The convolution of a partial Steiner triple system and a group

Abstract

We define the operation of convolution, which associates with a partial Steiner triple system and an abelian group a new partial Steiner triple system, and we determine conditions under which some fundamental geometric properties remain invariant under the operation of convolution.

Semi-Pappus configurations; combinatorial generalizations of the Pappus configuration

“…Each y ∈ {0, 1, 2} yields the For every 3-element line L the structure L C 2 is a Veblen configuration (cf. [8]). Therefore, each line L of P n yields a Veblen subconfiguration of M n = P n C 2 (comp 2.3).…”

“…[8]) that a Veblen subconfiguration of M n either has the form L C 2 , or it is a suitably indexed Veblen subconfiguration of P n . In view of 3.2, the class of Veblen subconfigurations of M n is the set V = {L C 2 : L is a line of P n }.…”

Combinatorial generalizations of generalized quadrangles of order (2, 2)

“…In another representation, R is the convolution R = G 2 (X ) C 2 of the combinatorial Grassmannian and the group C 2 , where |X | = n (cf. [10]). Let us consider an embedding ν of R into the projective space PG(n, 2); as in 2.6, every Veblen subconfigu- [10]) then the obtained structure of improper points is the convolution H 2 (X ) C 2 enlarged by a single point and a family of n 4 lines through this point.…”

Elementary Characterizations of Some Classes of Reducts of Affine Spaces

“…Such a generalization was introduced e.g. in [12] (reducts of even more general type were studied in [5,10]). Let A = F n , L, be the n-dimensional (analytical) affine space defined over a division ring F. For 1 ≤ t < n we denote by H t the family of the t-hyperplanes of A.…”

“…Let us fix L ∈ H k for some integer 1 ≤ k < n. After that let us define L * = {M ∈ L: L M } and L = L \ L * . The structure AS(n, k, F) = A− − (∅, L * ) = F n , L was called in [12] an affine slit space 1 . Let H t be the class of the t-dimensional strong subspaces of AS(n, k, F); clearly, H t ⊂ H t .…”