Pure Appl. Algebra (2022)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Brock University, St. Catharines, ON, Canada

    Michael Winter

Authors
  1. Michael Winter
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Winter .

Editor information

Editors and Affiliations

  1. Deutsches Zentrum für Luft- und Raumfahrt, Augsburg, Germany

    Roland Glück

  2. Aix-Marseille University, Marseille, France

    Luigi Santocanale

  3. Brock University, St Catharines, ON, Canada

    Michael Winter

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Winter, M. (2023). Relational Algebraic Approach to the Real Numbers the Additive Group. In: Glück, R., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2023. Lecture Notes in Computer Science, vol 13896. Springer, Cham. https://doi.org/10.1007/978-3-031-28083-2_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-28083-2_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-28082-5

  • Online ISBN: 978-3-031-28083-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation