Perspectives on the Yang-Baxter Equation in Bck-Algebras

We present set-theoretical solutions of the Yang-Baxter equation in BCK–algebras. Some solutions in BCK−algebras are not solutions in other structures (such as MV −algebras). Related to our investigations we also consider some new structures: Boolean coalgebras and a unified braid condition – quantum Yang-Baxter equation. Finally, we will see how poetry has accompanied the development / history of the Yang–Baxter equation.


Introduction
The Yang-Baxter equation was first discovered by Nobel laureate C.N. Yang in theoretical physics ( [1]) and by R.J. Baxter in statistical mechanics ( [2,3]). It turned out to be one of the main equations in mathematical physics, integrable systems, quantum algebraic systems, the theory of quantum groups, quantum computing, knot theory, braided categories, etc (see [4]). Yang initially considered the matrix equation F (x)G(x + y)F (y) = G(y)F (x + y)G(x), and found an explicit solution where F (x) and G(x) are rational functions. Other versions of this equation were later proved to be very useful, and many scientists have used the axioms of various algebraic structures (Hopf algebras, Yetter-Drinfeld categories, quandles, group actions, Lie (super)algebras, (co)algebra structures, Boolean algebras, relations on sets and so on) in order to obtain solutions for these versions of the Yang-Baxter equation ( [5]). F.F. Nichita et al. obtained results on Jordan algebras and Jordan coalgebras, and related them with the Yang-Baxter equations (see, for example, [6,7] and the references therein). Constructions of quantum gates and link invariants from solutions of the Yang-Baxter equation were described in [8,9]. Some solutions for the Yang-Baxter equation in MV-algebras, Wajsberg-algebras, MTL-algebras, weak implication algebras, and lattice effcet algebras were investigated in [10,11,12,13].
In this paper we consider BCK-algebras, which were introduced by Y. Imai and K. Iseki ([14]). BCK-algebras are generalizations of the notion of algebraic sets with subtraction and the notion of implication algebra ( [14,15]). These notions are derived in two different ways: one of them is based on set theory and the other is based on non-classical and classical propositional calculus system.
In the next section we will recall some fundamental definitions, lemmas and theorems which are needed for constructing solutions of Yang-Baxter equation in BCKalgebras. We will also define Boolean coalgebras. In Section 3, we will present explicit set-theoretical solutions. Our propositions, lemmas and theorems will hopefully give a new perspective on Yang-Baxter equation (in BCK-algebras). We also propose a braid-quantum Yang-Baxter equation, whose solutions include both solutions of the braid equation and solutions to the quantum Yang-Baxter equation. Finally, a section on poetry related to the Yang-Baxter equation concludes our paper.

Preliminaries
Throughout of this section, we give fundamental definitions, lemmas and theorems about the structure of BCK− algebras. These notions are taken from [16].
for each x, y ∈ A.
Lemma 2.2. The binary relation ≤ on A given by is a partial order on A with 1 as the biggest element.
As opposed to Lemma 2.2, the poset (A; ≤) has no particular property because any poset (P ; ≤) with 1 can be made a BCK−algebra by setting a → b := 1 for a ≤ b, and a → b := b otherwise for any a, b ∈ P . Definition 2.3. An algebra A = (A; →, 1) is said to be a bounded BCK − algebra, where (A; →, 1) is a BCK-algebra with the least element 0 such that 0 → x = x.
The commutative BCK−algebras can be characterized as join-semilattices by defining the ∨−operation as follows: x ∨ y := (x → y) → y. for each x, y ∈ A.
Definition 2.8. A commutative BCK-algebra is a BCK-algebra that satisfies the identity x y = y x for each x, y ∈ A.
Theorem 2.9. Let A = (A; →, 1) be a BCK−algebra. Define a unary operation N k on the section [k, 1] = {x ∈ A : k ≤ x} for each a ∈ A by Then the structure Ω(A) = (A; , N k , 1) is satisfies the following quasi-identities: for each a, b, c ∈ A.
Lemma 2.10. Let (A; ) be defined as Theorem 2.9. The binary relation ≤ is defined by x ≤ y if and only if x y = y.
Then, the binary relation ≤ is a partial order on A. Moreover, x y is the least upper bound of x and y. Dually, x y is the greatest lower bound of x and y Lemma 2.11. Let A = (A; →, 1) be a BCK−algebra. The binary operation is defined as Theorem 2.9. Then the following statements are equivalent to each other.
for each x, y, z ∈ A then, it is called a positive implicative BCK−algebra.
for each x, y, z ∈ A then, it is called a negative implicative BCK−algebra.
At the end of this preliminary section, let us define a new structure which will be used in our search for solutions to the Yang-Baxter equation. Further investigations in the framework of BCK-algebras will continue in the future.
Definition 2.14. A Boolean coalgebra is defined as a 6-tuple C = (C, ∨, ∆, N, 0, 1), where ∨, N, 0 and 1 have the usual properties. (So, ∨ is an associative and commutative operation, N is an involution, The new structure is ∆ : C → C × C, ∆(a) = (a 1 , a 2 ), and we require: For an arbitrary Boolean algebra, we can associate a Boolean coalgebra with ∆(a) = (a ∧ c, a ∧ N (c)). Also, if we recall that a → b = N (a) ∨ b, we obtain a BCK-algebra with the following property:

A Perspective on Yang-Baxter Equation in BCK-Algebras
In this part of this paper, we present some set-theoretical solutions of Yang-Baxter equation in BCK-algebras. Moreover, we define new operators on BCK-algebras, then we obtain new solutions by using thse operators.
Let K be a vector space over the Field Q. We define the twist map by (p ⊗ q) = q ⊗ p, where : K ⊗ K → K ⊗ K. Besides, the identity map of this vector space is defined I : K → K. As a Q−linear map, we define R 12 = R ⊗ I, R 23 = I ⊗ R and Definition 3.1.
[17] A Yang-Baxter operator is an invertible Q−linear map R : K ⊗ K → K ⊗ K and it verifies the braid condition (known as the "Yang-Baxter equation" or the "braid condition") If R verifies the equation (2), then τ • R and R • τ supply the quantum Yang-Baxter equation (known as QY BE):   (2) and (3) lead each to solutions for the following "braidquantum Yang-Baxter equation": Obviously, finding all solutions for the braid-quantum Yang-Baxter equation is an open problem. A first step to solve it would be to construct some solutions for it, and to make a small analysis on those solutions which are not solutions neither for the braid condition, nor for the quantum Yang-Baxter equation.
Back to BCK−algebras, we recall the following definition.
Definition 3.4. [17] Let P be any set. The mapping S(p, q) = (p , q ) is defined from P × P to P × P . The mapping S satisfies the Yang-Baxter equation (or equivalently, "S is a set-theoretical solution of the Yang-Baxter equation") if it holds the following equation which is also equivalent to where S 12 : P × P × P → P × P × P, S 12 (p, q, r) = (p , q , r), S 23 : P × P × P → P × P × P, S 23 (p, q, r) = (p, q , r ), S 13 : P × P × P → P × P × P, S 13 (p, q, r) = (p , q, r ). Now, we may handle verifying the Yang-Baxter equation in BCK-algebras. First of all, we give the following lemma which is needed for further processing of this work.
Lemma 3.6. Let (A, →, 0, 1) be a bounded BCK-algebra. Then, the mapping S(x, y) = (y → 0, x → 0) verifies the braid condition on this structure.  Proof. We define S 12 and S 23 as follows: We show that the equilibrium S 12 • S 23 • S 12 = S 23 • S 12 • S 23 are satisfied for each (x, y, z) ∈ A × A × A. By the help of the Definition 2.1, Lemma 2.4 (c) and (e) and Proposition 3.5 (2) and (3), we have Thus, the Yang-Baxter equation is satisfied in BCK−algebras. The mapping S(x, y) = ((x → 0) → y, 0) is a set-theoretical solution of it on these structures. Lemma 3.9. Let (A, →, 1) be a commutative BCK-algebra. Then, the mapping S(x, y) = ((x → y) → y, y) verifies the braid condition on this structure. Therefore, the Yang-Baxter equation has a set-theoretical solution in BCK-algebras.
Since the Equation (7) is equal to the Equation (8) for all a, b, c ∈ X, we obtain that the Yang-Baxter equation is satisfied in this structure. The mapping S(a, b) = (((b → 0) → (a → 0)) → 0, 0) is a set-theoretical solution of this equation on this structure, whereas it is not a set-theoretical solution of the Yang-Baxter equation in bounded BCK-algebras.
"A piece of literature Is meant for the millennium But its ups and downs are known Already in the authors heart." In THOUGHTS ON MY FIRST THEOREM, F.F. Nichita describes in a poetic manner the beginning of the unification theory of algebras and coalgebras structures in the framework of Yang-Baxter equations ( [20]): (...) The small particle was captured... The common piece of information...
The two streams arrived on my table from overseas were unified...
Returning from a daily walk, the above author finds out that his office is full with literature works, but there is no contradiction in this mixture: A POST-MODERN MANIFEST Once... after a promenade, I gorged a "pomegranate":