A Note on Embeddable F2-Geometries

✍ Scribed by Stefan Heiss

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 122 KB
• Volume: 92
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1999.3051

✦ Synopsis

An embeddable F 2 -geometry 1 with embedding rank er(1)=4 is given, which has no generating set of size 4.

2000 Academic Press

1. Introduction

Let 1=(P, L) be a point-line geometry with points P and lines L ( P 3 ) (i.e., every line has three points). A subset S P is called a subgeometry of 1 if S contains every line l # L with |l & S| >1. A subset G P is said to generate 1, if P is the only subgeometry of 1 containing G. The generating rank gr(1) is the minimal size of a generating set of points of 1.

Next, identify P with a basis of some vector space V over F 2 , i.e., V= { :

For v # V the weight of v= ? p p is given by wt(v)= |[ p # P | ? p {0]|. Every line l=[ p, q, r] # L defines a vector l = p+q+r of weight three in V. Finally let C=(l | l # L) and E=VÂC. Then 1=(P, L) is an embeddable F 2 -geometry if and only if wt \ : l # L * l l + 2 O : l # L * l l =0. (V)