A note on Garside monoids and Braces

TL;DR

This paper explores the connection between Garside monoids, a class of algebraic structures, and braces, introducing new types of braces and demonstrating how lcm-monoids induce left -braces, enriching the algebraic framework.

Contribution

It introduces the concept of left -braces and shows that every lcm-monoid induces such a brace, linking Garside monoids with brace theory.

Findings

Every lcm-monoid induces a left -brace.

Every Gaussian group induces a partial left brace.

The class of lcm-monoids includes Gaussian, quasi-Garside, and Garside monoids.

Abstract

A left brace is a triple $(\mathcal{B},+,\cdot)$, where $(\mathcal{B},+)$ is an abelian group, $(\mathcal{B},\cdot)$ is a group, and there is a left-distributivity-like axiom that relates between the two operations in $\mathcal{B}$. In analogy with a left brace, we define a left $\mathscr{M}$-brace to be a triple $(\mathcal{B},+,\cdot)$, where $(\mathcal{B},+)$ is a commutative monoid, $(\mathcal{B},\cdot)$ is a monoid, and the axiom of left distributivity holds. A lcm-monoid $M$ is a left-cancellative monoid such that $1$ is the unique invertible element in $M$, and every pair of elements in $M$ admit a lcm with respect to left-divisibility. The class of lcm-monoids contains the Gaussian, quasi-Garside and Garside monoids. We show that every lcm-monoid induces a left $\mathscr{M}$-brace. Furthermore, we show that every Gaussian group induces a partial left brace.