×

A geometric approach to universal quasigroup identities. (English) Zbl 0797.05025

Summary: We construct the accompanying identity \(\widehat{\mathcal I}\) of a given quasigroup identity \({\mathcal I}\). After that we deduce the main result: \({\mathcal I}\) is isotopically invariant (i.e., for every quasigroup \({\mathbb{Q}}\) it holds that if \({\mathcal I}\) is satisfied in \({\mathbb{Q}}\) then \({\mathcal I}\) is satisfied in every quasigroup isotopic to \({\mathbb{Q}}\)) if and only if it is equivalent to \(\widehat{\mathcal I}\) (i.e., for every quasigroup \({\mathbb{Q}}\) it holds that in \({\mathbb{Q}}\) either \({\mathcal I}\), \(\widehat{\mathcal I}\) are both satisfied or both not).

MSC:

05B30 Other designs, configurations
20N05 Loops, quasigroups