Journal of Generalized Lie Theory and Applications

Bruck decomposition for endomorphisms of quasigroups

Abstract

In 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 $e$.

Article information

Source
J. Gen. Lie Theory Appl., Volume 3, Number 3 (2009), 191-196.

Dates
First available in Project Euclid: 6 August 2010

https://projecteuclid.org/euclid.jglta/1281106538

Digital Object Identifier
doi:10.4303/jglta/S090305

Mathematical Reviews number (MathSciNet)
MR2534024

Zentralblatt MATH identifier
1179.20062

Subjects