IP Loops of Small Order
Asif Ali and John Slaney
DefinitionAn IP loop is a groupoid (L,*) such that
A group is an associative IP loop. A Steiner loop is an IP loop in which for every x we have x = x. IP loops are thus a strong generalisation of both groups and Steiner loops. This class of loops is also important in that the Moufang nucleus (the set of a in L such that a*((x*y)*a) = (a*x)*(y*a) for all x, y in L) behaves as a nilpotency function for this class. Moreover, the power sets of IP loops form semiassociative relation algebras.
Small IP loopsClick on the following links to download the IP loops of order up to 13. One representative of each isomorphism class is given, in (numerical) lexicographic order.
Computer Sciences Laboratory, Australian National University