Journal of Generalized Lie Theory and Applications

Bruck decomposition for endomorphisms of quasigroups

Péter T. Nagy and Peter Plaumann

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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

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

First available in Project Euclid: 6 August 2010

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20N05: Loops, quasigroups [See also 05Bxx]

Group theory Generalizations of groups Loops Quasigroups


Nagy, Péter T.; Plaumann, Peter. Bruck decomposition for endomorphisms of quasigroups. J. Gen. Lie Theory Appl. 3 (2009), no. 3, 191--196. doi:10.4303/jglta/S090305.

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