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An extension formula for right Bol loops arising from Bol reflections

Mario Galici, Gabor P. Nagy

Abstract

We study a new extension formula for right Bol loops. We prove the necessary or sufficient conditions for the extension to be right Bol. We describe the most important invariants: right multiplication group, nuclei, and center. We show that the core is an involutory quandle which is the disjoint union of two isomorphic involutory quandles. We also derive further results on the structure group of the core of the extension.

An extension formula for right Bol loops arising from Bol reflections

Abstract

We study a new extension formula for right Bol loops. We prove the necessary or sufficient conditions for the extension to be right Bol. We describe the most important invariants: right multiplication group, nuclei, and center. We show that the core is an involutory quandle which is the disjoint union of two isomorphic involutory quandles. We also derive further results on the structure group of the core of the extension.

Paper Structure

This paper contains 10 sections, 17 theorems, 74 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Loop folders
  4. Moufang loops by Chein extension
  5. Nets and collineations
  6. Bol reflections
  7. Algebraic properties of the extension
  8. Nuclei and center of the extension
  9. The core of the extension of a right Bol loop
  10. Acknowledgement

Key Result

Theorem 1.1

$\widetilde{L}$ is a right Bol loop if and only if $L$ is a right Bol loop with $x^2\in Z(L)$ for every $x\in L$. Moreover, the following are equivalent:

Figures (1)

  • Figure 1: Bol reflection through the line $h_d$

Theorems & Definitions (38)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Chein
  • Definition 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 28 more