Semi-planar Steiner quasigroups of cardinality 3n

It is well known that for each n ≡ 1 or 3 (mod 6) there is a planar Steiner quasigroup (briefly, squag) of cardinality n (Doyen (1969) and Quackenbush (1976)). A simple squag is semi-planar if every triangle either generates the whole squag or the 9-element subsquag (Quackenbush (1976)). In fact, Quakenbush has stated that there should be such semi-planar squags. In this paper, we construct a semi-planar squag of cardinality 3n for all n > 9 and n ≡ 1 or 3 (mod 6). For n = 9, we give a construction for a semi-planar squag of cardinality 27 which is not planar. Steiner triple systems are in 1–1 correspondence with the squags (see Quackenbush (1976)). In this article, the Steiner triple system associated with a semi-planar squag will be called semi-planar. Consequently, we may say that there is a semi-planar Steiner triple system of cardinality m which is not planar for all m > 9 and m ≡ 3 or 9 (mod 18). Quackenbush has also proved that the variety generated by a finite simple planar squag covers the variety of all medial squags. Similarly, it is easy to show that the variety generated by a finite semi-planar squag also covers the variety of all medial squags.