in a fixed dimension $m$. In case $m=0$ and $k$ is the codimension of $W$ we get a pretty well known space of linear complements (cf. [@lincomp] and also [@blunck-havlicek]).

Now we turn back to our major question: is it possible to recover the underlying space $\A$ and the horizon $\W$ in the complement $\fixoutf(\A,\W)$? This is not a completely new question and there are some papers devoted to such recovery problem. In [@slitgras] projective Grassmannians are successfully recovered from complements of their Grassmann substructures. The concept of two-hole slit space is introduced in [@sulima]. It is a point-line space whose point set is the complement of the set of points of two fixed complementary subspaces, not hyperplanes, in a projective space and the line set is the set of all those lines which do not intersect any of these two subspaces. This is not exactly what we have used to call a complement as lines are taken differently. Nevertheless, using very similar methods to ours the recovery problem has been solved here in incidence geometry settings. It has been also solved in chain geometry associated with a linear group approach. In [@partaffine] a partial linear space with parallelism embeddable 