Groupoids, root systems and weak order I

TL;DR

This paper introduces the concept of rootoids, a new algebraic structure inspired by Coxeter groups, which generalizes root systems and weak orders using groupoids and Boolean rings, laying the groundwork for further theoretical development.

Contribution

It defines and explores the foundational properties of rootoids, a novel structure extending Coxeter group concepts with new categorical and algebraic frameworks.

Findings

Rootoids embed weak orders as order ideals in complete ortholattices

Basic definitions and examples of rootoids are established

Main results involving categories and functor rootoids are deferred to future work

Abstract

This is the first of a series of papers which define and study structures called rootoids, which are groupoids equipped with a representation in the category of Boolean rings and with an associated 1-cocycle. The axioms for rootoids are abstracted from formal properties of Coxeter groups with their root systems and weak orders. They imply that each of the weak orders of a rootoid embeds as an order ideal in a complete ortholattice. This first paper is concerned only with the most basic definitions, facts and examples; the main results, which are new even for Coxeter groups, will be stated and proved in subsequent papers. They involve certain categories of rootoids and especially a notion of functor rootoid.