The Structure of Non-Associative Finite Invertible Loops (NAFIL)*

TL;DR

This paper explores the structure of finite loops called NAFIL, focusing on their classification into composite and non-composite types, and introduces block products as a key class of composite NAFIL loops.

Contribution

It provides a structural analysis of NAFIL loops and introduces block products as a significant class of composite NAFIL loops.

Findings

NAFIL loops are classified into composite and non-composite types.

Block products are introduced as a new class of composite NAFIL loops.

Structural properties of NAFIL loops are analyzed.

Abstract

The NAFIL is a finite loop in which every element has a unique (two-sided)inverse. NAFIL loops can be classified into two types: composite (with at least one non-trivial subsystem) and non-composite or plain (without any non-trivial subsystem). This paper deals with the structure of these loops. In particular we shall introduce an important class of composite NAFIL loops called block products.