Skip to main content
SpringerLink
Log in

Type-n Arrow Categories

  • Conference paper
  • First Online:
Relational and Algebraic Methods in Computer Science (RAMICS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10226))

Abstract

It has been shown that the arrow category of type-2 fuzzy relations with respect to an arrow category \({\mathcal A}\) can be defined as the Kleisli category of \({\mathcal A}\) for a monad based on the concept of the extension of an object. In this paper we want to continue the study of higher-order arrow categories by showing two major results. First, we are going to remove the ad-hoc notion of an extension of an object completely from the construction of higher-order arrow categories. The second result establishes that the newly constructed higher-order arrow category has sufficient structure for constructing further higher-order arrow categories, i.e., that the process of moving from type-n to type-(n+1) arrow categories can always be iterated.

M. Winter—The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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, vol. 39. North-Holland Mathematical Library, Amsterdam (1990)

    MATH  Google Scholar 

  2. Furusawa, H.: Algebraic formalizations of fuzzy relations and their representation theorems. Ph.D.-thesis, Department of Informatics, Kyushu University, Japan (1998)

    Google Scholar 

  3. Furusawa, H., Kawahara, Y., Winter, M.: Dedekind categories with cutoff operators. Fuzzy Sets Syst. 173, 1–24 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jónsson, B., Tarski, A.: Boolean algebras with operators, I, II. Amer. J. Math. 73, 891–939 (1951). 74, 127–162 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  5. Kawahara, Y., Furusawa, H.: Crispness and representation theorems in dedekind categories. DOI-TR 143, Kyushu University (1997)

    Google Scholar 

  6. Kawahara, Y., Furusawa, H.: An algebraic formalization of fuzzy relations. Fuzzy Sets Syst. 101, 125–135 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Olivier, J.P., Serrato, D.: Squares and rectangles in relational categories - three cases: semilattice, distributive lattice and Boolean non-unitary. Fuzzy Sets Syst. 72, 167–178 (1995)

    Article  MATH  Google Scholar 

  9. Schmidt, G., Hattensperger, C., Winter, M.: Heterogeneous relation algebras. In: Brink, C., Kahl, W., Schmidt, G. (eds.) Relational Methods in Computer Science. Advances in Computing Sciences, pp. 39–53. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  10. Schmidt, G., Ströhlein, T.: Relations and Graphs. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  11. Schmidt, G.: Relational Mathematics. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  12. Tarski, A.: On the calculus of relations. J. Symbolic Logic 6, 73–89 (1941)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  14. Winter, M.: A new algebraic approach to \(L\)-fuzzy relations convenient to study crispness. INS Inf. Sci. 139, 233–252 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Winter, M.: Goguen Categories – A Categorical Approach to \(L\)-Fuzzy Relations. Springer, Berlin (2007)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Winter, M.: Membership values in arrow categories. Fuzzy Sets Syst. 267, 41–61 (2015)

    Article  MathSciNet  Google Scholar 

  18. Winter, M.: Higher-order arrow categories. In: Höfner, P., Jipsen, P., Kahl, W., Müller, M.E. (eds.) RAMICS 2014. LNCS, vol. 8428, pp. 277–292. Springer, Cham (2014). doi:10.1007/978-3-319-06251-8_17

    Chapter  Google Scholar 

  19. Winter, M., Jackson, E.: Categories of relations for variable-basis fuzziness. Fuzzy Sets Syst. 298, 222–237 (2016)

    Article  MathSciNet  Google Scholar 

  20. Winter, M., Jackson, E., Fujiwara, Y.: Type-2 fuzzy controllers in arrow categories. In: Höfner, P., Jipsen, P., Kahl, W., Müller, M.E. (eds.) RAMICS 2014. LNCS, vol. 8428, pp. 293–308. Springer, Cham (2014). doi:10.1007/978-3-319-06251-8_18

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Brock University, St. Catharines, ON, L2S 3A1, 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. Data61, CSIRO, Sydney, New South Wales, Australia

    Peter Höfner

  2. CNRS, Lyon, France

    Damien Pous

  3. University of Sheffield, Sheffield, United Kingdom

    Georg Struth

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Winter, M. (2017). Type-n Arrow Categories. In: Höfner, P., Pous, D., Struth, G. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2017. Lecture Notes in Computer Science(), vol 10226. Springer, Cham. https://doi.org/10.1007/978-3-319-57418-9_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-57418-9_19

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57417-2

  • Online ISBN: 978-3-319-57418-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation