The Homology Groups of a Partial Trace Monoid Action

TL;DR

This paper investigates the homology groups of models of concurrency, reducing complex homology calculations to more manageable forms and providing an algorithm for computing these groups in specific systems.

Contribution

It introduces a method to compute homology groups of partial trace monoid actions by reducing Baues-Wirsching homology to Leech homology and constructing a cubical complex for trace monoids.

Findings

Homology groups can be reduced to Leech homology groups.

A cubical complex construction enables homology computation.

An algorithm for calculating homology groups of CE nets is provided.

Abstract

The aim of this paper is to investigate the homology groups of mathematical models of concurrency. We study the Baues-Wirsching homology groups of a small category associated with a partial monoid action on a set. We prove that these groups can be reduced to the Leech homology groups of the monoid. For a trace monoid with an action on a set, we will build a cubical complex of free Abelian groups with homology groups isomorphic to the integral homology groups of the action category. It allows us to solve the problem posed by the author in 2004 of the constructing an algorithm for computing homology groups of the CE nets. We describe the algorithm and give examples of calculating the homology groups.