Self-orthogonal semisymmetric quasigroups

Parastrophic-orthogonal Quasigroups

Annotated translation of Parastrophic-orthogonal quasigroups, Acad. Nauk Moldav. SSR, Inst. Mat.s Vychisl. Tsentrom, Kishinev, 1983, prepared by A.D.Keedwell and P.Syrbu based on the original Russian and on an earlier English translation supplied to the rst author by Belousov himself. The notion of orthogonality plays an important role in the theory of Latin squares, and consequently also in th...

Quasigroups, right quasigroups and category coverings

The category of modules over a fixed quasigroup in the category of all quasigroups is equivalent to the category of representations of the fundamental groupoid of the Cayley diagram of the quasigroup in the category of abelian groups. The corresponding equivalent category of coverings, and the generalization to the right quasigroup case, are also described.

Semisymmetric graphs from polytopes

Every finite, self-dual, regular (or chiral) 4-polytope of type {3, q, 3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edgebut not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.

A guide to self-distributive quasigroups, or latin quandles

We present an overview of the theory of self-distributive quasigroups, both in the two-sided and one-sided cases, and relate the older results to the modern theory of quandles, to which self-distributive quasigroups are a special case. Most attention is paid to the representation results (loop isotopy, linear representation, homogeneous representation), as the main tool to investigate self-dist...

Self-orthogonal designs