Related papers: A note on Garside monoids and Braces

An AG-monoid is an AG-groupoid (a groupoid satisfying the identity called left invertive law $(xy)z=(zy)x$) and having a left identiy. In this paper we enumerate AG-monoids algebraically and then implement them in GAP to compute them…

Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been…

We give an explicit geometric argument that Artin's braid group $B_n$ is right-orderable. The construction is elementary, natural, and leads to a new, effectively computable, canonical form for braids which we call left-consistent canonical…

The binary products of right, left or double division in semigroups that are semilattices of groups give interesting groupoid structures that are in one to one correspondence with semigroups that are semilattices of groups. This work is…

The main objective of this paper is to deepen the relationship between skew left braces and trifactorised groups that encodes the information about skew left braces, their structure, their quotients, and their homomorphisms.

Fix a commutative monoid $(T,+,0)$, a commutative monoid $(\Gamma,+,0_\Gamma)$, and a map \[ (a,\alpha,b,\beta,c)\longmapsto a\,\alpha\,b\,\beta\,c\in T \] which is additive in each variable and associative in the ternary sense. A left…

We present a construction of left braces of right nilpotency class at most two based on suitable actions of an abelian group on itself with an invariance condition. This construction allows us to recover the construction of a free right…

Much study has been done on semigroups which are unions of groups. There are several ways in which a union of groups can be made into a semigroup in which each of the component groups arises as subgroups of the constructed semigroup. An…

A classification up to isomorphism of all left braces of order $p^3$, where $p$ is any prime number, is given. To this end, we first classify all the left braces of order $p$ and $p^2$, and then we construct explicitly the hypothesis…

We define a bi-skew brace to be a set $G$ with two group operations $\star$ and $\circ$ so that $(G, \circ, \star)$ is a skew brace with additive group $(G, \star)$ and also with additive group $(G, \circ)$. If $G$ is a skew brace, then $G$…

In this paper we introduce in a braided setting the notion of left module for an invertible 1-cocycle and we prove some categorical equivalences between categories of modules associated to an invertible 1-cocycle and categories of modules…

We construct a quasi-Garside monoid structure for the free group. This monoid should be thought of as a dual braid monoid for the free group, generalising the constructions by Birman-Ko-Lee and by the author of new Garside monoids for Artin…

Garside calculus is the common mechanism that underlies a certain type of normal form for the elements of a monoid, a group, or a category. Originating from Garside's approach to Artin's braid groups, it has been extended to more and more…

The submonoid of the $3$-strand braid group $\mathcal{B}_3$ generated by $\sigma_1$ and $\sigma_1 \sigma_2$ is known to yield an exotic Garside structure on $\mathcal{B}_3$. We introduce and study an infinite family $(M_n)_{n\geq 1}$ of…

In this article, we show that a group $G$ is the union of two proper subsemigroups if and only if $G$ has a nontrivial left-orderable quotient. Furthermore, if $G$ is the union of two proper semigroups, then there exists a minimum normal…

We study some relations between left cancellative left semi-braces and other existing algebraic structures. In particular, we show that every left semi-brace arises from a left seminear-ring, extending the correspondence given by Rump…

In this note, we define the Burnside ring of a monoid, generalizing the construction for groups. After giving foundational definitions, we characterize transitive M-sets and their automorphisms, then prove a structure theorem for a broad…

Motivated by the Brou\'e conjecture on blocks with abelian defect groups for finite reductive groups, we study "parabolic" Deligne-Lusztig varieties and construct on those which occur in the Brou\'e conjecture an action of a braid monoid,…

Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial $n$ may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to…

A subsemigroup S of a semigroup Q is a left order in Q and Q is a semigroup of left quotients of S if every element of Q can be expressed as a# b where a and b are elements of S and if, in addition, every element of S that is square…