Geodesic
A geometric approach to universal quasigroup identities
Archivum mathematicum, Tome 29 (1993) no. 1-2, pp. 97-103

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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).
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).
Classification : 05B30, 20N05
Keywords: 3-webs and their coordinatizing quasigroups; isotopic quasigroups and loops; identities invariant under isotopies; accompanying identities 
Havel, V. J. A geometric approach to universal quasigroup identities. Archivum mathematicum, Tome 29 (1993) no. 1-2, pp. 97-103. http://geodesic.mathdoc.fr/item/ARM_1993_29_1-2_a11/
@article{ARM_1993_29_1-2_a11,
     author = {Havel, V. J.},
     title = {A geometric approach to universal quasigroup identities},
     journal = {Archivum mathematicum},
     pages = {97--103},
     year = {1993},
     volume = {29},
     number = {1-2},
     mrnumber = {1242632},
     zbl = {0797.05025},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1993_29_1-2_a11/}
}
TY  - JOUR
AU  - Havel, V. J.
TI  - A geometric approach to universal quasigroup identities
JO  - Archivum mathematicum
PY  - 1993
SP  - 97
EP  - 103
VL  - 29
IS  - 1-2
UR  - http://geodesic.mathdoc.fr/item/ARM_1993_29_1-2_a11/
LA  - en
ID  - ARM_1993_29_1-2_a11
ER  - 
%0 Journal Article
%A Havel, V. J.
%T A geometric approach to universal quasigroup identities
%J Archivum mathematicum
%D 1993
%P 97-103
%V 29
%N 1-2
%U http://geodesic.mathdoc.fr/item/ARM_1993_29_1-2_a11/
%G en
%F ARM_1993_29_1-2_a11

[1] Brožíková, E.: On universal quasigroup identities. Mathematica Bohemica 117 (1992), 20-32. | MR

[2] Evans, T.: Identical relation in loops, I. Journ. Austral. Math. Soc. 12 (1971), 275-286. | MR

[3] Falconer, E.: Isotopy invariants in quasigroups. Trans. Amer. Math. Soc. 151 (1970), 511-526. | MR | Zbl

[4] Movsisjan, J. M.: Introduction to the theory of algebras with hyperidentities. (in Russian), Erevan, 1986. | MR

[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.