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Left quasigroups and reductive spaces. (English) Zbl 0870.20052
Let $$(Q,\cdot,\setminus)$$ be a left $$F$$-quasigroup with the defining left $$F$$-identity $$x\cdot(y\cdot z)=(x\cdot y)\cdot(ex\cdot z)$$ where $$ex=x\setminus x$$. This quasigroup is called correct if the right translation $$\rho_{ex}\colon y\to y\cdot ex$$ is invertible.
It is shown that any correct left $$F$$-quasigroup generates a unique reductive space with a specific reductant. This space is called natural $$F$$-space of the quasigroup. Any reductive $$F$$-space may be considered, in a canonical way, as a correct left $$F$$-quasigroup whose natural $$F$$-space is isomorphic to the initial one. The work is related to a paper by L. V. Sabinin and L. Sabinina [Algebras Groups Geom. 12, No. 2, 127-137 (1995; Zbl 0831.20097)].
##### MSC:
 20N05 Loops, quasigroups 53A60 Differential geometry of webs