Extended triple systems: geometric motivations and algebraic constructions

Abstract

Extended triple systems (or ETSs for short) generalize the Steiner triple systems: they are provided with a collection of (unordered) triples (( x, y, z)) in which multiple points are allowed. We still have the characterizing fact that any pair of points ( x, y) lies in a unique (( x, y, z)). This notion is thereby perfectly suitable for describing the situation of the cubic curves or cubic surfaces. The triples may be set under the form ( x, y, x∘ y) and then the mid-point binary law x∘ y makes the set of points into a totally symmetric quasigroup. By choosing an origin u one gets some loop operation x∗y=u∘(x∘y). This algebraic approach is used so as to state structure theorems for special subcategories; for instance the entropic (or abelian) ETS, whose triples can be set under the form ( x, y, a− x− y) in the underlying set of some abelian group. By replacing the abelian group by some commutative Moufang loop in which the fixed element a is central with respect to the associativity, we obtain the wider category of the terentropic ETS. A 3-identity characterization of their related symmetric quasigroups is given. We call them Manin quasigroups. One may restate Manin's structure theorems in combinatorial terms as follows: in a suitable factor set of a cubic hypersurface the three-place relation of collinearity gives rise to an ETS which splits as a direct product B× H of a binary ETS B by some Hall triple system H. The difficult problem of finding eventually a cubic hypersurface whose related ETS is not entropic is not answered yet, as far as we know. But it may be reduced to the finding a surface whose related ETS is the famous 81-point triple system constructed by Marshall Hall Jr. We show that there exists exactly three non entropic Manin quasigroups of (minimum) order 81. Besides we present an exterior-algebra process that can be used for describing an important subcategory of Hall triple systems.