A differential bialgebra associated to a set theoretical solution of the Yang-Baxter equation

For a set theoretical solution of the Yang-Baxter equation $(X,\sigma )$, we define a d.g. bialgebra $B=B(X,\sigma )$, containing the semigroup algebra $A=k\{X\}\mathrm{/}⟨xy=zt:\sigma (x,y)=(z,t)⟩$, such that $k\otimes_A B\otimes_Ak$and$$\mathrm{H}\mathrm{o}\mathrm{m}$_{A-A}(B,k)$ are respectively the homology and cohomology complexes computing biquandle homology and cohomology defined in \cite{CJKS} and other generalizations of cohomology of rack-quanlde case (for example defined in \cite{CES}). This algebraic structure allow us to show the existence of an associative product in the cohomology of biquandles, and a comparison map with Hochschild (co)homology of the algebra $A$.