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On the gyrogroups of Ungar. (English. Russian original) Zbl 0865.20054

Russ. Math. Surv. 50, No. 5, 1095-1096 (1995); translation from Usp. Mat. Nauk 50, No. 5, 251-252 (1995).
The author shows that the “gyrogroups” of A. Ungar [Aequationes Math. 47, No. 2-3, 240-254 (1994; Zbl 0799.20032)] are Bruck loops, i.e. Bol loops characterized by \(x+(y+(x+z))=[x+(y+x)]+z\) which satisfy also the Bruck identity \(x+(y+(x+z))=(x+y)+(x+y)\) (a result which was also established by A. Kreuzer [J. Geom. 47, No. 1-2, 86-93 (1993; Zbl 0788.20035)]) and that some of Ungar’s axioms are superfluous. Moreover he remarks that Ungar’s example \((D_c,\oplus)\) of the gyrogroup in the complex disk \(D_c:=\{x\in\mathbb{C}\mid|x|<c\}\), \(c>0\) with \(x\oplus y:=(x+y)c^2\cdot(c^2+\overline xy)^{-1}\) was also given by A. I. Nesterov in his dissertation in 1989.

MSC:

20N05 Loops, quasigroups
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