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GS-quasigroups. (English) Zbl 0715.20044

An idempotent quasigroup \((Q,\cdot)\) is said to be a golden section (GS-) quasigroup if it satisfies the identities \(a(ab\cdot c)\cdot c=b\), \(a\cdot (a\cdot bc)c=b\). The author introduces the notion of a parallelogram as a quadruple \((a,b,c,d)\in Q^ 4\) such that there are elements \(p,q\in Q\) such that \(ap=bq\), \(dp=cq\). After considering the space of all parallelograms the representation theorem for GS-quasigroups is deduced. This theorem asserts that a GS-quasigroup \((Q,\cdot)\) exists even together with a commutative group \((Q,+)\) possessing an automorphism \(\phi\) satisfying \((\phi\circ\phi)-\phi-id=0\). The correspondence between GS-quasigroups and their associated commutative groups is described in detail.
Reviewer: V.Havel

MSC:

20N05 Loops, quasigroups
51A15 Linear incidence geometric structures with parallelism
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