prob003: quasigroup existence

proposed by Toby Walsh
tw@cs.york.ac.uk
with assistance from Kostas Stergiou and Mark Stickel.

Specification

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*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.