On Steiner Quasigroups of Cardinality

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) x 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.