Non-commutative spaces with transitive translation groups (Q753137)

The purpose of this paper is to investigate skewaffine spaces with transitive translation group. A skewaffine space is like an affine space but the line joining x and y might be different from the line joining y and x. A typical example is given by a group G acting transitively on a set X. The line joining x and y, x,y\(\in X\), is the orbit of (x,y) under G. If \(X=G\) this is called the group space \({\mathcal F}(G).\) The central objects in the paper under review are the ``coarsenings''. This is a method to construct new spaces by using equivalence relations on the set of ideal points. There are many results on these coarsenings which yield characterizations of the group spaces above. For example: A space possesses a transitive translation group G iff it is a coarsening of the space \({\mathcal F}(G)\). Furthermore one can show that the strongly skewaffine spaces with transitive translation group G are 1-1 correspondence with the Schur rings over G. This brings new methods into this interesting area of geometry.