Set-theoretic solutions of the Yang-Baxter equation associated to weak braces

Abstract

We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang-Baxter equation. Specifically, a weak (left) brace is a non-empty set $S$ endowed with two binary operations $+$ and $\circ$ such that both $(S,+)$ and $(S,\circ )$ are inverse semigroups and they hold \begin{align*} a \circ \left(b+c\right) = a\circ b - a +a\circ c \qquad \text{and} \qquad a\circ a^- = - a + a, \end{align*} for all $a,b,c\in S$, where $-a$ and $a^{-}$ are the inverses of $a$ with respect to $+$ and $\circ$, respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution $r$ associated to an arbitrary weak brace $S$ has a behavior close to bijectivity, namely $r$ is a completely regular element in the full transformation semigroup on $S\times S$. In addition, we provide some methods to construct weak braces.

29 pages. The name of the structure and the title of the paper have changed. Accepted for publication in Semigroup Forum

References in corpus (3)

• Bi-skew braces and Hopf Galois structures
• Nilpotency in left semi-braces
• Inverse semi-braces and the Yang-Baxter equation

Cited by in corpus (7)

• Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation
• Extensions and automorphisms of Rota-Baxter groups
• Solutions of the Yang-Baxter equation and strong semilattices of skew braces
• Deformed solutions of the Yang-Baxter equation associated to dual weak braces
• Cohomology and Extensions of Relative Rota-Baxter groups
• Affine structures on groups and semi-braces
• Hopf bracoids