Subsquags and Normal Subsquags

Abstract

Quackenbush [5] has studied the properties of squags or “Steiner quasigroups”, that is, the corresponding algebra of Steiner triple systems. He has proved that if a finite squag $(P;\cdot )$ contains two disjoint subsquags $(P_1;\cdot )$ and $(P_2;\cdot )$ with cardinality $|P_1|=|P_2|=\frac{1}{3}|P|$, then the complement $P_3=P–(P_1\cup P_2)$ is also a subsquag and the three subsquags $P_1,P_2$ and $P_3$ are normal. Quackenbush then asks for an example of a finite squag of cardinality $3n$ with a subsquag of cardinality $n$, but not normal. In this paper, we construct an example of a squag of cardinality $3n$ with a subsquag of cardinality $n$, but it is not normal; for any positive integer $n\ge 7$ and $n\equiv 1$ or $3$ (mod $6$).