The Γ-Equivalence Partition

IVP prescribes one module structure: the Γ-equivalence partition. This page explains that object — what the equivalence relation is, what properties its classes have, and how multi-driver elements fit.

The principle itself — its one-sentence statement and the four conditions — is stated on the structural formulation page. Here we work with the object the principle produces.

1 The equivalence relation

Define the relation $\tilde{\Gamma}$ on the element set $E$ by:

$e_1 \sim_\Gamma e_2 \iff \Gamma(e_1) = \Gamma(e_2)$

Two elements are change-driver-equivalent when they are assigned the same set of change drivers. This relation is an equivalence relation: it is reflexive, symmetric, and transitive. Therefore it partitions the element set into equivalence classes.

The Γ-equivalence partition $\mathcal{P}_\Gamma = E / \tilde{\Gamma}$ is the set of these classes. Each class collects exactly the elements that respond to the same change drivers.

2 Purity and completeness

A module drawn from this partition has two properties.

Purity. The module contains no element from outside its Γ-equivalence class. Every element in the module is Γ-equivalent to every other. When a change driver activates, every element affected lives in this module — and nothing unrelated lives here.
Completeness. The module contains every element of its Γ-equivalence class. Nothing that should be here has been placed elsewhere. When a change driver activates, no element that must change is scattered across other modules.

Purity is the module-level form of Separation (IVP-3): no foreign elements. Completeness is the module-level form of Unification (IVP-4): no missing elements. A module that is both pure and complete is maximally cohesive with respect to the change-driver rule. It is the unit that contains exactly the elements a change driver forces to change together, and nothing else. The formal account of purity and completeness as correctness relative to a partitioning rule is given in the published paper on the nature of cohesion [1], and the change-driver-assignment instantiation of the schema is developed in the paper on causal cohesion [2].

3 Multi-driver elements

Not every element is pure. An adapter that must satisfy both the domain’s rule and a foreign interface has $|\Gamma(e)| > 1$ . Its driver set is irreducible: it genuinely needs all of its drivers.

An element is single-driver or multi-driver. The word “impure” applies to a module, not an element: a module is impure when it groups elements that are not Γ-equivalent. A module containing a multi-driver element is necessarily impure, because that element answers to drivers its pure neighbours do not. The adapter is well-designed; the impurity it forces on its module is necessary, not a mistake.

4 The partition is what IVP produces

Because IVP prescribes exactly this partition, the partition theorem is the result that the compliant structure is unique — the structure is determined once $\Gamma$ is read off the domain.

The formal development. The definitions, theorems, and proofs behind this page are published in:

On the Nature of Cohesion: Cohesion as Correctness Relative to a Partitioning Rule — Zenodo, 2026 — doi:10.5281/zenodo.21703193
Causal Cohesion — Zenodo, 2026 — doi:10.5281/zenodo.21703156

All published papers

References

[1]

Y. Loth, “On the nature of cohesion: Cohesion as correctness relative to a partitioning rule.” Zenodo, Jul. 2026. doi: 10.5281/zenodo.21703193.

[2]

Y. Loth, “Causal cohesion.” Zenodo, Jul. 2026. doi: 10.5281/zenodo.21703156.