Tobias Schlemmer | Searcharxiv

arXiv2021

Factorisations of some partially ordered sets and small categories

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.

math.GR↗

arXiv2014

Linear extensions of partial orders on Abelian groups

Partially ordered groups, also known as po-groups, are groups with a compatible partial order. Results from M.I. Zajceva and H.-H. Teh are combined in order to provide a full characterisation of linear order extensions of a given order on a group. In contrast to Teh this approach provides a method to discuss linear orders of different abelian rank in a uniform manner. This will be achieved by modelling the linear orders using hyperplanes in a real vector spaces. Among some additional remarks a construction of an archimedian directed order is given for every torsion free abelian group.

math.GR↗