Graphical view on linear extensions of finite posets
Milan Studený, Václav Kratochvíl
TL;DR
The paper establishes a precise graphical characterization of finite posets by linking the set of linear extensions ${\cal L}(P)$ to geodetically convex sets in the permutohedral graph, thereby providing a purely graphical cryptomorphic description of posets. Using Galois connections, it shows that a non-empty set of total orders on $N$ arises from some poset iff it is geodetically convex, and it develops a graded lattice of these convex sets with a height function tied to edge-label diversity. It also relates the height to poset dimension and discusses alternative representations via braid cones and finite topologies, connecting combinatorial, geometric, and topological perspectives. The work suggests new algorithmic angles for poset-related problems and situates these results within the broader landscape of cryptomorphic poset representations and polyhedral geometry.
Abstract
One of possible cryptomorphic definitions of a partially ordered set (= a poset) $P$ on a non-empty finite basic set $N$ is in terms of the set ${\cal L}(P)$ of all its linear extensions, that is, in terms of the set of total orders of $N$ consonant with $P$. Any total order of $N$ can be interpreted as a node of a particular graph, called the permutohedral graph (over $N$), because it is indeed the graph of a certain polytope in $\mathbb{R}^{N}$, known as the permutohedron. It is shown in the paper that a non-empty set of total orders of $N$ equals to ${\cal L}(P)$ for some poset $P$ on $N$ iff it is a geodetically convex set in the permutohedral graph. This result means that a purely graphical concept of geodetical convexity in this graph is a cryptomorphic definition of a finite poset. In particular, the lattice of geodetically convex sets in this graph is graded and its height function is described in graphical terms. A counter-example, however, shows that the height function does not correspond to the usual graphical diameter, relating this matter to a combinatorial concept of the dimension of a poset. Two alternative cryptomorphic views on a poset $P$ on $N$ are also briefly commented. The geometric counterpart is its full-dimensional braid cone in $\mathbb{R}^{N}$, while a combinatorial alternative is a topology on $N$ distinguishing points, often referred as a distributive lattice.