Duality for pairs of upward bipolar plane graphs and submodule lattices

Abstract

Let $G$ and $H$ be acyclic, upward bipolarly oriented plane graphs with the same number $n$ of edges. While $G$ can symbolize a flow network, $H$ has only a controlling role. Let $\phi$ and $\psi$ be bijections from $\{1, \dots, n\}$ to the edge set of $G$ and that of $H$, respectively; their role is to define, for each edge of $H$, the corresponding edge of $G$. Let $b$ be an element of an Abelian group $\mathbb A$. An $n$-tuple $(a_1$, $\dots$, $a_n)$ of elements of $\mathbb A$ is a solution of the paired-bipolar-graphs problem $P:=(G,H$, $\phi,\psi$, $\mathbb A, b)$ if whenever $a_i$ is the ``all-or-nothing-flow'' capacity of the edge $\phi(i)$ for $i=1, \dots, n$ and $\vec e$ is a maximal directed path of $H$, then by fully exploiting the capacities of the edges corresponding to the edges of $\vec e$ and neglecting the rest of the edges of $G$, we have a flow process transporting $b$ from the source (vertex) of $G$ to the sink of $G$. Let $P':=(H',G'$, $\psi',\phi'$, $\mathbb A, b)$, where $H'$ and $G'$ are the ``two-outer-facet'' duals of $H$ and $G$, respectively, and $\psi'$ and $\phi'$ are defined naturally. We prove that $P$ and $P'$ have the same solutions. This result implies George Hutchinson's self-duality theorem on submodule lattices.