Homology Groups of Pipeline Petri Nets

TL;DR

This paper investigates the homology groups of elementary Petri nets used in pipeline systems, revealing their structure in various dimensions and establishing that directed homology groups are trivial.

Contribution

It provides a detailed analysis of the homology groups of elementary Petri nets, including their dimensions and the behavior of directed homology groups, which was previously unexplored.

Findings

Integral homology groups in dimensions 0 and 1 are isomorphic to the integers.

Homology groups in other dimensions are zero.

Directed homology groups are zero in all dimensions.

Abstract

We study homology groups of elementary Petri nets for the pipeline systems. We show that the integral homology groups of these nets in dimensions 0 and 1 equal the group of integers, and they are zero in other dimensions. We prove that directed homology groups of elementary Petri nets are zero in all dimensions.