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On noncontinuous bisymmetric strictly monotone operations

Gergely Kiss

TL;DR

The paper investigates regularity of bisymmetric, strictly increasing binary operations without assuming reflexivity. It constructs discontinuous, weighted-quasi-sum type operations $F(x,y)=f^{-1}(\alpha f(x)+\beta f(y))$ on an interval using a Cantor-type perfect set, showing that bisymmetry and monotonicity do not imply continuity outside the reflexive setting, and extends the construction to $n$-ary operations. A two-point reflexivity result is proved: if $F(a,a)=a$ and $F(b,b)=b$ for $a<b$, then $F$ is continuous on $[a,b]$ and has a quasi-arithmetic representation there. The work thus reveals a sharp dichotomy between local rigidity under two-point reflexivity and global irregularity in the non-reflexive regime, and it raises open problems for global continuity and multivariate characterization of bisymmetric, monotone operations.

Abstract

In this paper, we construct a bisymmetric, strictly increasing, symmetric binary operation $F$ on an interval which is not continuous. This answers a natural question in the study of bisymmetric and mean-type operations by showing that continuity may fail even for non-reflexive operations of the form \[ F(x,y)=f^{-1}\bigl(αf(x)+βf(y)\bigr), \] where $f$ is a bijection between an interval and a perfect, nowhere dense fractal-type set, and $α,β>0$ with $α+β\neq 1$. As a consequence, we also obtain a noncontinuous, associative, strictly increasing, symmetric operation on an interval. We generalize these constructions to the multivariate case. We also prove a complementary result: if a symmetric, bisymmetric, strictly increasing operation is reflexive at the endpoints of an interval, then it must be continuous and coincide with a quasi-arithmetic mean on that interval.

On noncontinuous bisymmetric strictly monotone operations

TL;DR

The paper investigates regularity of bisymmetric, strictly increasing binary operations without assuming reflexivity. It constructs discontinuous, weighted-quasi-sum type operations on an interval using a Cantor-type perfect set, showing that bisymmetry and monotonicity do not imply continuity outside the reflexive setting, and extends the construction to -ary operations. A two-point reflexivity result is proved: if and for , then is continuous on and has a quasi-arithmetic representation there. The work thus reveals a sharp dichotomy between local rigidity under two-point reflexivity and global irregularity in the non-reflexive regime, and it raises open problems for global continuity and multivariate characterization of bisymmetric, monotone operations.

Abstract

In this paper, we construct a bisymmetric, strictly increasing, symmetric binary operation on an interval which is not continuous. This answers a natural question in the study of bisymmetric and mean-type operations by showing that continuity may fail even for non-reflexive operations of the form where is a bijection between an interval and a perfect, nowhere dense fractal-type set, and with . As a consequence, we also obtain a noncontinuous, associative, strictly increasing, symmetric operation on an interval. We generalize these constructions to the multivariate case. We also prove a complementary result: if a symmetric, bisymmetric, strictly increasing operation is reflexive at the endpoints of an interval, then it must be continuous and coincide with a quasi-arithmetic mean on that interval.
Paper Structure (11 sections, 14 theorems, 92 equations)

This paper contains 11 sections, 14 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Noncontinuous strictly increasing bisymmetric operations
  4. Weighted quasi-sums on general sets
  5. The main binary counterexample
  6. Multi-variate extensions
  7. Basic notions for --ary operations
  8. Weighted quasi--sums in variables
  9. Discontinuous bisymmetric --ary operations
  10. Local reflexivity at two points and continuity
  11. Concluding remarks and open problems

Key Result

Theorem 2.1

A function $F:I^2\to I$ is continuous, reflexive, partially strictly monotone, symmetric, and bisymmetric if and only if there exists a continuous strictly monotone function $f:I\to\mathbb{R}$ such that In this case $F$ is called a quasi–arithmetic mean generated by $f$.

Theorems & Definitions (28)

  • Theorem 2.1: Aczél, 1948; quasi–arithmetic means
  • Theorem 2.2: Aczél–Dhombres; quasi–sums
  • Theorem 2.3: Aczél
  • Theorem 2.4: Burai–Kiss–Szokol, 2021
  • Theorem 2.5: Burai–Kiss–Szokol, 2023
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • ...and 18 more