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The class of semisymmetric quasigroups is determined by the identity (yx)y = x. We prove that the universal multiplication group of a semisymmetric quasigroup Q is free over its underlying set and then specify the pointstabilizers of an action of this free group on Q. A theorem of Smith indicates that Beck modules over semisymmetric quasigroups are equivalent to modules over a quotient of the integral group algebra of this stabilizer. Implementing our description of the quotient ring, we provide some examples of semisymmetric quasigroup extensions. Along the way, we provide an exposition of the quasigroup module theory in more general settings.

We define a Mendelsohn triple system (MTS) with selfdistributive quasigroup multiplication and order coprime with 3 to be distributive, non-ramified (DNR). We classify, up to isomorphism, all DNR MTS and enumerate isomorphism classes (extending the work of Donovan, Griggs, McCourt, Opršal, and Stanovský). The classification is accomplished via the representation theory of the Eisenstein integers,Containing the class of DNR MTS is that of MTS with an entropic (linear over an abelian group) quasigroup operation. Partial results on the classification of entropic MTS with order divisible by 3 are given, and a complete classification is conjectured. We also prove that for any entropic MTS, the qualities of being non-ramified, pure, and self-orthogonal are equivalent. We introduce the varieties RE and LE of (resp. right and left) Eisenstein quasigroups, whose respective linear representation theories correspond to the alternative presentation Z[X]/(X 2 + X + 1) of the Eisenstein integers.

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