Die Varietät der auflösbaren Ternare der Ordnung 2. (The variety of soluble Steiner ternary systems of order 2) (Q913246)

Associated to Steiner systems is an algebra T with a ternary operation q satisfying (a) \(q(x,y,z)=q(x,z,y)=q(z,x,y)\); (b) \(q(x,x,y)=y\); (c) \(q(x,y,q(x,y,z))=z\). It is called Boolean if (d) \(q(a,x,q(a,y,z))=q(x,y,z)\). The algebra is soluble of order n if there is a descending Boolean series \(1=\theta_ 0\supseteq \theta_ 1\supseteq...\supseteq \theta_ n=0\) of congruence relations. The variety of soluble algebras of length at most n is denoted by \(\alpha_ n\). A basis for the latter is found when \(n=2\).