Rules of Three for commutation relations

TL;DR

This paper explores the Rule of Three, a phenomenon where certain commutation relations in rings depend only on their validity for subsets of size one, two, and three, simplifying the understanding of these relations.

Contribution

It introduces and formalizes the Rule of Three, demonstrating that verifying relations for small subsets suffices for larger subsets in ring theory.

Findings

The Rule of Three applies to a wide class of commutation relations.

Verifying relations for subsets of size up to three is sufficient for all larger subsets.

This simplifies the process of establishing commutation relations in rings.

Abstract

We investigate the following surprisingly widespread phenomenon which we call The Rule of Three: in order for a particular kind of commutation relation to hold for subsequences of elements of a ring labeled by any subset of indices, it is enough that these relations hold for subsets of size one, two, and three.