proposed by | Toby Walsh tw@cs.york.ac.uk |
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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.