On semi-planar Steiner quasigroups

A Steiner triple system (briefly ST) is in 1-1 correspondence with a Steiner quasigroup or squag (briefly SQ) B. Ganter, H. Werner, Co-ordinatizing Steiner systems, Ann. Discrete Math. 7 (1980) 3-24; C.C. Lindner, A. Rosa, Steiner quadruple systems: A survey, Discrete Math. 21 (1979) 147-181]. It is well known that for each n≡1 or 3 (mod 6) there is a planar squag of cardinality n J. Doyen, Sur la structure de certains systems triples de Steiner, Math. Z. 111 (1969) 289-300]. Quackenbush expected that there should also be semi-planar squags R.W. Quackenbush, Varieties of Steiner loops and Steiner quasigroups, Canad. J. Math. 28 (1976) 1187-1198]. A simple squag is semi-planar if every triangle either generates the whole squag or the 9-element squag. The first author has constructed a semi-planar squag of cardinality 3n for all n>3 and n≡1 or 3 (mod 6) M.H. Armanious, Semi-planar Steiner quasigroups of cardinality 3n, Australas. J. Combin. 27 (2003) 13-27]. In fact, this construction supplies us with semi-planar squags having only nontrivial subsquags of cardinality 9. Our aim in this article is to give a recursive construction as n→3n for semi-planar squags. This construction permits us to construct semi-planar squags having nontrivial subsquags of cardinality >9. Consequently, we may say that there are semi-planar (or semi-planar ) for each positive integer m and each n≡1 or 3 (mod 6) with n>3 having only medial subsquags at most of cardinality 3^ν (sub-) for each ν∈{1,2,…,m+1}.