On the connection between many-valued contexts and general geometric structures

We study the connection between certain many-valued contexts and general geometric structures. The known one-to-one correspondence between attribute-complete many-valued contexts and complete affine ordered sets is used to extend the investigation to π-lattices, class geometries, and lattices with classification systems. π-lattices are identified as a subclass of complete affine ordered sets, which exhibit an intimate relation to concept lattices closely tied to the corresponding context. Class geometries can be related to complete affine ordered sets using residuated mappings and the notion of a weak parallelism. Lattices with specific sets of classification systems allow for some sort of “reverse conceptual scaling”.