New Structure of Skew Braces and their Ideals

Recall that a set B with binary operations + and ∗ is a left brace if (B,+) is an abelian group and the following version of distributive combined with associativity holds: (a + b + a ∗ b) ∗ c = a ∗ c + b ∗ c + a ∗ (b ∗ c), a ∗ (b + c) = a ∗ b + a ∗ c for all a, b, c ∈ B. Moreover (B, ◦) is a group where we define a ◦ b = a + b + a ∗ b. Skew braces have connections to several different topics. In particular, skew braces provide the right algebraic framework to study set-theoretic solutions to the Yang–Baxter equation. The goal of this paper is to construct and study braces (resp. skew -braces) with their ideals like strongly prime ideals as a tool for solutions to the Yang-Baxter equation via changing their conditions. We propose several problems in the theory of semi-braces and using new concepts. We hope that these problems will help to strengthen the interest in the theory of skew braces and set-theoretic solutions to the Yang-Baxter equation. Braces were introduced as a generalization of Jacobson radical rings. It turns out those braces allow us to use ring-theoretic and group-theoretic methods for studying involutive solutions to the Yang–Baxter equation. If braces are replaced by skew braces, then one can use similar methods for studying not necessarily involutive solutions. Here we will collect problems related to skew braces (resp. semi-braces) and set-theoretic solutions to the Yang-Baxter equation.