Direct products of affine partial linear spaces
type: Item
instance of: scholarly article
title: Direct products of affine partial linear spaces
zbMATH Open document ID: 0873.51001
author: Norman L. Johnson
author: Theodore G. Ostrom
published in: Journal of Combinatorial Theory. Series A
publication date: 1997-09-28T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51A15
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51E14
zbMATH DE Number: 909616
zbMATH Keywords: affine partial linear space
zbMATH Keywords: parallelism
zbMATH Keywords: translation nets
reviewed by: Helmut Salzmann
MaRDI profile type: Publication
OpenAlex ID: W2029555141
DOI: 10.1006/JCTA.1996.0066
description: scientific article; zbMATH DE number 909616