Graded sets, graded groups, and Clifford algebras

TL;DR

This paper introduces a broad framework for centrally graded sets and groups over arbitrary rings, explores their categorical properties, and examines the special case related to Clifford algebras and discrete Clifford groups.

Contribution

It generalizes the concept of graded groups and sets to arbitrary rings and establishes their structure as braided monoidal categories, with detailed analysis of the Z/2Z case linked to Clifford algebras.

Findings

Categories of graded sets and groups are braided monoidal.

The Z/2Z case relates to Clifford algebras and Salingaros Vee groups.

Basic properties of these categories are established.

Abstract

We define a general notion of centrally $\Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $\Gamma$ is an arbitrary (generalized) ring. The case $\Gamma$ = Z/2Z is studied in detail: it is related to Clifford algebras and their discrete Clifford groups (also called Salingaros Vee groups).