Inverse semi-braces and the Yang-Baxter equation

Extensions and Well's type exact sequence of skew braces

“…Then we have the maps ν : (H, •) → Aut(I, +)/N , μ : (H, +) → Aut(I, )/ Inn(I, +) and σ : (H, •) → Aut(I, •)/ Inn(I, •) defined by ν, µ and σ composing with natural projections respectively. We call the triplet χ := (ν, µ, σ) satisfying ( 9), (10), and (11) an action of H on I and corresponding triplet χ := (ν, μ, σ) will be called a coupling from H to I corresponding to (ν, µ, σ). Let χ = (ν, µ, σ) and χ ′ = (ν ′ , µ ′ , σ ′ ) be two actions.…”

“…) and (β 2 , τ 2 ) are 2-cocycles with the same action χ 2 . Then from ( 9), (10) and (11) it follows that β 3 (h 1 , h 2 ) ∈ Z(I, +) and τ 3 (h 1 , h 2 ) ∈ Soc(I) for all h 1 , h 2 ∈ H. If we take I to be trivial skew brace, then Z(I, +) = Soc(I) = Ann(I). Finally we get β 3 , τ 3 : H → Ann(I).…”

“…In 2007, Rump [20] introduced classical braces to study involutive and non-degenerate solutions of the Yang-Baxter equation. Later, Guarnieri and Vendramin [14] generalized this concept to skew brace to study the non-degenerate solution of the Yang-Baxter equation, which is further generalized to semi-braces by Catino, Colazzo, and Stefanelli in [11] to study non-bijective solutions of the Yang-Baxter equation. In [7], Ben David and Ginosar investigated extensions of bijective 1-cocycles.…”

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

“…Among the seminal papers, we mention those of Gateva-Ivanova and Van den Bergh [18], Gateva-Ivanova and Majid [19], Etingof, Schedler, and Soloviev [16], Lu, Yan, and Zhu [27]. For more details on the development of the studies to date, we refer the reader to the introduction of [10] and references therein. If S is a set, a map r : S × S −→ S × S satisfying the relation (r × id S ) (id S ×r ) (r × id S ) = (id S ×r ) (r × id S ) (id S ×r ) is said to be a set-theoretic solution of the Yang-Baxter equation, or shortly a solution, on S. If S and T are sets, two solutions r and s on S and T , respectively, are called equivalent if there exists a bijective map f : S → T such that ( f × f )r = s( f × f ), see [16].…”

A note on Garside monoids and Braces

“…There exist several extensions of the definition of a left brace, and in particular extensions for which the underlying set is a semigroup. Indeed, in [2], F. Catino, I. Colazzo and P. Stefanelli define a left semi-brace to be a triple (B, +, •), with (B, +) a left-cancellative semigroup, (B, •) a group, and a left-distributivity-like axiom, and in [3], F. Catino, M. Mazzotta and P. Stefanelli define a left inverse semi-brace to be a triple (B, +, •), with (B, +) a semigroup, (B, •) an inverse semigroup, and a left-distributivity-like axiom. In [9], we establish a connection between set-theoretic solutions and Thompson's group F , and in this context, we define a left partial brace to be a triple (B, ⊕, •) such that (B, ⊕) is a partial commutative monoid, (B, •) is an inverse monoid, and the left-distributivity axiom holds.…”