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Local symmetries in partially ordered sets

Christoph Minz

TL;DR

The paper develops a formal framework for local symmetries in finite posets by exploiting automorphisms and retract operations to build symmetry classes, thereby enabling refined enumeration. It systematically analyzes a hierarchy of local symmetries (via $(Q,r)$-generators and prime vs composite retracts) and applies these ideas to structured posets such as complete bipartite graphs, fences, polygons, and simplexes, revealing explicit counts and retraction patterns (e.g., to the 2-chain). By enumerating locally unsymmetric posets by layer and deriving exact formulas for several layer configurations, the work connects combinatorial structure to asymptotic behavior and to Kleitman–Rothschild orders relevant in causal-set theory. The study then links these combinatorial insights to physics by considering sprinklings in Minkowski spacetime, showing that typical causal sets arising from random sprinkling are total locally unsymmetric with probability 1, which has implications for modeling discrete spacetime in quantum gravity. Together, the results offer a robust toolkit for classifying posets by local symmetries and for distinguishing causal-set models from generic posets in physics.

Abstract

Partially ordered sets (posets) play a universal role as an abstract structure in many areas of mathematics. For finite posets, an explicit enumeration of distinct partial orders on a set of unlabelled elements is known only up to a cardinality of 16 (listed as sequence A000112 in the OEIS), but closed expressions are unknown. By considering the automorphisms of (finite) posets, I introduce a formulation of local symmetries. These symmetries give rise to a division operation on the set of posets and lead to the construction of symmetry classes that are easier to characterise and enumerate. Furthermore, we consider polynomial expressions that count certain subsets of posets with a large number of layers (a large height). As an application in physics, local symmetries or rather their absence helps to distinguish causal sets (locally finite posets) that serve as discrete spacetime models from generic causal sets.

Local symmetries in partially ordered sets

TL;DR

The paper develops a formal framework for local symmetries in finite posets by exploiting automorphisms and retract operations to build symmetry classes, thereby enabling refined enumeration. It systematically analyzes a hierarchy of local symmetries (via -generators and prime vs composite retracts) and applies these ideas to structured posets such as complete bipartite graphs, fences, polygons, and simplexes, revealing explicit counts and retraction patterns (e.g., to the 2-chain). By enumerating locally unsymmetric posets by layer and deriving exact formulas for several layer configurations, the work connects combinatorial structure to asymptotic behavior and to Kleitman–Rothschild orders relevant in causal-set theory. The study then links these combinatorial insights to physics by considering sprinklings in Minkowski spacetime, showing that typical causal sets arising from random sprinkling are total locally unsymmetric with probability 1, which has implications for modeling discrete spacetime in quantum gravity. Together, the results offer a robust toolkit for classifying posets by local symmetries and for distinguishing causal-set models from generic posets in physics.

Abstract

Partially ordered sets (posets) play a universal role as an abstract structure in many areas of mathematics. For finite posets, an explicit enumeration of distinct partial orders on a set of unlabelled elements is known only up to a cardinality of 16 (listed as sequence A000112 in the OEIS), but closed expressions are unknown. By considering the automorphisms of (finite) posets, I introduce a formulation of local symmetries. These symmetries give rise to a division operation on the set of posets and lead to the construction of symmetry classes that are easier to characterise and enumerate. Furthermore, we consider polynomial expressions that count certain subsets of posets with a large number of layers (a large height). As an application in physics, local symmetries or rather their absence helps to distinguish causal sets (locally finite posets) that serve as discrete spacetime models from generic causal sets.
Paper Structure (20 sections, 17 theorems, 85 equations, 5 figures, 2 tables)

This paper contains 20 sections, 17 theorems, 85 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Local symmetries in posets
  4. Symmetric elements in antichains
  5. Cyclic symmetries
  6. Local symmetry properties
  7. Classes of posets with local symmetries
  8. Posets prime retractable to the 2-chain
  9. Complete bipartite graphs and crown posets
  10. Fences and polygons
  11. Regular polytopes and dimensional reduction
  12. Local symmetries by complete bipartite graphs
  13. Enumeration of locally unsymmetric posets by layers
  14. Numbers of locally unsymmetric posets vs. all posets
  15. Posets with one layer less than their cardinality
  16. ...and 5 more sections

Key Result

Lemma 3.1

Let $P$ be a poset. For all elements $a, b \in P$,

Figures (5)

  • Figure 4.1: The first-layer and second-layer elements of polygonal posets (first row) correspond to the vertices (red) and edges (blue) of regular polygons (second row), respectively. Regular polygons have reflection symmetries (dash dotted axis) that are represented by lower fences (green, third row). Taking the quotient by the fence is equivalent to matching vertex and edge elements for all reflection axes that meet an edge. This results in one equivalence class (fourth row) for the vertices (red cycle) and one or two equivalence classes --- the conjugacy classes of the respective dihedral groups --- represented by the non-degenerate or degenerate star polygons (blue), respectively. Hence, the quotients are isomorphic to the 2-chain or the Vee-poset (fifth row). The additional element in each poset (first row) above the polygonal poset is the closure including the central 2-face (grey). The element below each poset (light grey) is the dual to the top face, it is the abstract $(-1)$- or least face (empty shape).
  • Figure 4.2: The simplices in dimensions 0 to 4 as posets (first row), where the dimension index of the faces is placed to the left of each layer. The rows below show all (but opposite) distinct subsets formed by the elements of two distinct layers. Green links mark examples of subsets that are reflected.
  • Figure 4.3: The square, 3-cube and 4-cube represented by their Hasse diagrams (first row), where the dimension of the faces is indicated to the left of each layer. The second row shows the posets that are obtained when taking a quotient by two different types of reflection (hyper)planes. Different to \ref{['fig:SimplexPosets']}, here only one of subposet is shown for each of the two reflections. The green subposet is reflected by to a hyperplane that cuts through edges, while the purple subposet is reflected by a hyperplane that meets vertices.
  • Figure 5.1: Distributions of the number of (locally unsymmetric) posets relative to the total number of (locally unsymmetric) posets vs. the number of layers (length of longest chain) in the posets, all for different cardinalities $n$ of the posets.
  • Figure 6.1: Choice of coordinates and construction of the region $I_t$ for a pair of separated elements in a sprinkle in Minkowski spacetime.

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Fences
  • Definition 2.4: Element properties
  • Remark
  • Definition 2.5: Subset properties
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 3.1: Common links
  • ...and 48 more