On 3-basic quasigroups and their congruences. (English) Zbl 0701.20044

The quadruple \((Q_ 1,Q_ 2,Q_ 3;A)\), where \(Q_ 1\), \(Q_ 2\), \(Q_ 3\) are non-void sets with the same cardinality and A is a map of \(Q_ 1\times Q_ 2\) onto \(Q_ 3\) is called a 3-basic quasigroup if in the equation \(A(a_ 1,a_ 2)=a_ 3\) any two of the elements \(a_ 1\in Q_ 1\), \(a_ 2\in Q_ 2\), \(a_ 3\in Q_ 3\), uniquely determine the remaining one. The set of all autotopies of a given 3-basic quasigroup forms a group, it is called the full autotopy group. A subgroup G of the full autotopy group of a given 3-basic quasigroup Q is said to be special if its component groups \(\Gamma_ 1\), \(\Gamma_ 2\), \(\Gamma_ 3\) from a 3-basic quasigroup \((\Gamma_ 1,\Gamma_ 2,\Gamma_ 3;*)\), where \(\alpha *\beta =\gamma\Leftrightarrow\) (\(\alpha\),\(\beta\),\(\gamma\))\(\in G\) for \(\alpha \in \Gamma_ 1\), \(\beta \in \Gamma_ 2\), \(\gamma \in \Gamma_ 3\). In this paper a one-to-one correspondence between special subgroups G and normal congruences of a given 3-basic quasigroup Q is proved. In the end of the paper the author shows that all results can be generalized to (n\(+1)\)-basic quasigroups.
Reviewer: I.Corovei


20N05 Loops, quasigroups
20N15 \(n\)-ary systems \((n\ge 3)\)