Small finite trimedial quasigroups

Abstract

One says that a groupoid (E,.) is medial (or: metabelian, bisymmetric, entropie..) if its binary law satisfies (x.a).(b.y) = (x.b).(a.y) identically. If one assumes only that this equality should be obeyed whenever the four arguments belong to any subgroupoid generated by 3 elements, (E,.) is said to be trimedial. The smallest possible order for a non-medial trimedial groupoid (resp. quasigroup) turns out to be 5 (resp. 81), and there are up to isomorphism exactly 35 non-medial trimedial quasi-groups of order 81. Only 8 of them are isotopic to L(1), free exponent 3 commutative Moufang loop on 3 generators, previously described by ZASSENHAUS and Marshall HALL Junior. The 27 remaining ones are isotopic to L(1) the only other non-associative commutative Moufang loop of order 81, whose exponent is 9. These results generalize known classifica-tion theorems concerning the two main special cases of trimedial quasigroups: the distributive quasigroups and the cubic hypersurface quasigroups.