Pseudo-embeddings and pseudo-hyperplanes

We generalize some known results regarding hyperplanes and projective embeddings of point-line geometries with three points per line to geometries with an arbitrary but finite number of points on each line. In order to generalize these results, we need to introduce the new notions of pseudo-hyperplane, (universal) pseudo-embedding, pseudo-embedding rank and pseudo-generating rank. We prove several connections between these notions and address the problem of the existence of (certain) pseudo-embeddings. We apply this to several classes of point-line geometries. We also determine the pseudo-embedding rank and the pseudo-generating rank of the projective space PG(n; 4) and the affine space AG(n; 4).