## 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

### MSC:

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