Affine geometry over free unitary modules
Let P be a set and L be a non-empty set of non-empty subsets of P. The elements of P are called ''points'' and those of L ''lines''. It is also assumed the existence of an antireflexive relation, non-neighboured, on P, and of an equivalence relation and parallelism on L, denoted by \(\phi\), \(\|\), respectively. The authors introduce appropriate geometrical axioms on the quadruplet \((P,L,\phi,\|)\) in order to define successively the concepts of generalized affine incidence structure (GAIS), generalized affine spaces (GAS), and generalized affine translation spaces (GATS), with a structure theorem being proven for this last. The main theorem reads as follows: The class of all affine Barbilian-spaces coincides with the class of all GATS which satisfy, in addition, the stretching-axiom, namely: for all p, q, r in P, with \(p\neq q\) and r in \(\overline{pq}\), there exists a p-dilatation mapping q onto r.
scientific article; zbMATH DE number 3929830