Graphical view on linear extensions of finite posets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917099693146112 |
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| author | Studený, Milan Kratochvíl, Václav |
| author_facet | Studený, Milan Kratochvíl, Václav |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_11785 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graphical view on linear extensions of finite posets Studený, Milan Kratochvíl, Václav Combinatorics 06A07 (Primary) 06A15, 62R01 (Secondary) 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. |
| title | Graphical view on linear extensions of finite posets |
| topic | Combinatorics 06A07 (Primary) 06A15, 62R01 (Secondary) |
| url | https://arxiv.org/abs/2511.11785 |