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Associative idempotent nondecreasing functions are reducible

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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.

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Notes

  1. The definition of extremality stems from [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. 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. A monotone function is strictly monotone if every inequality in the definition of monotonicity (see Eq. (2)) is strict.

References

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Science, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Gergely Kiss

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

    Gábor Somlai

Authors
  1. Gergely Kiss
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  2. Gábor Somlai
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Correspondence to Gergely Kiss.

Additional information

Communicated by Mikhail Volkov.

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.

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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

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