  We generalize parts of the theory of associative geometries developed by
Kinyon and the author in the framework of universal algebra: we prove that
certain associoid structures, such as pregroupoids and principal equivalence
relations, have a natural prolongation from a set to its the power set. We
reinvestigate the case of homogeneous pregroupoids (corresponding to the
projective geometry of a group) from the point of view of pairs of commuting
principal equivalence relations. We use the ternary approach to groupoids
developed by Anders Kock, and the torsors defined by our construction can be
seen as a generalisation of the known groups of bisections of a groupoid.
