Direct product of affine partial linear spaces, general approach (Q811832)

Let \({\mathcal Y}=(Y_i : i \in I)\) be an indexed family of sets. An agreement of \({\mathcal Y}\) is a function \(\varepsilon\) which assigns to every pair \((i,j) \in I \times I\) a bijection \(\varepsilon_{i,j}: Y_i \to Y_j\) such that \(\varepsilon_{i,i} = Id_{Y_i}\), \((\varepsilon_{i,j})^{-1}=\varepsilon_{j,i}\) and \(\varepsilon_{k,j} \circ \varepsilon_{i,k} = \varepsilon_{i,j}\) for all \(i,j,k \in I\). A partial linear space is a point-line incidence structure with the property that every line is incident with at least two points and every two points are incident with at most one line. A partial linear space with parallelism is a partial linear space together with an equivalence relation (parallelism) on the line set such that for every line \(L\) and every point \(x\), there exists a unique line through \(x\) parallel with \(L\). Let \((U_i:i \in I)\) be a family of partial linear spaces with parallelism and fix an agreement \(\delta\) of the family of sets of directions of lines of the \(U_i\). This enables us to consider parallelism between lines of different partial linear spaces. We can now construct the so-called direct product \(U\) of the family \((U_i : i \in I)\). The point set of \(U\) is the cartesian product of the point-sets of the \(U_i\). The line-set of \(U\) is the subset of the cartesian product of the line-sets of the \(U_i\) consisting of sequences of mutually parallel lines. Incidence is the natural one. The paper under review deals with the following kind of problem. Suppose the \(U_i\)'s satisfy property (P), does the direct product \(U\) also satisfy property (P)? The paper offers an answer for various possibilities of this property (P).