Previous |  Up |  Next


3-webs and their coordinatizing quasigroups; isotopic quasigroups and loops; identities invariant under isotopies; accompanying identities
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).
[1] Brožíková, E.: On universal quasigroup identities. Mathematica Bohemica 117 (1992), 20-32. MR 1154051
[2] Evans, T.: Identical relation in loops, I. Journ. Austral. Math. Soc. 12 (1971), 275-286. MR 0297915
[3] Falconer, E.: Isotopy invariants in quasigroups. Trans. Amer. Math. Soc. 151 (1970), 511-526. MR 0272932 | Zbl 0209.04701
[4] Movsisjan, J. M.: Introduction to the theory of algebras with hyperidentities. (in Russian), Erevan, 1986. MR 0877518
[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.
Partner of
EuDML logo