Associative idempotent nondecreasing functions are reducible

TL;DR

This paper proves that associative, idempotent, and nondecreasing functions defined on a chain are uniquely reducible to a composition of binary associative functions, clarifying their structural properties.

Contribution

It establishes the unique reducibility of associative, idempotent, and nondecreasing functions on chains, extending the understanding of their compositional structure.

Findings

Associative, idempotent, nondecreasing functions are uniquely reducible.

Summarizes known results for functions on chains.

Main result confirms unique reducibility property.

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.