$k$-seminets $(T,L_1,\dots,L_k)$ are introduced in [1], as a generalization of $k$-nets [4-5]. They are closely related to a special orthogonal system of partial quasigroups [1], as well as to the special codes [8-12], and to $r$-designs [13]. The article gives a characterization of $k$-seminets by means of $(T,L,\|)$-type objects, where $T=\emptyset$, $L\subset P(T)\backslash\{\emptyset\}$, $\|\subseteq L^2$, and $\|$ satisfies the axiom of Euclidean parallelism. Since affine planes and affine Sperner spaces are the objects of the same type, they are thus special $k$-seminets [2, 4-5]. Finally, it is shown that besides $k$-seminets $(T,L,\|)$ which are affine planes and affine Sperner spaces, there are other $k$-semlnets in which every two different points are collinear.