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arXiv · 2607.27872

The nucleus of a semisymmetric quasigroup

Andrew Richard Kozlik
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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.

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Andrew Richard Kozlik. 2026-07-30. The nucleus of a semisymmetric quasigroup. https://arxiv.org/abs/2607.27872

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