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Zbl 0715.20044
Volenec, Vladimir
GS-quasigroups. (English)
[J] Čas. Pěstování Mat. 115, No. 3, 307-318 (1990). ISSN 0528-2195

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 $(a,b,c,d)\in Q\sp 4$ 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.
[V.Havel]

MSC 2000:: *20N05 Loops (group theory)
51A15 Geometric structures with parallelism

Keywords: golden section quasigroups; idempotent quasigroups; parallelograms; representation theorem; GS-quasigroups; automorphisms; associated commutative groups

Cited in: Zbl 1101.20042 Zbl 1073.20062 Zbl 1016.20052 Zbl 0808.20056

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