Universal associative geometry

Wolfgang Bertram (1)

1 IECL - Institut Élie Cartan de Lorraine

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.

Mots clés: binary relation; associoid; bisection; universal algebra; relational composition; (semi)torsor; principal equivalence relation (principal bundle); (pre)groupoid

Domaines: Catégories et ensembles [math.CT]; Théorie des groupes [math.GR]

Wolfgang Bertram. Universal associative geometry. 2014. ⟨hal-01002611⟩