Polar graphs and corresponding involution sets, loops and Steiner triple systems.
This paper is part of the authors' investigations on regular involution sets \((P,\mathcal I)\), complete graphs with parallelism \((P,\mathcal E,\parallel)\) and involutorial difference loops \((P,+)\). After the section of notations and known results, the authors study characteristic configurations, graphs with polar triangles, substructures and automorphisms, and invariant polar involution sets and corresponding graphs and loops. In the last section they make the further assumption that the polar involution set \((P,\mathcal I)\) is invariant. In the paper there are the results: the polar graph \((P,\mathcal E,\parallel)\) satisfies the trapezium axiom, is self-distributive, \((P,+)\) is a Moufang loop, is an affine triple system; and: \((P,\mathcal E,\parallel)\) satisfies the quadrangle axiom, is a group, is an affine space.
scientific article; zbMATH DE number 5117891