In connection with the emerging theory of Garside categories, we develop the notions of a left-Garside category and of a locally left-Garside monoid. In this framework, the connection between the self-distributivity law LD and braids amounts to the result that a certain category associated with LD is a left-Garside category, which projects onto the standard Garside category of braids. This appr...

Starting from a certain monoid that describes the geometry of the left self-distributivity identity, we construct an explicit realization of the free left self-distributive system on any number of generators. This realization lives in the charged braid group, an extension of Artin’s braid group B∞ with a simple geometrical interpretation. AMS Classification: 20N02, 20F36, 08B20. Constructing ex...

Slim groupoids are groupoids satisfying x(yz) ≈ xz. We find all simple slim groupoids and all minimal varieties of slim groupoids. Every slim groupoid can be embedded into a subdirectly irreducible slim groupoid. The variety of slim groupoids has the finite embeddability property, so that the word problem is solvable. We introduce the notion of a strongly nonfinitely based slim groupoid (such g...

Building on a result by W.Rump, we show how to exploit the right-cyclic law (xy)(xz) = (yx)(yz) in order to investigate the structure groups and monoids attached with (involutive nondegenerate) set-theoretic solutions of the Yang–Baxter equation. We develop a sort of right-cyclic calculus, and use it to obtain short proofs for the existence both of the Garside structure and of the I-structure o...

Define an augmented LD-system, or ALD-system, to be a set equipped with two binary operations, one satisfying the left self-distributivity law x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z) and the other satisfying the mixed laws (x ◦ y) ∗ z = x ∗ (y ∗ z) and x ∗ (y ◦ z) = (x ∗ y) ◦ (x ∗ z). We solve the word problem of the ALD laws, and prove that every element in the parenthesized braid group B• of [2, 3, 5...

A groupoid is alternative if it satisfies the alternative laws x(xy) = (xx)y and x(yy) = (xy)y. These laws induce four partial maps on N+ × N+ (r, s) 7→ (2r, s− r), (r − s, 2s), (r/2, s + r/2), (r + s/2, s/2), that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that nth powers in a free alternative groupoid on one generator are w...

We investigate a group B• that includes Artin’s braid group B∞ and Thompson’s group F . The elements of B• are represented by braids diagrams in which the distances between the strands are not uniform and, besides the usual crossing generators, new rescaling operators shrink or strech the distances between the strands. We prove that B• is a group of fractions, that it is orderable, admits a non...

This paper is a survey of recent work about the action of braids on selfdistributive systems. We show how the braid word reversing technique allows one to use new self-distributive systems, leading in particular to a natural linear ordering of the braids. AMS Subject Classification: 20F36, 20N02. It has been observed for many years that there exist a connection between braids and left selfdistr...