Theory Update 70

Observe that the three qutrit tripled trefoil quandles are specified by two out of the three words in the quandle, since the third word is given by the quandle rules. A triple may be listed in any order, and this defines a larger set of $162$ triples. By construction, this set is invariant under permutations in $S_3$. A set with these properties is known as a symmetric quasigroup.

Manin wants to reconstruct projective surfaces from combinatorial information, and he shows that an example of an Abelian symmetric quasigroup is given by collinearity for plane cubic curves. A quasigroup is like a group, in that it has left and right inverses, but it may not have an identity element and it may not be associative. The multiplication table for a quasigroup is a Latin square. For example, the elements $X$, $Y$ and $Z$ of one trefoil quandle define the familiar circulant $3 \times 3$ matrix Latin square.

A quasigroup with identity is called a loop. A loop is a Moufang loop if it satisfies the rule $(xy)(zx) = x((yz)x)$. The non zero octonions form a nonassociative Moufang loop under ordinary multiplication. For the associative product of trefoil quandle triples, the Moufang rule is also obviously obeyed. There are not many nonassociative Moufang loops of low finite order. There is one of order $12$ and there are $5$ of order $81$. Associativity depends on the prime factorization of the loop order.