Set-theoretic solutions of the Yang-Baxter equation from inverse braces

Abstract

We introduce the algebraic structure of an inverse brace, namely, a triple $(S,+,\circ)$ such that both $(S,+)$ and $(S,\circ)$ are inverse semigroups and the following identity holds $a\circ(b+c)=a\circ b-a+a\circ c$, for all $ a, b, c \in S$, where $-a$ denotes the inverse of $a \in S$, with respect to $+$. In particular, every weak brace is an inverse brace. We investigate the fundamental properties of inverse braces and analyze the relationship between additive and multiplicative idempotents, characterizing the condition under which an inverse brace is a weak brace. Our main results concern the connection with set-theoretic solutions to the Yang-Baxter equation. Specifically, we provide a class of inverse braces that yield solutions and give several examples. Finally, we introduce constructions of inverse braces via the matched product and the strong semilattice of inverse braces. We show that these constructions preserve the conditions required to produce solutions, thereby providing a systematic method for generating new examples.