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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
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