The nucleus of a semisymmetric quasigroup

Abstract

A binary operation $\cdot$ which satisfies the identity $(x\cdot y)\cdot x=y$ is called a semisymmetric quasigroup. We show that the nucleus of a semisymmetric quasigroup is either empty or an elementary abelian 2-group coinciding with the centre, and that a semisymmetric quasigroup with a non-empty nucleus is necessarily a Mendelsohn loop, i.e. the loop associated with a Mendelsohn triple system. We derive necessary and sufficient conditions for the existence of a semisymmetric quasigroup of order $n$ with nucleus of order $m$. Furthermore, we characterize the nuclear elements of a Mendelsohn loop in terms of a particular orientation of the Pasch configuration in the associated triple system.

arXiv commentv2: simplified proof of Theorem 3.2, added acknowledgement