Set-theoretic solutions of the Yang-Baxter equation and regular *-semibraces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916328388952064 |
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| author | Liu, Qianxue Wang, Shoufeng |
| author_facet | Liu, Qianxue Wang, Shoufeng |
| contents | As generalizations of inverse semibraces introduced by Catino, Mazzotta and Stefanelli, Miccoli has introduced regular $\star$-semibraces under the name of involution semibraces and given a sufficient condition under which the associated map to a regular $\star$-semibrace is a set-theoretic solution of the Yang-Baxter equation. From the viewpoint of universal algebra, regular $\star$-semibraces are (2,2,1)-type algebras. In this paper we continue to study set-theoretic solutions of the Yang-Baxter equation and regular $\star$-semibraces. We first consider several kinds of (2,2,1)-type algebras that induced by regular $\star$-semigroups and give some equivalent characterizations of the statement that they form regular $\star$-semibraces. Then we give sufficient and necessary conditions under which the associated maps to these (2,2,1)-type algebras are set-theoretic solutions of the Yang-Baxter equation. Finally, as analogues of weak braces defined by Catino, Mazzotta, Miccoli and Stefanelli, we introduce weak $\star$-braces in the class of regular $\star$-semibraces, describe their algebraic structures and prove that the associated maps to weak $\star$-braces are always set-theoretic solutions of the Yang-Baxter equation. The result of the present paper shows that the class of completely regular, orthodox and locally inverse regular $\star$-semigroups is a source of possibly new set-theoretic solutions of the Yang-Baxter equation. Our results establish the close connection between the Yang-Baxter equation and the classical structural theory of regular $\star$-semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12533 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Set-theoretic solutions of the Yang-Baxter equation and regular *-semibraces Liu, Qianxue Wang, Shoufeng Group Theory Quantum Algebra 16T25, 20M17, 81R50, 16Y99 As generalizations of inverse semibraces introduced by Catino, Mazzotta and Stefanelli, Miccoli has introduced regular $\star$-semibraces under the name of involution semibraces and given a sufficient condition under which the associated map to a regular $\star$-semibrace is a set-theoretic solution of the Yang-Baxter equation. From the viewpoint of universal algebra, regular $\star$-semibraces are (2,2,1)-type algebras. In this paper we continue to study set-theoretic solutions of the Yang-Baxter equation and regular $\star$-semibraces. We first consider several kinds of (2,2,1)-type algebras that induced by regular $\star$-semigroups and give some equivalent characterizations of the statement that they form regular $\star$-semibraces. Then we give sufficient and necessary conditions under which the associated maps to these (2,2,1)-type algebras are set-theoretic solutions of the Yang-Baxter equation. Finally, as analogues of weak braces defined by Catino, Mazzotta, Miccoli and Stefanelli, we introduce weak $\star$-braces in the class of regular $\star$-semibraces, describe their algebraic structures and prove that the associated maps to weak $\star$-braces are always set-theoretic solutions of the Yang-Baxter equation. The result of the present paper shows that the class of completely regular, orthodox and locally inverse regular $\star$-semigroups is a source of possibly new set-theoretic solutions of the Yang-Baxter equation. Our results establish the close connection between the Yang-Baxter equation and the classical structural theory of regular $\star$-semigroups. |
| title | Set-theoretic solutions of the Yang-Baxter equation and regular *-semibraces |
| topic | Group Theory Quantum Algebra 16T25, 20M17, 81R50, 16Y99 |
| url | https://arxiv.org/abs/2407.12533 |