Title:

A geometric approach to universal quasigroup identities (English) 
Author:

Havel, V. J. 
Language:

English 
Journal:

Archivum Mathematicum 
ISSN:

00448753 (print) 
ISSN:

12125059 (online) 
Volume:

29 
Issue:

1 
Year:

1993 
Pages:

97103 
Summary lang:

English 
. 
Category:

math 
. 
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). (English) 
Keyword:

3webs 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 
. 
Date available:

20080606T21:24:06Z 
Last updated:

20120510 
Stable URL:

http://hdl.handle.net/10338.dmlcz/107470 
. 
Reference:

[1] Brožíková, E.: On universal quasigroup identities.Mathematica Bohemica 117 (1992), 2032. MR 1154051 
Reference:

[2] Evans, T.: Identical relation in loops, I.Journ. Austral. Math. Soc. 12 (1971), 275286. MR 0297915 
Reference:

[3] Falconer, E.: Isotopy invariants in quasigroups.Trans. Amer. Math. Soc. 151 (1970), 511526. 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. 
. 