Gabrieli, Elisabetta; Karzel, Helmut Point-reflection geometries, geometric K-loops and unitary geometries. (English) Zbl 0922.51006 Result. Math. 32, No. 1-2, 66-72 (1997). 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) Cited in 1 ReviewCited in 9 Documents MSC: 51F15 Reflection groups, reflection geometries 20N05 Loops, quasigroups Keywords:loops; reflection structures Citations:Zbl 0632.51019 PDFBibTeX XMLCite \textit{E. Gabrieli} and \textit{H. Karzel}, Result. Math. 32, No. 1--2, 66--72 (1997; Zbl 0922.51006) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.