Steiner Heptagon Systems

Steiner Heptagon Systems (SHS) of type 1, 2, and 3 are defined and the spectrum of type 2 SHSs (SHS2) is studied. It is shown that the condition \(n \equiv 1 \) { or } \(7 \pmod{14}\) is not only necessary but also sufficient for the existence of an SHS2 of order \(n\), with the possible exceptions of \(n=21\) and \(85\). This gives an interesting algebraic result since the study of SHS2s is equivalent to the study of quasigroups satisfying the identities \(x^2 = x\), \((yx)x = y\), and \((xy)(y(xy)) = (yx)(x(yx))\).