Generic coverings and injective set-theoretic solutions to the Yang–Baxter equation

A necessary criterion for injective solutions to the set-theoretical Yang–Baxter equation in terms of q -cycle sets is given, improving Soloviev’s syntactic criteria. Supporting evidence is provided that the new criterion is sufficient. Generalizing augmented rack solutions, q -cycle groups are revisited, and a general extension theorem for the underlying solution is established. Close relationships between injective solutions, generic coverings, and extensions of q -cycle groups, are obtained. Several characterizations of generic coverings are given.