The congruence structure of racks

We examine racks from an algebraic viewpoint, concentrating on conned;ions between racks and other, more throughly understood structures such as groups. The fundamental rack is a necessarily complicated object which does not lend itself to detailed study. We therefore focus our attention on congruences on racks which enable us to simplify their structure and study their representation. It is likely that new, easily calculable invariants of links may be derived in this way. (For example, a link in S3 is n-colourable if and only if the fundamental rack is congruent to the dihedral rack of order n F-R].) The main result states that the congruence structure on any rack may be described by considering certain subgroups of either of two groups naturally associated to the rack.