Set theoretic Yang-Baxter equation is the quantum Yang-Baxter equation in the Cartesian monoidal category of sets. Thus we talk about set-theoretic solutions of (quantum) Yang-Baxter equation.
Pioneering work
T. Gateva-Ivanova, M. Van den Bergh, Semigroups of I-type, J. Algebra 206 (1998) 97-112
based on some observations from
Tatiana Gateva-Ivanova, Noetherian properties of skew polynomial rings with binomial relations, Trans. Amer. Math. Soc. 343 (1994) 203-219, Skew polynomial rings with binomial relations, J. Algebra 185 (1996) 710-753
eventually lead to modern theory of braces and skew-braces.
Pavel Etingof, Travis Schedler, Alexandre Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100(2) (1999) 169-209 doi
W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (2007), no. 1, 153-170 doi
Tatiana Gateva-Ivanova, Shahn Majid, Matched pairs approach to set theoretic solutions of the Yang–Baxter equation, J. Algebra 319:4, 15 (2008) 1462-1529 doi
F. Cedó, E. Jespers, J. Okniński, Braces and the Yang–Baxter equation, Commun. Math. Phys. 327, 101–116 (2014) doi
Aryan Ghobadi, Skew braces as remnants of co-quasitriangular Hopf algebras in SupLat, J. Algebra 586 (2021) 607-642 doi
A dynamical version of set theoretic quantum Yang-Baxter is introduced using category theory in