Extensions of Yang–Baxter sets

This paper is a first step in constructing the category of braided sets and its closest relative, the category of Yang–Baxter sets. Our main emphasis is on the construction of morphisms and extensions of Yang–Baxter sets. This problem is important for the possible constructions of new solutions of the Yang–Baxter equation and the braid equation. Our main result is the description of a family of solutions of the Yang–Baxter equation on $B \otimes C$ and on $B \times C$, given two linear (set-theoretic) solutions $(B, R^B)$ and $(C, R^C)$ of the Yang–Baxter equation.