来自逆括号的杨 - 巴克斯特方程的集合论解
Set-theoretic solutions of the Yang-Baxter equation from inverse braces
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中文总结 AI 辅助
研究逆括号代数结构，通过其性质刻画与杨 - 巴克斯特方程集合论解的联系，给出产生解的逆括号类及示例，还介绍逆括号构造方法，为生成新示例提供系统方式。
中文摘要 AI 辅助
我们引入逆括号的代数结构，即三元组$(S, +, \circ)$，其中$(S, +)$和$(S, \circ)$都是逆半群，且对于所有$a, b, c \in S$，恒等式$a\circ(b + c)=a\circ b - a + a\circ c$成立，这里$-a$表示$a$关于$+$的逆元。特别地，每个弱括号都是逆括号。我们研究逆括号的基本性质，分析加法和乘法幂等元之间的关系，刻画逆括号成为弱括号的条件。主要结果涉及与杨 - 巴克斯特方程的集合论解的联系，给出了一类产生解的逆括号及示例。最后，通过匹配积和逆括号的强半格介绍逆括号的构造，表明这些构造保持产生解所需的条件，提供了生成新示例的系统方法。
英文摘要
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.
