Bruck decomposition for endomorphisms of quasigroups

Peter T. NAGY and Peter PL

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)=xnx is a group. More generally, we consider the variety of quasigroups which is dened 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.