Extended triple systems: Geometric motivations and algebraic constructions (Q1808771)

The author first reviews the basic facts about extended triple systems (ETSs) which generalise Steiner triple systems in the sense that multiple points are allowed. An algebraic interpretation of such triples turns an ETS into a symmetric quasigroup or, by adding an extra point, into a loop. Such an algebraic approach allows to prove structure theorems for entropic (abelian) ETSs. Furthermore, by replacing the abelian group by a commutative Moufang loop with a suitable choice of the added element, the entropic ETSs are obtained and a characterisation, via a 3-identity, of their related symmetric quasigroups is provided. Links with cubic surfaces are pointed out which show that an ETS can be obtained which splits into a direct product of a binary ETS and some Hall triple system \(H\). In the case \(H\) is the well-known 81-point system constructed by M. Hall jr., the author proves that there are exactly three non-entropic quasigroups of minimum order 81 related to cubic surfaces. Finally, exterior algebras are used to describe a subcategory of Hall triple systems.