Article in press
Authors:
Title:
Set-theoretical solutions for the Yang-Baxter equation in triangle algebras
PDFSource:
Discussiones Mathematicae - General Algebra and Applications
Received: 2021-11-07 , Revised: 2022-09-07 , Accepted: 2022-09-08 , Available online: 2023-06-07 , https://doi.org/10.7151/dmgaa.1431
Abstract:
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 $\widetilde{N}$ and $\mathfrak{M}$. 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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