Skip to main content
SpringerLink
Log in

A characterization of n-associative, monotone, idempotent functions on an interval that have neutral elements

  • Research Article
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

We investigate monotone idempotent n-ary semigroups and provide a generalization of the Czogala–Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element. We also present a complete characterization of idempotent monotone n-associative functions on an interval that have neutral elements.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. If I is bounded and we denote the end-points of I by m and M (\(m\ge M\)), then \(\bar{I}=[m,M]\). If I is not bounded from below (or above), then we let \(m=-\infty \) (\(M=+\infty \), respectively). For instance, \(\bar{\mathbb {R}}=\mathbb {R}\cup \{\pm \infty \}\).

  2. Let m and M be the boundary points of \(\bar{I}\). We use the convention that \(g(m-0)=M\) and \(g(M+0)=m\).

References

  1. Ackerman, N.L.: A characterization of quasitrivial \(n\)-semigroups, preprint

  2. Czogala, E., Drewniak, J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets Syst. 12, 249–269 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Devillet, J., Kiss, G., Marichal, J.-L., Teheux, B.: On conservative nondecreasing associative operations, working paper (2017)

  4. Dudek, W.A., Mukhin, V.V.: On \(n\)-ary semigroups with adjoint neutral element. Quasigroups Relat. Syst. 14, 163–168 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Encyclopedia of Mathematics and its Applications, vol. 127. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  6. Lehtonen, E., Marichal, J.-L., Teheux, B.: Associative string functions. Asian Eur. J. Math. 7(4), 1450059 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Martin, J., Mayor, G., Torrens, J.: On locally internal monotonic operations. Fuzzy Sets Syst. 137, 27–42 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ruiz-Aguilera, D., Torrens, J., De Baets, B., Fodor, J.: Some remarks on the characterization of idempotent uninorms. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) Computational Intelligence for Knowledge-Based Systems Design. Lecture Notes in Artificial Intelligence, vol. 6178, pp. 425–434. Springer, Berlin (2010)

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the referee, whose comments and suggestions highly improved the presentation of this article. One of the most important observations of the referee was that most of the results of Sect. 3 can be stated in a more general setting. The authors appreciate a lot the valuable suggestions of Mikhail Volkov about the final form of the paper.

Author information

Authors and Affiliations

  1. Mathematical Research Unit, Faculty of Science, University of Luxembourg, 2, avenue de l’Université, 4365, Esch-sur-Alzette, Luxembourg

    Gergely Kiss

  2. Faculty of Science, Institute of Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/c, Budapest, 1117, Hungary

    Gábor Somlai

Authors
  1. Gergely Kiss
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gábor Somlai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gergely Kiss.

Additional information

Communicated by Mikhail Volkov.

Supported by the project R-STR-1041-00-Z of the University of Luxembourg. Gergely Kiss was supported by the Hungarian Scientific Research Fund (OTKA) K104178 and the internal research project R-AGR-0500-MR03 of the University of Luxembourg. Gábor Somlai was supported by the Hungarian Scientific Research Fund (OTKA) K115799.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kiss, G., Somlai, G. A characterization of n-associative, monotone, idempotent functions on an interval that have neutral elements. Semigroup Forum 96, 438–451 (2018). https://doi.org/10.1007/s00233-017-9876-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-017-9876-3

Keywords

Search

Navigation