2026/07/30 by Andrew Richard Kozlik
Mathematics · #math.CO #msc:05B07 #msc:20N05
arxiv created 2026/07/30 · arxiv updated 2026/07/31
A binary operation ⋅ which satisfies the identity (x ⋅ y) ⋅ 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.