For a given (X, S, β), where S, β : X × X → X × X are set theoretical solutions of Yang-Baxter equation with a compatibility condition, we define an invariant for virtual (or classical) knots/links using non commutative 2-cocycles pairs (f, g) that generalizes the one defined in [FG2]. We also define, a group U f g nc = U f g nc (X, S, β) and functions π f , π g : X × X → U f g nc (X) governing all 2-cocycles in X.We exhibit examples of computations achieved using [GAP2015].

For a set theoretical solution of the Yang-Baxter equation (X, σ), we define a d.g. bialgebra B = B(X, σ), containing the semigroup algebra A = k{X}/ xy = zt : σ(x, y) = (z, t) , such that k ⊗ A B ⊗ A k and Hom A−A (B, k) are respectively the homology and cohomology complexes computing biquandle homology and cohomology defined in [CEGN, CJKS] and other generalizations of cohomology of rack-quanlde case (for example defined in [CES2]). 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.