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Point-reflection geometries, geometric K-loops and unitary geometries. (English) Zbl 0922.51006

The authors use unitary geometries to show the existence of reflection geometries to which correspond \(K\)-loops with an incidence fibration \(F\) [E. Zizioli, J. Geom. 30, 144-156 (1987; Zbl 0632.51019)] where \(F\) consists of proper subloops (other than groups). This is in sharp contrast to ordinary hyperbolic spaces, where those subloops are even commutative subgroups.
Reviewer: H.Havlicek (Wien)

MSC:

51F15 Reflection groups, reflection geometries
20N05 Loops, quasigroups

Citations:

Zbl 0632.51019
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References:

[1] F.B Achmann: Zur Begründung der Geometrie aus dem Spiegelungsbegriff. Math. Ann. 123, 341-344, (1951). · Zbl 0043.35106
[2] H. Karzel: Recent developments on absolute geometries and algebraization by K-loops. Submitted to Discrete Mathematics. (1996). · Zbl 0941.51002
[3] H. Karzel, A. Konrad: Reflection groups and K-loops. J.Geom. 52, 120-129, (1995). · Zbl 0821.51011
[4] C.F. Manara, M. MarchI: On a class of reflection geometries. 1st. Lomb. Rend. Sc. A 125,2, 203-217, (1991) · Zbl 0798.51017
[5] H.A. Schmidt: Die Dualität von Inzidenz und Senkrechtstehen in der absoluten Geometrie. Math. Ann. 118, 609-625, (1941/1943). · Zbl 0027.41801
[6] G. Thomsen: Grundlagen der Elementargeometrie in gruppenalgebraischer Behandlung. Hamburger Math. Einzelschr. 15, (1933) · JFM 59.0549.01
[7] H. Wiener: Über Grundlagen und Außau der Geometrie. Jber DMV 1, 45-48, (1890/91).
[8] H. Wiener: Sechs Abhandlungen über das Rechnen mit Spiegelungen. Abdrucke aus den Ber. der math.-phys. Kl. der Kgl. Sachs. Ges. der Wiss. 1890, 1891, 1893, Leipzig 1893.
[9] E.Zizioli: Fibered incidence loops and kinematic loops. J.Geom. 30, 144-156, (1987). · Zbl 0632.51019
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