# IP Loops of Small Order

## Asif Ali and John Slaney

### Definition

An IP loop is a groupoid (L,*) such that
• * is left and right cancellative: for all x, y in L there exist unique z1, z2 in L such that x*z1 = y = z2*x.

• There is an element e in L such that for all x in L e*x = x = x*e;

• For every element x there is some element x such that for every y we have x *(x*y) = y = (y*x)* x.

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 loops

Click 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.

 Order 2 ( .gz ) 1 group 0 non-groups Order 3 ( .gz ) 1 group 0 non-groups Order 4 ( .gz ) 2 groups 0 non-groups Order 5 ( .gz ) 1 group 0 non-groups Order 6 ( .gz ) 2 groups 0 non-groups Order 7 ( .gz ) 1 group 1 non-group Order 8 ( .gz ) 5 groups 3 non-groups Order 9 ( .gz ) 2 groups 5 non-groups Order 10 ( .gz ) 2 groups 45 non-groups Order 11 ( .gz ) 1 group 48 non-groups Order 12 ( .gz ) 5 groups 2679 non-groups Order 13 ( .gz ) 1 group 10341 non-groups

Computer Sciences Laboratory, Australian National University