003: Quasigroup Existence

Proposed by Toby Walsh

An order m quasigroup is a Latin square of size m. That is, a m by m multiplication table in which each element occurs once in every row and column. For example, “` 1 2 3 4 4 1 2 3 3 4 1 2 2 3 4 1 ““ is an order 4 quasigroup. A quasigroup can be specified by a set and a binary multiplication opertor, * defined over this set. Quasigroup existence problems determine the existence or non-existence of quasigroups of a given size with additional properties. Certain existence problems are of sufficient interest that a naming scheme has been invented for them. We define two new relations, *321 and *312 by $a *321 b = c$ iff $c*b=a$ and $a *312 b = c$ iff $b*c=a$.

QG1.m problems are order m quasigroups for which if $a*b=c$ $a*b=c*d$ and $a *321 b = c *321 d$ then $a=c$ and $b=d$.

QG2.m problems are order m quasigroups for which if a*b=c*d and a *312 b = c *312 d then a=c and b=d.

QG3.m problems are order m quasigroups for which $(a*b)*(b*a) = a$.

QG4.m problems are order m quasigroups for which $(b*a)*(a*b) = a$.

QG5.m problems are order m quasigroups for which $((b*a)*b)*b = a$.

QG6.m problems are order m quasigroups for which $(a*b)*b = a*(a*b)$.

QG7.m problems are order m quasigroups for which $(b*a)*b = a*(b*a)$.

For each of these problems, we may additionally demand that the quasigroup is idempotent. That is, a*a=a for every element a.