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The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry,

The Yang-Baxter equation first appeared in theoretical physics, in a paper by Yang [

In this paper we present qualitative results concerning the (set-theoretical) Yang-Baxter equation. We first consider the solutions arising from relations. For any relation on a given set we construct a map. We give necessary and sufficient conditions for this map to be a solution to the set-theoretical Yang-Baxter equation. In

Let V be a vector space over a field k, which is algebraically closed and of characteristic zero.

holds in the automorphism group of

where R_{ij} means R acting on the i-th and j-th component.

Let T be the twist map,

Finding all solutions of the Yang-Baxter equation is a difficult task far from being resolved. Nevertheless many solutions of these equations have been found during the last 30 years and the related algebraic structures have been studied.

Reference [

We call (3) the set-theoretical Yang-Baxter equation.

It is obvious that from a solution to (3), one could obtain a solution to (1), by considering the vector space generated by the set X, and linearly extending the map S.

A lesser known example of solutions for the set-theoretical Yang-Baxter equation is the following. Given a binary relation R on X (

^{op} is an equivalence relation on X and the complement relation of R is a strict partial order on each class of R ^{op} (where R^{op} is the opposite relation of R).

Let A be a k-algebra, and

Then, according to [

(i) The above operator is connected to the theory of entwining structures and corings (see [

(ii) Using the method of [

(iii) Other generalizations and properties for

The Kaplansky’s tenth conjecture about the classification of finite dimensional Hopf algebras was proved in negative by references [

According to [

The author of [

Some directions for future research are: Finding the smallest dimension of a vector space for which Theorem 2 holds, and the study of solutions derived from relations for other non-linear equations from Quantum Group Theory. For example, these equations might be: Pentagonal equation, Long equation, Frobenius-separability equation (see [

The author would like to thank Sorin Dascalescu, Tomasz Brzezinski and the referees.