On the structure of left braces satisfying the minimal condition for subbraces (Q6639806)

In the study of algebraic structures (e.g., groups and rings), it is natural to try and characterize those that satisfy the minimal condition (which is equivalent to the descending chain condition) on their sub-structures when ordered by inclusion. For example, a group is \textit{Artinian} if it satisfies the minimal condition on subgroups, and a ring is \textit{Artinian} if it satisfies the minimal condition on both left and right ideals.\N\NIn this paper, the authors considered the minimal condition in the setting of skew braces. Recall that a skew (left) brace \(A = (A,+,\cdot)\) is a set equipped with two group operations \(+\) and \(\cdot\) such that\N\[\Na\cdot (b+c) = a\cdot b - a + a\cdot c\N\]\Nholds for all \(a,b,c\in A\). Here \(+\), despite the notation, is not assumed to be commutative. Previously, \textit{E. Jespers} et al. [Adv. Math. 385, Article ID 107767, 20 p. (2021; Zbl 1473.16028)] defined \(A\) to be \textit{Artinian} if it satisfies the minimal condition on ideals. To avoid confusion, the authors in the present paper defined \(A\) to be \textit{\(s\)-Artinian} if it satisfies the minimal condition on sub-skew braces.\N\NRecall that a \textit{Chernikov group} is a group that is a finite extension of an abelian Artinian group. It is known that a soluble group is Artinian if and only if it is a Chernikov group (see (4.2.11) and (5.4.23) of [\textit{D. J. S. Robinson}, A course in the theory of groups, Springer, New York (1982; Zbl 0483.20001)], for example). For a long time, Chernikov groups were the only known examples of Artinian groups.\N\NIn this paper, the authors restricted their attention to \textit{weakly soluble} skew braces (Definition 1.1), a notion that was introduced in [\textit{A. Ballester-Bolinches} et al., Adv. Math. 455, Article ID 109880, 27 p. (2024; Zbl 07902804)]. They prove the following result (Theorem A and Corollary 1.2) that is somewhat similar to the case of groups.\N\NTheorem. Let \(A = (A,+,\cdot)\) be a skew brace.\N\begin{itemize}\N\item[1.] If \(A\) is \(s\)-Artinian and weakly soluble, then \((A,+)\) and \((A,\cdot)\) are Chernikov groups.\N\item[2.] If either \((A,+)\) or \((A,\cdot)\) is a Chernikov group, then \(A\) is \(s\)-Artinian.\N\end{itemize}\NAs a consequence, in the case that \(A\) is weakly soluble, one obtains that \((A,+)\) is a Chernikov group if and only if \((A,\cdot)\) is a Chernikov group.\N\NThe authors also proved a structure theorem for infinite weakly soluble skew braces whose additive group is an abelian Chernikov group (Theorem B).