Duality for pairs of upward bipolar plane graphs and submodule lattices

We present and prove the main result in Sections 2—4, intended to be readable for most mathematicians. Section 5, an application of the preceding sections, presupposes a modest familiarity with some fundamental concepts from (universal) algebra and, mainly, from lattice theory.

Sections 2–4 prove a duality theorem, Theorem 1, for some pairs of finite, oriented planar graphs. The first graph, $G$, is a flow network with edge capacities belonging to a fixed Abelian group. The other graph plays a controlling role: each of its maximal paths determines a set of edges of $G$ to be used at their full capacities while neglecting the rest of the edges; see Section 2 for a preliminary illustration.

Section 5 applies Theorem 1 to give a new and elementary proof of George Hutchinson’s self-duality theorem on identities that hold in submodule lattices; our approach is simpler (mainly conceptually simpler) than the earlier ones.

The aforementioned two parts of the paper are interdependent. The second part, Section 5, is based upon the first part (Sections 2–4), while the necessity for a suitable tool in the second part led to the creation of the first part.

Before delving into the technicalities of Section 3, consider the following example.

In Figure 1, $G$ and $H$ are oriented graphs. With the convention that every edge is upward oriented (like in the case of Hasse diagrams of partially ordered sets), the arrowheads are omitted. The subscripts $1,\dots,17$ supply a bijective correspondence between the edge set of $G$ and that of $H$. We can think of $G$ as a hypothetical concrete system in which the arcs (i.e., the edges) are transit routes, pipelines, fiber-optic cables, or freighters (or passenger vehicles) traveling on fixed routes, etc. The numbers in colored geometric shapes are the capacities of the arcs of $G$. (Even though we repeat these numbers on the arcs of $H$, they still mean the capacities of the corresponding arcs of $G$; the arcs of $H$ have no capacities.) These numbers are “all-or-nothing-flow” capacities, that is, each arc should be either used at full capacity or avoided; this stipulation is due to physical limitations or economic inefficiency. (However, there can be parallel arcs with different capacities; see, for example, $e_{13}$ and $e_{15}$.) The vertices of $G$ are repositories (or warehouses, depots, etc.). In contrast to $G$, the graph $H$ is to provide visual or digital information within a hypothetical control room. Each maximal directed path of $H$ defines a method to transport exactly $6$ units (such as pieces, tons, barrels, etc.) of something from $\textup{source}(G)$ to $\textup{sink}(G)$ without changing the final contents of other repositories. For example, $(e^{\prime}_{9}$, $e^{\prime}_{15}$, $e^{\prime}_{10})$ is a maximal directed path of $H$; its meaning for $G$ is that we use exactly the arcs $e_{9}$, $e_{15}$, and $e_{10}$ of $G$. Namely, we use $e_{9}$ to transport $3$ (units of something) from $\textup{source}(G)$ to $v_{1}$, $e_{15}$ to transport 6 from $v_{1}$ to $\textup{sink}(G)$, and $e_{10}$ to transport 3 from $\textup{source}(G)$ to $v_{1}$. Depending on the physical realization of $G$, we can use $e_{9}$, $e_{15}$ and $e_{10}$ in this order, in any order, or simultaneously. No matter which of the ten maximal paths of $H$ we choose, the result of the transportation is the same. The negative sign of $-3$ at $e_{5}$ and $e_{8}$ means that the arc is to transport 3 in the opposite direction (that is, downward). The scheme of transportation just described is very adaptive. Indeed, when choosing one of the ten maximal paths of $H$, several factors like speed, cost, the operational conditions of the edges, etc. can be taken into account.

First, we recall some, mostly well-known and easy, concepts and fix our notations. They are not unique in the literature, but we try to use the most expressive ones. We go mainly after Auer at al. [1]¹¹1At the time of writing, freely available at http://dx.doi.org/10.1016/j.tcs.2015.01.003 . and Di Battista at al. [7]²²2At the time of writing, freely available at https://doi.org/10.1016/0925-7721(94)00014-X .. In the present paper, every graph is assumed to be finite and directed. Sometimes, we say digraph to emphasize that our graphs are directed. A (directed edge) $e$ of a graph starts at its tail, denoted by $\textup{{tail}}(e)$, and ends at its head, denoted by $\textup{{head}}(e)$. Occasionally, we say that $e$ goes from $\textup{{tail}}(e)$ to $\textup{{head}}(e)$; see the middle of Figure 1. We can also say that $e$ is an outgoing edge from $\textup{{tail}}(e)$ and an incoming edge into $\textup{{head}}(e)$. For a vertex $c$, let $\textup{inc}(c)$ and $\textup{out}(c)$ stand for the set of edges incoming into $c$ and that of edges outgoing from $c$, respectively. Sometimes, $\textup{{head}}(e)$ is denoted by an arrowhead put on $e$. The vertex set (set of all vertices) and the edge set of a graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. The graph containing no directed cycle is said to be acyclic. Such a graph has no loop edges, since there is no cycle of length 1, and it is oriented, that is, $\textup{inc}(\textup{{tail}}(e))\cap\textup{out}(\textup{{head}}(e))=\emptyset$ for all $e\in E(G)$. A vertex $c\in V(G)$ is a source or a sink if $\textup{inc}(c)=\emptyset$ or $\textup{out}(c)=\emptyset$, respectively. A bipolarly oriented graph or, briefly saying, a bipolar graph is an acyclic digraph that has exactly one source, has exactly one sink, and has at least two vertices. For such a graph $G$, $\textup{source}(G)$ and $\textup{sink}(G)$ denote the source and the sink of $G$, respectively. The uniqueness of $\textup{source}(G)$ and that of $\textup{sink}(G)$ imply that in a bipolar graph $G$,

Next, guided by Section 2 and Figure 1, we introduce the concept of a paired-bipolar-graphs problem. This problem with one of its solutions forms a paired-bipolar-graphs scheme. For sets $X$ and $Y$, $X^{Y}$ denotes the set of functions from $Y$ to $X$.

For example, Figure 1 determines $\textup{PBGP}(G,H,$ $\varphi,\psi,$ $\mathbb{Z},6)$, where $\mathbb{Z}$ is the additive group of integers. As the numbers in colored geometric shapes form a solution, the figure defines a paired-bipolar-graphs scheme, too. Even though we do not use the following two properties of the figure, we mention them. First, the paired-bipolar-graphs problem defined by the figure has exactly one solution. Second, if $k\in{\mathbb{N}^{+}}:=\{1,2,3,\dots\}$, we change $\mathbb{Z}$ to the $(2k)$-element additive group of integers modulo $2k$, and $b$ is (the residue class of) $1$ rather than 6, then the problem determined by the figure has no solution.

The next section says more about paired-bipolar-graphs problems but only for specific bipolar graphs, including those in Figure 1.

For digraphs $G_{1}$ and $G_{2}$, a pair $(\gamma,\chi)$ of functions is an isomorphism from $G_{1}$ onto $G_{2}$ if both $\gamma\colon V(G_{1})\to V(G_{2})$ and $\chi\colon E(G_{1})\to E(G_{2})$ are bijections, and for $e\in E(G_{1})$, $\gamma(\textup{{tail}}(e))=\textup{{tail}}(\chi(e))$ and $\gamma(\textup{{head}}(e))=\textup{{head}}(\chi(e))$. Hence, $V(G_{i})$ and $E(G_{i})$ have been abstract sets and, in essence, a graph $G_{i}$ has been the system $(V(G_{i}),E(G_{i}),\textup{tail},\textup{head})$ so far. However, in case of a plane graph $G$, $V(G)$ is a finite subset of the plane $\mathbb{R}^{2}$ and $E(G)$ consists of oriented Jordan arcs (i.e., homeomorphic images of $[0,1]\subseteq\mathbb{R})$ such that each arc $e\in E(G)$ goes from a vertex $\textup{{tail}}(e)\in V(G)$ to a vertex $\textup{{head}}(e)\in V(G)$; see the middle part of Figure 1. On the other hand, $G$ is a planar graph if it is isomorphic to a plane graph. Note the difference: a plane graph is always a planar graph but not conversely.

The boundary $\textup{bnd}(G)$ of a plane graph consists of those arcs of $G$ that can be reached (i.e., each of their points can be reached) from any sufficiently distant point of the plane by walking along an open Jordan curve crossing no arc of the graph. Usually, we cannot define the boundary of a planar (rather than plane) graph.

Next, let $G$ be an upward bipolar plane graph. The arcs of $G$ divide the plane into regions. Exactly one of these regions is geometrically unbounded; we call the rest of the regions inner facets. Take a Jordan curve $C$ such⁵⁵5We stipulate that $C$ has exactly one point at infinity and, if possible, $C$ is a projective line. that $C$ connects $\textup{source}(G)$ and $\textup{sink}(G)$ in the projective plane and the affine part $C^{\prime}$ (the set of those points of $C$ that are not on the line at infinity) lies in the unbounded region. Then $C^{\prime}$ divides the unbounded region into two parts called outer facets . In Figure 2, $C^{\prime}$ is the union of the two thick dotted half-lines. The facets of $G$ are its inner facets and the two outer facets. In Figure 2, any two facets of $G$ sharing an arc are indicated by different colors (or by distinct shades in a grey-scale version).

Note that there are isomorphic upward bipolar plane graphs $G_{1}$ and $G_{2}$ such that ${G_{1}^{\textup{du}}}$ and ${G_{2}^{\textup{du}}}$ are non-isomorphic; this is why we cannot define the dual of an upward bipolarly oriented planar graph. Observe that the dual of an upward bipolar plane graph is a bipolar graph⁷⁷7This is why Definition 3 deviates from the literature, where ${G^{\textup{du}}}$ has only one outer facet, the outer region. Fact 1, to be formulated later, asserts more., so the following definition makes sense.

Next, based on (3.5), (3.7), and (4.1), we state our main theorem and a corollary.

This theorem, to be proved soon, trivially implies the following statement.

An arc $\{(x(t),y(t)):0\leq t\leq 1\}$ in the plane is strictly ascending if $y(t_{1})<y(t_{2})$ for all $0\leq t_{1}<t_{2}\leq 1$. A plane graph is ascending if all its arcs are strictly ascending. Platt [11] proved⁸⁸8Indeed, as $\textup{source}(G)$ and $\textup{sink}(G)$ are on $\textup{bnd}(G)$, we can connect them by a new arc without violating planarity. Furthermore, we can add parallel arcs to any arc. Thus, Platt’s result applies. the following result, mentioned also in Auer at al. [1].

The paragraph on pages 272–273 in [9] gives a detailed account on the contribution of each of the two authors of [9]. In particular, the self-duality theorem, to be recalled soon, is due exclusively to George Hutchinson. Thus, we call it Hutchinson’s self-duality theorem, and we reference Hutchinson [9] in connection with it. A similar strategy applies when citing his other exclusive results from [9].

The original proof of the self-duality theorem is deep. It relies on Hutchinson [8], which belongs mainly to the theory of abelian categories, on the fourteen-page-long Section 2 of Hutchinson and Czédli [9], and on the nine-page-long Section 3 of Hutchinson [9]. A second proof given by Czédli and Takách [6] avoids Hutchinson [8] and abelian categories, but relying on the just-mentioned Sections 2 and 3, it is still complicated. No elementary proof of Hutchinson’s self-duality theorem has previously been given; in light of Remark 1, we present such a proof here.

By a module $M$ over a ring $R$ with $1$ we always mean a unital left module, that is, $1m=m$ holds for all $m\in M$. The lattice of all submodules of $M$ is denoted by $\textup{Sub}(M)$. For $X,Y\in\textup{Sub}(M)$, $X\leq Y$ and $X\wedge Y$ means $X\subseteq Y$ and $X\cap Y$, respectively, while $X\vee Y$ is the submodule generated by $X\cup Y$. A lattice term is built from variables and the operation symbols $\vee$ and $\wedge$. For lattice terms $p$ and $q$, the string “$p=q$” is called a lattice identity. For example, $x_{1}\wedge(x_{2}\vee x_{3})=(x_{1}\wedge x_{2})\vee(x_{1}\wedge x_{3})$ is a lattice identity; in fact, it is one of the two (equivalent) distributive laws. To obtain the dual of a lattice term, we interchange $\vee$ and $\wedge$ in it. For example, the dual of