Bruck decomposition for endomorphisms of quasigroups

TL;DR

This paper explores the structure of certain quasigroups using Bruck's extension method, focusing on cases where the endomorphism map's image forms a group, and characterizes these quasigroups within specific varieties.

Contribution

It introduces a characterization of quasigroups with endomorphic $e$-maps via Bruck decomposition, expanding understanding of their algebraic structure.

Findings

Constructs examples of LF-quasigroups with group images of $e$

Characterizes quasigroups where $e$ is an endomorphism

Describes subvarieties with group images of $e$

Abstract

In the year 1944 R. H. Bruck has described a very general construction method which he called the extension of a set by a quasigroup. We use it to construct a class of examples for LF-quasigroups in which the image of the map $e(x)=x\backslash x$ is a group. More generally, we consider the variety of quasigroups which is defined by the property that the map $e$ is an endomorphism and its subvariety where the image of the map $e$ is a group. We characterize quasigroups belonging to these varieties using their Bruck decomposition with respect to the map $e$.