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Title: A geometric approach to universal quasigroup identities  (English)
Author: Havel, V. J.
Language: English
Journal: Archivum Mathematicum
ISSN: 0044-8753 (print)
ISSN: 1212-5059 (online)
Volume: 29
Issue: 1
Year: 1993
Pages: 97-103
Summary lang: English
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Category: math
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Summary: In the present paper we construct the accompanying identity $\hat{I}$ of a given quasigroup identity $I$. After that we deduce the main result: $I$ is isotopically invariant (i.e., for every guasigroup $Q$ it holds that if $I$ is satisfied in $Q$ then $I$ is satisfied in every quasigroup isotopic to $Q$) if and only if it is equivalent to $\hat{I}$ (i.e., for every quasigroup $Q$ it holds that in $Q$ either $I, \hat{I}$ are both satisfied or both not).
Keyword: 3-webs and their coordinatizing quasigroups
Keyword: isotopic quasigroups and loops
Keyword: identities invariant under isotopies
Keyword: accompanying identities
MSC: 05B30
MSC: 20N05
idZBL: Zbl 0797.05025
idMR: MR1242632
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Date available: 2008-06-06T21:24:06Z
Last updated: 2012-05-10
Stable URL: http://hdl.handle.net/10338.dmlcz/107470
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Reference: [1] Brožíková, E.: On universal quasigroup identities.Mathematica Bohemica 117 (1992), 20-32. MR 1154051
Reference: [2] Evans, T.: Identical relation in loops, I.Journ. Austral. Math. Soc. 12 (1971), 275-286. MR 0297915
Reference: [3] Falconer, E.: Isotopy invariants in quasigroups.Trans. Amer. Math. Soc. 151 (1970), 511-526. Zbl 0209.04701, MR 0272932
Reference: [4] Movsisjan, J. M.: Introduction to the theory of algebras with hyperidentities.(in Russian), Erevan, 1986. MR 0877518
Reference: [5] Belousov, V. D."$^\dag $"V. D. Belousov ($^\ast $20.2.1925, $^\dag $23.7.1988) was a leader of the Russian research in quasigroup theory.: Algebraic nets and quasigroups.(in Russian), Kishinev, 1971.
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