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The Geometry of Point Reflections and Quasigroups

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Abstract

This is a survey of results from the rich theory of medial and more general quasigroups and its close connection with the geometry of point-reflections. The emphasis is on questions regarding the simplest axiomatization, in the sense of the minimal number of variables appearing in the identtties, on dependence or independence of axioms, and on representation theorems in the style of Toyoda.

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Notes

  1. M. Takasaki [48] has provided (as reported by Y. Kawada in his Zentralblatt review of [48]) a model satisfying the axioms S1, S4, S11, and S12, but not S6 (his multiplication is written such that ab is meant to signify the reflection of a in b, so that all of his axioms need to be re-written so that all xy become \(y\cdot x\)). Structures satisfying S1, S4, and S11 have been recently considered in [10].

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Acknowledgements

This paper was written while the second author was a Fulbright Scholar at Yerevan State University. The first author’s research was partially supported by the State Committee of Science of the Republic of Armenia, Grants 10-3/1-41, 18T-1A306. Thanks are due to Stephan Schulz for assistance with the automatic theorem prover E, to Tomáš Kepka for pointing out the existence of nonassociative commutative Moufang loops of order 81, to Michael Kinyon for having provided the model of independence of S6 mentioned in Sect. 3.4 and for having answered several questions, and to the anonymous referee for many improvements and for having provided greatly simplified proofs carried out by the automatic theorem prover PROVER9.

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  1. Faculty of Mathematics, Yerevan State University, Alex Manoogian 1, Yerevan, Armenia

    Yuri Movsisyan

  2. School of Mathematical and Natural Sciences (MC 2352), Arizona State University - West Campus, P. O. Box 37100, Phoenix, AZ, 85069-7100, USA

    Victor Pambuccian

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  1. Yuri Movsisyan
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  2. Victor Pambuccian
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Correspondence to Victor Pambuccian.

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Heinrich Wefelscheid in memoriam.

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Movsisyan, Y., Pambuccian, V. The Geometry of Point Reflections and Quasigroups. Results Math 75, 132 (2020). https://doi.org/10.1007/s00025-020-01264-7

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