Direct products of affine partial linear spaces (Q1919671)

An \textit{affine} partial linear space is one with a parallelism partitioning the set of lines such that each parallel class simply covers the point set; it is called a \textit{net} if any two non-parallel lines intersect. In order to define a direct product of a family of affine partial linear spaces \(A_k\), for each \(k\) a bijection \(\sigma_k\) must be specified from the set \(C_k\) consisting of the parallel classes of \(A_k\) to a fixed such set \(C_0\). The product \(\prod_k A_k\) is again an affine partial linear space, it is a net if each \(A_k\) is a net, and it is said to be regular if each \(\sigma_k\) is an isomorphism \(A_k \cong A_0\). Coordinatizations are introduced. The long Section 4 deals with translation nets and Baer subplanes. A direct product or direct sum of (abelian) translation nets is again an (abelian) translation net. If some point of an abelian translation net \(M\) is contained in \(3\) distinct Baer subplanes, then \(M\) is a regular direct product. Powers of finite translation planes admit fairly large collineation groups (Theorem 5.5). Each power of a Desarguesian plane is covered by subplanes with a common point, and any subplane-covered net is a sum of identical Desarguesian planes. Direct sums of subplanes in an André plane are studied in Section 7. Finally, there is a clever construction of nets which cannot be extended to an affine plane.