Partition loops and affine geometries

Summary

A partition group (G, Π) is a group G together with a family Π of subgroups such that every element of G, other than the identity, is in exactly one subgroup of Π. Partition groups play an important role in the study of translation affine parallel structures (for short translation a.p. structures) and, in particular, of translation Sperner spaces (see André [1]). The notion of partition group has been generalised in many ways. Havel (see [5], [6], and many other papers) has introduced the notion of parallelisable partition of groupoids and of partition in many other algebraic structures and has studied the correspondent incidence structures. Wahling, generalising the concept of incidence-group introduced by Ellers and Karzel, has studied the structure of incidence loops and of incidence groupoids (see [13], [14]). In the present Note we consider a suitable class of loops with a partition Π (by means of non-trivial subgroups) such that all Sperner spaces and many a.p. structures can be represented by loops of this class (see also [2] and [7]). In §§1,2, we construct suitable partition loops in order to represent various types of a.p. structures in the “nicest” way. The case of the existence of a group of translations is particularly considered. Conversely, in §§3,4, we initiate the study of a.p. structures represented by a given partition loop. The geometrical meaning of left, middle and right nuclei is introduced.