Dynkin Systems and the One-Point Geometry

In this note I demonstrate that the collection of Dynkin systems on finite sets assembles into a Connes-Consani F₁-module, with the collection of partitions of finite sets as a sub-module. The underlying simplicial set of this F₁-module is shown to be isomorphic to the delooping of the Krasner hyperfield K, where 1+1=\0, 1\. The face and degeneracy maps of the underlying simplicial set of the F₁-module of partitions correspond to merging partition blocks and introducing singleton blocks, respectively. I also show that the F₁-module of partitions cannot correspond to a set with a binary operation (even partially defined or multivalued) under the ``Eilenberg-MacLane'' embedding. These results imply that the n-fold sum of the Dynkin F₁-module with itself is isomorphic to the F₁-module of the discrete projective geometry on n points.