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Geometry of quasigroups. (English) Zbl 0723.20041
Quasigroups and loops: theory and applications, Sigma Ser. Pure Math. 8, 197-203 (1990).
[For the entire collection see Zbl 0704.00017.]
This paper does not contain new issues, but summarizes results of the author’s and K. Strambach’s earlier articles [Adv. Math. 49, 1-105 (1983; Zbl 0518.20064); Math. Z. 185, 465-485 (1984; Zbl 0543.51006)].
The main starting notions are: 3-web \({\mathcal W}\), its associated quasigroup or loop \({\mathcal Q}\), group \(\Pi\) of all projectivities in \({\mathcal W}\) and collineation group \(\Sigma\) of \({\mathcal W}\). The first part is devoted to von Staudt’s point of view, concretely the connections between some properties of \({\mathcal Q}\) and their equivalents of \(\Pi\) are shown. In the second part Klein’s point of view is followed and the connections between some properties of \({\mathcal Q}\) and corresponding properties of \(\Sigma\) are exhibited.
MSC:
20N05 Loops, quasigroups
53A60 Differential geometry of webs
51A45 Incidence structures embeddable into projective geometries