On uniconnected solutions of the Yang-Baxter equation and Dehornoy's class (Q6588187)

The paper under review studies indecomposable involutive set-theoretical solutions of the Yang-Baxter equation with specific focus their permutation group structures and their classification. The authors build upon foundational work that identified involutive solutions as central to the classification problem using tools such as twisted unions and retractions. See for example [\textit{M. Castelli} et al., J. Pure Appl. Algebra 223, No. 10, 4477--4493 (2019; Zbl 1412.16023); \textit{P. Etingof} et al., Duke Math. J. 100, No. 2, 169--209 (1999; Zbl 0969.81030); \textit{L. Vendramin}, J. Pure Appl. Algebra 220, No. 5, 2064--2076 (2016; Zbl 1337.16028)].\N\NThe paper primarily focuses on the subclass of uniconnected solutions within all indecomposable involutive set-theoretical solutions of the Yang-Baxter equation. A solution is called uniconnected if its permutation group acts transitively. The authors study those solutions whose permutation group is nonabelian of size \(q^np\) with a normal subgroup of size \(p\), where \(q\) and \(p\) are two district primes. They show that indecomposable solutions with minimal non-cyclic group are uniconnected. The results the authors obtain underscore the centrality of uniconnectedness -- the solutions characterizing these classes.