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,σ)(X,σ), we define a d.g. bialgebra B=B(X,σ)B=B(X,σ), containing the semigroup algebra A=k{X}/〈xy=zt:σ(x,y)=(z,t)〉A=k{X}/〈xy=zt:σ(x,y)=(z,t)〉, such that k⊗AB⊗Akk⊗AB⊗Ak and HomA−A(B,k)HomA−A(B,k) are respectively the homology and cohomology complexes computing biquandle homology and cohomology defined in [2] and [5] and other generalizations of cohomology of rack-quandle case (for example defined in [4]). This algebraic structure allows 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.