Nonassociative Moufang Research Articles

Investigation of subalgebra lattices by means of Hasse constants

Hasse constants and their basic properties are introduced to facilitate the connection between the lattice of subalgebras of an algebra \( \mathcal{C} \) and the natural action of the automorphism group Aut( \( \mathcal{C} \)) on \( \mathcal{C} \). These constants are then used to describe the lattice of subloops of the smallest nonassociative simple Moufang loop.

Minimally Nonassociative Commutative Moufang Loops

A commutative Moufang loop which is not associative has the property that all proper subloops are associative if and only if the loop can be generated by three elements.

Small finite trimedial quasigroups

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.