Steiner systems and Stein quasigroups
A Steiner system \(\text{S} (2,4,v)\) is a nonempty set \(V\) of points with \(v\) elements and a set \(B\) of lines (four-element subsets of \(V)\) with the property that any two different points are incident with exactly one line. A Stein quasigroup in the sense of Dénes-Keedwell is a quasigroup with \(x(xy) = yx\) for any elements \(x\) and \(y\). On the other hand, a Stein quasigroup in the sense of Ganter-Werner, written \(\text{SQG} (v)\), is a set of \(v\) elements such that \(xx=x\), \((xy)y=yx\), \((yx)y=x\) for any elements \(x\) and \(y\). Each \(\text{SQG} (v)\) is a Stein quasigroup in the sense of Dénes- Keedwell, but the other implication is not true. The paper discusses properties of \(\text{SQG} (v)\) and shows as main result: Each \(\text{SQG} (v)\) induces a Steiner system \(\text{S} (2,4,v)\) and vice versa. Some applications to the theory of Latin squares are given.
scientific article; zbMATH DE number 554056
scientific article