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Unraveling Functional Equations in Composition Algebra: Resolving Conjectures and Examining Implications

Daniel Kawai, Bruno Leonardo Macedo Ferreira

Abstract

We address the conjectures left by the recent article by Ferreira et al. titled ``Commuting maps and identities with inverses on alternative division rings.'' We also present an example showing the necessity of the conditions of the results that answer the conjectures.

Unraveling Functional Equations in Composition Algebra: Resolving Conjectures and Examining Implications

Abstract

We address the conjectures left by the recent article by Ferreira et al. titled ``Commuting maps and identities with inverses on alternative division rings.'' We also present an example showing the necessity of the conditions of the results that answer the conjectures.
Paper Structure (4 sections, 6 theorems, 103 equations, 1 figure)

This paper contains 4 sections, 6 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Alterntative Division Rings
  4. Split Octonion Algebras

Key Result

Theorem 1.1

Let $D$ be a noncommutative alternative division ring with characteristic different from $2$, and let $f : D \rightarrow D$ be an additive map satisfying the identity vukman: for for every $x \in D^\times$. Then $f(x) = 0$ for all $x \in D$.

Figures (1)

  • Figure 1: Multiplicative table of $\mathbb{O}_{\mathfrak{s}}$

Theorems & Definitions (14)

  • Theorem 1.1
  • Conjecture 1.1
  • Conjecture 1.2
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Definition 4.1
  • ...and 4 more