Overview
This work addresses the characterization of quasi-arithmetic means, focusing on refining established theorems by Kolmogorov–Nagumo–de Finetti and Aczel–Maksa. The core contribution involves demonstrating that continuity, often a prerequisite in these characterizations, arises automatically from other specified properties. The research specifically investigates $n$-variable operations on a real interval.
Research Context
The classical theorems in question provide frameworks for understanding quasi-arithmetic means from two distinct perspectives. The Kolmogorov–Nagumo–de Finetti theorem characterizes compatible families of means, particularly those satisfying the replacement axiom. In parallel, the Aczel–Maksa theorem characterizes bisymmetric means of a fixed arity. These established results have historically relied on certain regularity assumptions, including continuity.
Approach
The methodology employed in this research centers on a recursive construction derived from properties like bisymmetry. This construction operates on $n$-adic rationals. Following this recursive step, a dense-domain continuity argument is utilized. This two-part approach forms the basis for establishing the automatic emergence of continuity.
Findings
The primary finding of this research indicates that any operation defined on a real interval that possesses the properties of being reflexive, symmetric, bisymmetric, and partially strictly increasing, and involves $n$ variables, is inherently continuous. Consequently, such an operation is characterized as quasi-arithmetic. This result effectively removes the explicit requirement for continuity in this specific characterization, as it is shown to follow from the other stipulated properties.
Extending this method, the research also yields a regularity-free version of the Kolmogorov–Nagumo–de Finetti theorem. This revised theorem applies to compatible families of strictly increasing symmetric means, where the traditional requirement for regularity is no longer necessary for their characterization.
The proof underpinning these findings is constructed from the recursive process on $n$-adic rationals, enabled by bisymmetry, and subsequently confirmed through a dense-domain continuity argument.
Why This Matters
This research refines foundational understanding in the theory of means by demonstrating that continuity is an inherent property under certain conditions, rather than a separate assumption. This simplification in characterization has implications for theoretical work involving means and related mathematical operations, potentially streamlining future analytical approaches.
Key Limitations Mentioned by Researchers
The source does not explicitly mention limitations of the research.