The convolution of a partial Steiner triple system and a group (Q860237)

An operation called convolution of a partial Steiner triple system and an abelian group is defined as follows. For a partial Steiner triple system \({\mathcal M} = (S, {\mathcal L})\) and an abelian group \(G = (G, +,0)\) a new incidence structure on the cartesian product \(X = S \times G\) is declared by defining the set \(\mathcal G\) of blocks as \(\{(a_1, g_1),\dots,(a_{m},g_{m})\} \in {\mathcal G}\) if and only if \(\{a_ {1},\dots,a_{m}\} \in {\mathcal L}\), \(a_{1},\dots,a_{m}\) are pairwise distinct, and \(g_{1} + \dots + g_{m} = 0\). The result \(\mathcal K\) again is a partial Steiner triple system. Conditions are investigated under which fundamental geometric properties are preserved under the operation of convolution. For example, if the convolution \(\mathcal K\) is Vebleian, Pappian, Desarguesian, or Fanoian, the same is true for \( \mathcal M\). If \(\mathcal M\) is Vebleian and \(2G = 0\), then \(\mathcal K\) is Vebleian as well. If \(\mathcal M\) is Pappian, so is \(\mathcal K\). If \(6G = 0 \), \(\mathcal M\) is Desarguesian and Fanoian, then \(\mathcal K\) is Desarguesian. The paper contains a lot more results in this spirit.