Skip to main content

Associative idempotent nondecreasing functions are reducible

Abstract

An n-ary associative function is called reducible if it can be written as a composition of a binary associative function. We summarize known results when the function is defined on a chain and is nondecreasing. Our main result shows that associative idempotent and nondecreasing functions are uniquely reducible.

This is a preview of subscription content, access via your institution.

Notes

  1. 1.

    The definition of extremality stems from [2].

  2. 2.

    Adjoining a neutral element to X for an n-associative function F means to define an n-associative function \(\bar{F}\) on the set \(X \cup \{ e\} \) such that \(e\notin X\) is a neutral element for \(\bar{F}\) and \(\bar{F}(x_1 , \dots , x_n)=F(x_1 , \dots , x_n)\) for all \(x_1 , \dots , x_n \in X\).

  3. 3.

    In [2] a mean \(\mu :(\bigcup _{n\in \mathbb {N}}\mathbb {R}^{n})\rightarrow \mathbb {R}\) was called extremal if for all elements \(a_1,a_2,\dots ,a_n\in \mathbb {R}\) with \(a_1\le a_2 \le \dots \le a_n\), we have \(\mu (a_1, a_2, \dots , a_n)=\mu (a_1, a_n)\).

  4. 4.

    A monotone function is strictly monotone if every inequality in the definition of monotonicity (see Eq. (2)) is strict.

References

  1. 1.

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

  2. 2.

    Bennett, C.D., Holland, W.C., Székely, G.J.: Integer valued means. Aequat. Math. 88, 137–149 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Couceiro, M., Marichal, J.-L.: Aczélian \(n\)-ary semigroups. Semigroup Forum 85, 81–90 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Czogała, E., Drewniak, J.: Associative monotonic operations in fuzzy set theory. Fuzzy Sets Syst. 12(3), 249–269 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Devillet, J., Kiss, G., Marichal, J.-L.: Characterizations of quasitrivial symmetric nondecreasing associative operations, arXiv:1705.00719

  6. 6.

    Dörnte, W.: Untersuchengen über einen verallgemeinerten Gruppenbegriff. Math. Z. 29, 1–19 (1928)

    Article  MATH  Google Scholar 

  7. 7.

    Dudek, W.A., Mukhin, V.V.: On topological \(n\)-ary semigroups. Quasigroups Relat. Syst. 3, 373–388 (1996)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    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 

  9. 9.

    Kiss, G., Somlai, G.: A characterization of \(n\)-associative, monotone, idempotent functions on an interval that have neutral elements. Semigroup Forum 3, 438–451 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Post, E.L.: Polyadic groups. Trans. Am. Math. Soc. 48, 208–350 (1940)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for the valuable comments and suggestions which improved the quality of this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Gergely Kiss.

Additional information

The research was supported by the internal research project R-AGR-0500 of the University of Luxembourg. The first author was partially supported by the Hungarian Scientific Research Fund (OTKA) K124749. The second author was partially supported by the Hungarian Scientific Research Fund (OTKA) K115799.

Communicated by Mikhail Volkov.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kiss, G., Somlai, G. Associative idempotent nondecreasing functions are reducible. Semigroup Forum 98, 140–153 (2019). https://doi.org/10.1007/s00233-018-9973-y

Download citation

Keywords

  • n-ary semigroup
  • Associativity
  • Reducible
  • Extremal
  • Quasitrivial
  • Idempotent
  • Neutral element