On noncontinuous bisymmetric strictly monotone operations

Imagine you are a chef trying to create the perfect "average" dish. You have a rulebook (mathematics) that says: "If you mix two ingredients in a specific way, the result should be predictable, fair, and smooth."

For decades, mathematicians believed that if your mixing rule followed two golden laws—Bisymmetry (mixing order doesn't matter) and Strict Monotonicity (adding more of an ingredient always makes the dish "stronger")—then your mixing process would have to be smooth and continuous. They thought you couldn't have a "jumpy" or broken recipe.

This paper, by Gergely Kiss, is like a culinary prankster who says, "Actually, you can have a broken recipe that still follows the rules!" Here is the breakdown of his discovery using simple analogies.

1. The Two Golden Rules

To understand the paper, we need to know the two rules the "chef" (the mathematical operation) must follow:

Bisymmetry (The Fairness Rule): Imagine you have four ingredients: A, B, C, and D.
- Rule 1: Mix A and B, then mix C and D, and finally mix those two results together.
- Rule 2: Mix A and C, then mix B and D, and finally mix those two results together.
- The Rule: Both methods must yield the exact same final dish. It doesn't matter how you group the ingredients; the outcome is consistent.
Strict Monotonicity (The "More is Better" Rule): If you increase the amount of one ingredient, the final dish must get "stronger" (or larger). You can't add more sugar and have the cake get smaller.

2. The Old Belief: "Smoothness is Automatic"

Historically, mathematicians (like the famous J. Aczél) proved that if you follow these rules AND your recipe is "reflexive" (meaning if you mix an ingredient with itself, you just get that ingredient back, like $5 + 5 = 5$ in a specific averaging sense), then your recipe must be smooth. There are no gaps, no jumps, and no sudden breaks. It's like a perfectly flowing river.

3. The Big Discovery: The "Fractal" Recipe

Kiss asks: "What if we drop the 'reflexive' rule? What if mixing an ingredient with itself changes it?"

He constructs a counter-example. He builds a mixing machine that:

Follows the Fairness Rule (Bisymmetry).
Follows the "More is Better" Rule (Strict Monotonicity).
Is completely broken (Discontinuous).

The Analogy: The Cantor Set (The Swiss Cheese)
To build this, Kiss uses a mathematical object called a "Cantor set." Imagine a long loaf of bread (an interval).

You cut out the middle third.
You cut out the middle third of the remaining pieces.
You keep doing this forever.
What's left is a "dust" of crumbs. It's full of holes (gaps), but it's still infinite. It's a fractal.

Kiss builds his mixing machine so that it only works on these "crumbs."

He creates a special map (a function $f$ ) that takes normal numbers and squashes them onto these fractal crumbs.
He mixes the numbers using the fractal crumbs.
Because the crumbs have holes everywhere, the mixing process jumps around wildly. If you change the input by a tiny amount, the output might jump across a huge gap in the fractal.

The Result: The machine is fair and always gets "stronger," but it is jumpy. It is not smooth. It proves that without the "reflexive" rule, you can have a perfectly valid mathematical operation that is chaotic and discontinuous.

4. The Twist: One Point vs. Two Points

The paper also explores what happens if we do add the "reflexive" rule back in, but only at specific points.

One Point Reflexivity: If the machine works perfectly when you mix a number with itself at one specific spot (say, the number 0), the machine can still be broken and jumpy everywhere else. One anchor isn't enough to hold the whole ship steady.
Two Point Reflexivity: However, if the machine works perfectly at two different spots (say, 0 and 10), something magical happens. The "Fairness" and "More is Better" rules force the machine to become smooth and continuous everywhere between those two points.
- Metaphor: Imagine a tightrope. If you pin the rope down at one point, it can still flap wildly in the wind. But if you pin it down at two points, the rope between them is forced to be straight and smooth.

5. Why This Matters

This paper is a "reality check" for mathematicians.

Before: They thought, "If it's fair and consistent, it must be smooth."
Now: They know, "Not necessarily! Unless you pin it down with reflexivity, it can be a wild, jumpy fractal."

It shows that continuity (smoothness) is not a free gift given by fairness; it is a special property that requires specific conditions (like being reflexive at two points) to unlock.

Summary

The Problem: Can you have a fair, consistent, and always-increasing mixing rule that is not smooth?
The Answer: Yes! Kiss built one using a "fractal dust" of numbers.
The Lesson: Smoothness isn't automatic. You need "anchors" (reflexivity at two points) to force the math to behave nicely. Without them, the math can be as wild and broken as a shattered mirror, even if it follows the basic rules of fairness.

Here is a detailed technical summary of the paper "On noncontinuous bisymmetric strictly monotone operations" by Gergely Kiss.

1. Problem Statement and Context

The paper addresses a fundamental question in the theory of functional equations and mean-type operations: Does the combination of bisymmetry, strict monotonicity, and symmetry imply continuity?

Historical Background:
- Aczél's Theorems: Historically, it was established (by Aczél, Kolmogorov, Nagumo, de Finetti) that continuous, symmetric, reflexive, and bisymmetric operations are quasi-arithmetic means (of the form $F(x,y) = f^{-1}(\frac{f(x)+f(y)}{2})$ ).
- Regularity Improvement: Later work (Burai, Kiss, Szokol) showed that if an operation is reflexive (i.e., $F(x,x)=x$ ), symmetric, bisymmetric, and strictly increasing, continuity is automatic. Thus, reflexivity acts as a "regularizing" condition that forces the operation to be continuous and quasi-arithmetic.
The Gap: The status of non-reflexive operations remained unclear. Specifically, for operations of the weighted quasi-sum form $F(x,y) = f^{-1}(\alpha f(x) + \beta f(y))$ where $\alpha + \beta \neq 1$ , it was unknown whether continuity could be derived from bisymmetry and monotonicity alone, or if discontinuous examples existed.
Core Question: Can one construct a bisymmetric, strictly increasing (and potentially symmetric) binary operation on a real interval that is discontinuous?

2. Methodology

The author employs a constructive approach using set-theoretic and algebraic independence to build counterexamples.

Algebraic Independence: The construction relies on a countable subfield $\mathbb{F} \subset \mathbb{R}$ (generated by the parameters $\alpha, \beta$ and $\mathbb{Q}$ ). The core of the construction is a set $K$ of real numbers that is linearly independent over $\mathbb{F}$ .
Cantor-Type Perfect Sets:
- The author constructs a perfect, nowhere dense set $H_0$ (a Cantor-like set) within a closed interval such that every finite subset of $H_0$ is linearly independent over $\mathbb{F}$ .
- This set is iteratively expanded to form a larger set $H$ by taking linear combinations with coefficients $\alpha$ and $\beta$ : $H_{i+1} = H_i \cup (\alpha H_i + \beta H_i)$ .
- Key Property: Due to the linear independence and the condition $\alpha + \beta \neq 1$ , the set $H$ remains "thin" (nowhere dense) and does not contain any intervals.
The Generating Function ( $f$ ):
- A strictly increasing bijection $f: I \to H_1$ is constructed, where $H_1$ is a modified version of $H$ (specifically, $H$ with certain boundary points removed to ensure the operation is well-defined).
- This function $f$ maps a standard interval $I$ onto a "fractal-type" nowhere dense set $H_1$ .
The Operation: The operation is defined as:
$F(x, y) = f^{-1}(\alpha f(x) + \beta f(y))$
Because the image of $f$ is nowhere dense, the operation $F$ inherits the discontinuity of the mapping between the interval and the fractal set.

3. Key Contributions and Results

A. Existence of Discontinuous Bisymmetric Operations (Theorem 3.4)

The paper proves that for any $\alpha, \beta > 0$ with $\alpha + \beta \neq 1$ , there exists a bisymmetric, strictly increasing operation $F: I^2 \to I$ that is discontinuous.

Strong Discontinuity: The operation is not just discontinuous at a point; every section $F(x_0, \cdot)$ and $F(\cdot, y_0)$ is discontinuous everywhere on the interval.
Symmetry: If $\alpha = \beta$ , the constructed operation is also symmetric. This is a crucial result, as it shows that even symmetry does not force continuity in the absence of reflexivity.

B. Discontinuous Associative Operations (Theorem 3.8)

By setting $\alpha = \beta = 1$ , the author constructs an operation that is:

Associative: $F(F(x,y), z) = F(x, F(y,z))$ .
Strictly Increasing.
Discontinuous.
This provides a negative answer to the question of whether continuity is necessary for Aczél's characterization of associative quasi-sums.

C. One-Point Reflexivity (Remark 3.9)

The author demonstrates that even one-point reflexivity (where $F(c,c)=c$ for a single point $c$ ) is insufficient to force continuity. By extending the associative example to include a boundary point $0 $where$ F(0,0)=0$, the operation becomes reflexive at one point but remains discontinuous everywhere else.

D. Multivariate Extensions (Theorem 4.1)

The construction is generalized to $n$ -ary operations ( $n \geq 2$ ). For coefficients $\alpha_i > 0$ where $\sum \alpha_i \neq 1$ , there exist discontinuous, bisymmetric, strictly increasing $n$ -ary operations. This extends the counterexamples to higher dimensions.

E. The Two-Point Reflexivity Principle (Theorem 5.1)

In the "opposite direction," the paper proves a strong rigidity result:

If a symmetric, bisymmetric, and strictly increasing operation is reflexive at two distinct points ( $a$ $a$ and $b$ $b$ ), then:
1. It is continuous on the segment $[a, b]$ .
2. It coincides with a quasi-arithmetic mean on that segment.
Significance: This establishes a sharp dichotomy. One reflexive point is too weak to enforce regularity, but two reflexive points (in the symmetric case) are sufficient to restore full continuity and the standard quasi-arithmetic structure.

4. Significance and Implications

Refutation of Implicit Continuity: The paper definitively shows that continuity cannot be omitted from Aczél's characterization theorems for non-reflexive operations. Bisymmetry and strict monotonicity alone do not guarantee continuity, even if symmetry is present.
Regularity Dichotomy: The work highlights a critical threshold in the theory of means:
- Non-reflexive: "Wild" behavior is possible; discontinuous, bisymmetric, symmetric, and monotone operations exist.
- Reflexive (2 points): "Rigid" behavior is enforced; continuity and quasi-arithmetic representation are automatic.
Algebraic vs. Order-Theoretic: The results illustrate that purely algebraic properties (bisymmetry, associativity) combined with order properties (monotonicity) are insufficient to guarantee topological regularity (continuity) without the specific constraint of reflexivity at multiple points.
Open Problems: The paper leaves open the "Global Problem on Continuity" (Question 6.3): Is every reflexive (but not necessarily symmetric) bisymmetric operation continuous? The author notes that without symmetry, standard symmetrization techniques fail, making this a major open challenge.

Summary Conclusion

Gergely Kiss constructs a family of "pathological" operations using Cantor-type sets and linear independence to prove that bisymmetry and strict monotonicity do not imply continuity unless reflexivity is imposed at two points. This resolves a long-standing ambiguity in the theory of functional equations, clarifying that the regularizing power of bisymmetry is strictly dependent on the presence of reflexivity.