GARSIDE AND QUADRATIC NORMALISATION: A SURVEY

Abstract. Starting from the seminal example of the greedy normal norm in braid monoids, we analyze the mechanism of the normal form in a Garside monoid and explain how it extends to the more general framework of Garside families. Extending the viewpoint even more, we then consider general quadratic normalisation procedures and characterise Garside normalisation among them. This text is an essentially self-contained survey of a general approach of nor-malisation in monoids developed in recent years in collaboration with several co-authors and building on the seminal example of the greedy normal form of braids independently introduced by S.Adjan [1] and by M.El-Rifai and H.Morton [22]. The main references are the book [17], written with F.Digne, E.Godelle, D.Kram-mer, and J.Michel, the recent preprint [19], written with Y.Guiraud, and, for algorithmic aspects, the article [16], written with V.Gebhardt. If M is a monoid (or a semigroup), and S is a generating subfamily of M, then, by definition, every element of M is the evaluation of some S-word.