Projective remoteness planes (Q1912364)

The author gives a set of axioms involving incidence and remoteness (the negation of neighboring) leading to a class of Barbilian planes called projective remoteness planes. It is shown that a transvection projective remoteness plane \(P\) can be coordinatized by a ring \(R\) which is a two-sided inverse ring. Moreover it is proved that \(P\) is a transvection plane iff \(P\) is isomorphic to \(P(G,N)\), the plane associated with a group \(G\) of Steinberg type parametrized by \(R\) and with \(N\) a certain subgroup of \(G\). The author investigates also necessary and sufficient conditions on \(P\) for \(R\) to be alternative, associative or commutative. The notion of elementary \(n\)-basis set is introduced and constructions of projective and affine remoteness planes are given. Finally a plane with reflections determining a system of rotations is shown to have commutative, associative coordinates.