Left Distributive Quasigroups and Gyrogroups

J. Math. Sci. Univ. Tokyo
Vol. 8 (2001), No. 1, Page 1--16.

Issa, A. Nourou
Left Distributive Quasigroups and Gyrogroups
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The connection between gyrogroups and some types of left distributive quasigroups is established by means of isotopy considerations. Any quasigroup of reflection is isotopic to some gyrocommutative gyrogroup and any left distributive quasigroup satisfying some specific condition is isotopic to some nongyrocommutative gyrogroup. The geometry of reductive homogeneous spaces and the semidirect product for homogeneous loops are used to produce local exact decompositions of groups and local gyrocommutative gyrogroups.

Keywords: Quasigroups, loops, gyrogroups, homogeneous symmetric spaces, semidirect and gyrosemidirect products, gyrovector spaces

Mathematics Subject Classification (2000): Primary 20N05; Secondary 53C35
Mathematical Reviews Number: MR1818902

Received: 2000-05-30