Garside families and Garside germs
  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. 
Patrick Dehornoy (LMNO), Francois Digne (LAMFA), Jean Michel (IMJ)
math.GR