Dynkin Systems and the One-Point Geometry
Jonathan Beardsley
TL;DR
This work develops a detailed bridge between Dynkin systems on finite sets, Connes–Consani’s $\\mathbb{F}_1$-module framework, and discrete projective geometries via plasmas and the Krasner hyperfield $\\mathbb{K}$. It proves that the $n$-point discrete geometry corresponds to the $n$-fold coproduct of the Dynkin $\\mathbb{F}_1$-module and that the delooping of $\\mathbb{K}$ coincides with the delooping of the Dynkin module, while showing partitions form a proper substructure not realizable as a plasma module. The paper also provides a concrete description of the simplicial structure of $B\\widehat{H}\\mathbb{K}$, with explicit counts of low-dimensional simplices and a correspondence to 7-tuples in $\\mathbb{K}$ that encode associativity constraints. The results illuminate characteristic-one geometry and suggest refined links between combinatorial geometries, hyperfield-based algebra, and $\\mathbb{F}_1$-homotopy theory, with potential implications for how discrete geometries are modeled in this framework.
Abstract
In this note I demonstrate that the collection of Dynkin systems on finite sets assembles into a Connes-Consani $\mathbb{F}_1$-module, with the collection of partitions of finite sets as a sub-module. The underlying simplicial set of this $\mathbb{F}_1$-module is shown to be isomorphic to the delooping of the Krasner hyperfield $\mathbb{K}$, where $1+1=\{0,1\}$. The face and degeneracy maps of the underlying simplicial set of the $\mathbb{F}_1$-module of partitions correspond to merging partition blocks and introducing singleton blocks, respectively. I also show that the $\mathbb{F}_1$-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 $\mathbb{F}_1$-module with itself is isomorphic to the $\mathbb{F}_1$-module of the discrete projective geometry on $n$ points.