{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T20:56:45Z","timestamp":1773867405968,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,2]],"date-time":"2024-12-02T00:00:00Z","timestamp":1733097600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This manuscript presents set-theoretical solutions to the Yang\u2013Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to analyze these algebraic interactions, while an algorithm is introduced to automate the verification process, facilitating broader applications in quantum mechanics and mathematical physics. Additionally, the Yang\u2013Baxter equation is applied to spin transformations in quantum mechanical spin-12 systems, with transformations like rotations and reflections modeled using GE-algebras. A Cayley table is used to represent the algebraic structure of these transformations, and the proposed algorithm ensures that these solutions are consistent with the Yang\u2013Baxter equation, offering new insights into the role of GE-algebras in quantum spin systems.<\/jats:p>","DOI":"10.3390\/axioms13120846","type":"journal-article","created":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T10:07:10Z","timestamp":1733306830000},"page":"846","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Set-Theoretical Solutions for the Yang\u2013Baxter Equation in GE-Algebras: Applications to Quantum Spin Systems"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8296-2796","authenticated-orcid":false,"given":"Ibrahim","family":"Senturk","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Ege University, 35100 \u0130zmir, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6514-4027","authenticated-orcid":false,"given":"Tahsin","family":"Oner","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Ege University, 35100 \u0130zmir, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3441-6496","authenticated-orcid":false,"given":"Abdullah Engin","family":"\u00c7al\u0131k","sequence":"additional","affiliation":[{"name":"Department of Physics, Faculty of Science, Ege University, 35100 \u0130zmir, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2514-3935","authenticated-orcid":false,"given":"H\u00fcseyin","family":"\u015eirin","sequence":"additional","affiliation":[{"name":"Department of Physics, Faculty of Science, Ege University, 35100 \u0130zmir, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3352-4770","authenticated-orcid":false,"given":"Metin","family":"Bilge","sequence":"additional","affiliation":[{"name":"Department of Physics, Faculty of Science, Ege University, 35100 \u0130zmir, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0921-6054","authenticated-orcid":false,"given":"Neelamegarajan","family":"Rajesh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Rajah Serfoji Government College, Thanjavur 613204, Tamilnadu, India"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1312","DOI":"10.1103\/PhysRevLett.19.1312","article-title":"Some exact results for the many-body problem in one dimension with repulsive delta-function interaction","volume":"19","author":"Yang","year":"1967","journal-title":"Phys. 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