Graphical view on linear extensions of finite posets
Milan Studen\'y, V\'aclav Kratochv\'il

TL;DR
This paper characterizes finite posets through geodetically convex sets in the permutohedral graph, linking order theory with graph convexity and polyhedral geometry.
Contribution
It establishes a new cryptomorphic characterization of finite posets using geodetical convexity in the permutohedral graph, connecting order theory with graph and geometric concepts.
Findings
A set of total orders equals the linear extensions of a poset iff it is geodetically convex in the permutohedral graph.
The lattice of geodetically convex sets is graded, with a height function described graphically.
The height function differs from the graphical diameter, relating to the poset's dimension.
Abstract
One of possible cryptomorphic definitions of a partially ordered set (= a poset) on a non-empty finite basic set is in terms of the set of all its linear extensions, that is, in terms of the set of total orders of consonant with . Any total order of can be interpreted as a node of a particular graph, called the permutohedral graph (over ), because it is indeed the graph of a certain polytope in , known as the permutohedron. It is shown in the paper that a non-empty set of total orders of equals to for some poset on 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…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Digital Image Processing Techniques · Topological and Geometric Data Analysis
