Local symmetries in partially ordered sets

TL;DR

This paper introduces local symmetries in finite posets, enabling a division operation and symmetry class construction that facilitate enumeration and characterization, with applications in physics for distinguishing causal sets.

Contribution

It presents a novel formulation of local symmetries in posets, leading to new methods for enumeration and analysis of their structure.

Findings

Defined local symmetries via automorphisms of posets

Developed a division operation and symmetry classes for posets

Applied to distinguish causal sets 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.