Geodesic
Set-theoretical solutions for the Yang-Baxter equation in triangle algebras
Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 15-42

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In this study, we give some fundamental set-theoretical solutions of Yang-Baxter equation in triangle algebras and state triangle algebras. We prove that the necessary and sufficient condition for certain mappings to be set-theoretical solutions of Yang-Baxter equation on these structures is that these structures must be also MTL-(state) triangle algebras, BL-(state) triangle algebras or RL-(state) triangle algebras. In accordance with these, we recursively introduce new operators N and 𝔐. Then, we define the notion of formula on triangle algebra as a classical logic structure. Moreover, we state the relationship of transferring of set-theoretical solutions of Yang-Baxter equation among (MTL,BL, RL)-(state) triangle algebras and state (MTL,BL, RL)-(state) triangle algebras. Then, we give a scheme to explain clearly these relations.
Keywords: triangle algebra, Yang-Baxter equation, set-theoretical solution, residuated lattice, state operator, (MTL,BL, RL)-triangle algebras 
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Senturk, Ibrahim; Oner, Tahsin; Borumand Saeid, Arsham. Set-theoretical solutions for the Yang-Baxter equation in triangle algebras. Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 15-42. http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a1/