Factorisations of some partially ordered sets and small categories

TL;DR

This paper develops a framework using orbit categories to factorize partially ordered sets while preserving their antisymmetry, and explores graph-based representations for these structures with applications in mathematical music theory.

Contribution

It introduces a novel approach to factorize and represent partially ordered sets via orbit categories and graph models, maintaining order properties.

Findings

Orbit categories can represent automorphism group actions on posets.

Graph and relation models can encode poset structures with antisymmetry.

Applications demonstrated in mathematical music theory examples.

Abstract

Orbits of automorphism groups of partially ordered sets are not necessarily congruence classes, i.e. images of an order homomorphism. Based on so-called orbit categories a framework of factorisations and unfoldings is developed that preserves the antisymmetry of the order Relation. Finally some suggestions are given, how the orbit categories can be represented by simple directed and annotated graphs and annotated binary relations. These relations are reflexive, and, in many cases, they can be chosen to be antisymmetric. From these constructions arise different suggestions for fundamental systems of partially ordered sets and reconstruction data which are illustrated by examples from mathematical music theory.