Segre products and Segre morphisms in a class of Yang-Baxter algebras

Let $(X,r_X)$ and $(Y,r_Y)$ be finite nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation, and let $A_X=A(\mathbf{\text{k}},X,r_X)$ and $A_Y=A(\mathbf{\text{k}},Y,r_Y)$ be their quadratic Yang-Baxter algebras over a field $\mathbf{\text{k}}\mathrm{.}$ We find an explicit presentation of the Segre product $A_X\circ A_Y$ in terms of one-generators and quadratic relations. We introduce analogues of Segre maps in the class of Yang-Baxter algebras and find their images and their kernels. The results agree with their classical analogues in the commutative case.