The Homology Groups of a Partial Trace Monoid Action

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.

💡 Research Summary

The paper investigates the homology of mathematical models of concurrency, focusing on the Baues‑Wirsching homology of categories arising from partial actions of trace monoids on sets. After introducing the necessary categorical background—left and right fibres of functors, the category of factorizations, and the Yoneda embedding—the author proves a key structural lemma (Lemma 1.1) stating that each connected component of the right fibre over an object in the factorization category possesses an initial object. This property enables a reduction of the Baues‑Wirsching homology of the left fibre (h_D/X) to a direct sum over the values of a coefficient functor on those initial objects (Proposition 1.3).

The work then turns to monoid actions. By viewing a monoid (M) as a one‑object category and a right (M)-set (X) as a functor (X: M^{op}\to\mathbf{Set}), the associated action category (K(X)) (the opposite of the left fibre (h_M^*/X)) is defined. The author shows that for any convex subcategory of (K(X)), the Baues‑Wirsching homology coincides with the Leech homology of the underlying monoid (Sections 2.2 and 2.3). This reduction is crucial because Leech homology is a well‑studied invariant of monoids, with existing computational techniques.

Specializing to trace monoids—free partially commutative monoids defined by a set of generators together with a commutation relation—the paper constructs a cubical chain complex (C_\bullet) of free abelian groups. Each (C_n) is generated by (n)-dimensional cubes corresponding to compatible sequences of generators, and the boundary maps are given explicitly in terms of the monoid’s relations. Theorem 3.1 proves that the homology of this complex is naturally isomorphic to the integral Baues‑Wirsching homology of the trace‑monoid action category. Moreover, Theorem 3.2 establishes that this isomorphism persists when restricting to any convex subcategory, ensuring that the cubical model captures all relevant homological information.

The theoretical results are leveraged to design an algorithm for computing the homology groups of the augmented category of a partial trace‑monoid action. The algorithm proceeds as follows:

1. Input the presentation of the trace monoid (generators and commutation pairs) and the right action on a finite set (X).
2. Build the category of factorizations (F(K(X))) and identify the initial objects of each right fibre using Lemma 1.1.
3. Assemble the cubical chain complex (C_\bullet) by enumerating admissible cubes; the number of cubes in each dimension is finite because the action is on a finite set.
4. Compute the integer matrices representing the boundary operators; these matrices are sparse due to the partial commutativity.
5. Perform Smith normal form reduction to obtain the invariant factor decomposition of each homology group (H_n).

The algorithm is implemented and applied to several examples, most notably to condition/event (CE) nets, a class of Petri nets used to model concurrent systems. By interpreting a CE net as a partial action of a trace monoid on a set of markings, the algorithm yields the full homology groups (H_0, H_1, H_2,\dots). The paper presents concrete calculations showing that higher homology groups (e.g., (H_2) and (H_3)) can be non‑trivial, containing copies of (\mathbb{Z}) and finite cyclic groups. These results answer Open Problem 1 from the author’s 2004 work, which asked for a method to compute all homology groups of CE nets, not just the first one.

In the discussion, the author emphasizes that the reduction to Leech homology provides a conceptual bridge between concurrency theory and classical algebraic topology. The cubical complex offers a combinatorial model that is both theoretically sound and computationally tractable, allowing existing linear‑algebra software to be employed. Moreover, the paper observes that even for a partial trace‑monoid action on a single point, the homology groups can be arbitrarily large, as any finite abelian group can appear in dimensions (n>2). This highlights the expressive power of the model and suggests further research directions, such as exploring homological dimensions of more general asynchronous transition systems.

Overall, the paper delivers a complete framework: it establishes the theoretical equivalence between Baues‑Wirsching and Leech homology for monoid actions, constructs an explicit cubical chain complex for trace‑monoid actions, and translates these insights into a practical algorithm that successfully computes the full homology of CE nets and related concurrent structures.