Introduction: Non-Associative Finite Invertible Loops

TL;DR

This paper introduces non-associative finite invertible loops (NAFIL), exploring their properties and significance across various mathematical and physical fields, serving as a foundation for further theoretical development.

Contribution

It provides an introductory overview of NAFIL loops, highlighting their diverse applications and setting the stage for future research in their theoretical framework.

Findings

NAFIL loops include IP, Moufang, and Bol loops

NAFIL loops are involved in combinatorics, geometries, and physics

This paper initiates the study of NAFIL loop theory

Abstract

Non-associative finite invertible loops (NAFIL) are loops whose every element has a unique two-sided inverse. Not much is known about the class of NAFIL loops which includes the familiar IP (Inverse Property), Moufang, and Bol loops. Our studies have shown that they are involved in such diverse fields as combinatorics, finite geometries, quasigroups and related systems, Cayley algebras, as well as in theoretical physics. This paper presents and introductions to the class of NAFIL loops as the starting point for the development of the theory of these interesting structures.