Set-theoretic solutions of the Yang–Baxter equation associated to weak braces

Abstract

We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang–Baxter equation. Specifically, a weak (left) brace is a non-empty set S endowed with two binary operations $$+$$ + and $$\circ $$ ∘ such that both $$(S,+)$$ ( S , + ) and $$(S, \circ )$$ ( S , ∘ ) are inverse semigroups and $$\begin{aligned} a \circ \left( b+c\right) = \left( a\circ b\right) - a + \left( a\circ c\right) \qquad \text {and} \qquad a\circ a^- = - a + a \end{aligned}$$ a ∘ b + c = a ∘ b - a + a ∘ c and a ∘ a - = - a + a hold, for all $$a,b,c \in S$$ a , b , c ∈ S , where $$-a$$ - a and $$a^-$$ a - are the inverses of a with respect to $$+$$ + and $$\circ $$ ∘ , respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution r associated to an arbitrary weak brace S has a behavior close to bijectivity, namely r is a completely regular element in the full transformation semigroup on $$S\times S$$ S × S . In addition, we provide some methods to construct weak braces.

Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation

“…As proved in [13,Theorem 8], the additive semigroup of any weak brace S is a Clifford semigroup. Moreover, to our knowledge, the multiplicative semigroup of a weak brace is not a Clifford semigroup (see, for instance, [13,Example 3]). Thus, the following question arises.…”

“…The aim of this preliminary section is to give essential results on the structures of weak braces, recently introduced in [13] to find solutions of the Yang-Baxter equation. To this purpose, we initially recall some basics on Clifford semigroups, covered in detail in the books [19,25,28].…”

“…However, one can find some instances of weak braces having also a Clifford semigroup as a multiplicative structure among [12,13,14]. Definition 2.…”

“…A generalisation of the skew brace that determines a solution has been newly introduced in [13], and it is named weak brace. To give the definition, we firstly recall that a Clifford semigroup (S, •) is an inverse semigroup, i.e., a semigroup for which every a ∈ S admits a unique a −1 ∈ S satisfying aa −1 a = a and a −1 a −1 = a −1 , having central idempotents.…”

“…for all a, b, c ∈ S, where −a and a − denote the inverses of a with respect to + and •, respectively, and we assume that the multiplication has higher precedence than the addition. As demonstrated in [13,Thereom 10], the additive semigroup of any weak brace is a Clifford semigroup. Moreover, if both (S, +) and (S, •) are groups, then (S, +, •) is a skew brace; in addition, if (S, +) is abelian, then (S, +, •) is a brace [32].…”