# 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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1. 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. 3.

In  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.

## Author information

Authors

### Corresponding author

Correspondence to Gergely Kiss.