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.

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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.

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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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Correspondence to Gergely Kiss.

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.

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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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Keywords

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