Garside families and Garside germs

TL;DR

This paper explores Garside families, providing new characterizations and criteria for their normal decompositions, thereby extending the classical theory of braid groups and Garside structures.

Contribution

It introduces multiple new characterizations of Garside families, enhancing understanding of their structure and normal decompositions.

Findings

New criteria for Garside family existence

Extended the theory of Garside monoids and groups

Provided characterizations for normal decompositions

Abstract

Garside families have recently emerged as a relevant context for extending results involving Garside monoids and groups, which themselves extend the classical theory of (generalized) braid groups. Here we establish various characterizations of Garside families, that is, equivalently, various criteria for establishing the existence of normal decompositions of a certain type.