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.

Disclosure

“ents The author would like to thank Aleš Drápal for pointing out the possible con- nection of Lemma 3.5 with LCC loops and for suggesting the investigation of the binary operation in Proposition 3.9. The author acknowledges the use of Claude Fable 5 (Anthropic) in developing the statements and proofs of Propositions 3.10 and 3.12, the discussion following Proposition 2.5 and in conducting the literature review, which brought the papers [7], [8] and [14] to the author’s attenti”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T19:13:18.583699Z

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