On Steiner Quasigroups of Cardinality \(21\)

Abstract

In \([12]\) Quackenbush has expected that there should be subdirectly irreducible Steiner quasigroups (squags), whose proper homomorphic images are entropic (medial). The smallest interesting cardinality for such squags is \(21\). Using the tripling construction given in \([1]\) we construct all possible nonsimple subdirectly irreducible squags of cardinality \(21\) \((SQ(21)s)\). Consequently, we may say that there are \(4\) distinct classes of nonsimple \(SQ(21)s\), based on the number \(n\) of sub-\(SQ(9)s\) for \(n = 0, 1, 3, 7\). The squags of the first three classes for \(n = 0, 1, 3\) are nonsimple subdirectly irreducible having exactly one proper homomorphic image isomorphic to the entropic \(SQ(3)\) (equivalently, having \(3\) disjoined sub-\(SQ(7)s)\). For \(n = 7\), each squag \(SQ(21\)) of this class has \(3\) disjoint sub-\(SQ(7)s\) and \(7\) sub-\(SQ(9)s\), we will see that this squag is isomorphic to the direct product \(SQ(7)\) \(\times\) \(SQ(3)\). For \(n = 0\), each squag \(SQ(21)\) of this class is a nonsimple subdirectly irreducible having three disjoint sub-\(SQ(7)s\) and no sub-\(SQ(9)s\). In section \(5\), we describe an example for each of these classes. Finally, we review all well-known classes of simple \(SQ(21)s\).