Semiboolean SQS-skeins
A 3-quasigroup is a pair \((S,q)\), where \(S\) is a set and \(q:S^ 3 \to S\) is a mapping such that, in the equation \(q(x,y,z)=w\), any three variables uniquely determine the fourth. An SQS-skein is a 3-quasigroup satisfying the four identities \[ q(x,x,y) = y,\;q(x,y,z) = q(x,z,y),\;q(x,y,z) = q(y,z,x) \text{ and } q \bigl( x,y,q(x,y,z) \bigr) = z. \] It is well known that the SQS-skeins are in one-to-one correspondence with the Steiner quadruple systems. An SQS-skein is called boolean if there is a boolean group \((S,+)\) such that \(q(x,y,z) = x+y+z\). The Steiner quadruple systems corresponding to boolean SQS-skeins are exactly the Steiner quadruple systems whose quadruples are the planes of an affine space of order 2. It is not too difficult to see that an SQS-skein is boolean if and only if \(q(x,u,q(y,u,z)) = q(x,y,z)\). M. H. Armanious asked whether or not the class of all boolean SQS-skeins is characterized by the equation \(q(x,u,q(y,u,z)) = q(q(x,u,y),u,z)\). The author answers this question in the negative. He calls SQS-skeins satisfying the above equation semiboolean and shows that the class of all semiboolean SQS- skeins consists exactly of all SQS-skeins corresponding to Steiner quadruple systems all of whose derived Steiner triple systems are projective geometries of order 2. Examples of such Steiner quadruple systems that are not affine geometries of order 2 were known before, but the author gives an example described in a more algebraic way.
scientific article; zbMATH DE number 429045