There shouldn't be any relations in general. Given an algebra $A$, there can be multiple different co-multiplications on $A$ to make it a bialgebra, or even a Hopf algebra. For instance, let $M(G)$ be the set of functions $G\rightarrow k$ where $G$ is a finite group and $k$ is a field. This is a $k$-algebra using pointwise multiplication. Every different group structure on $G$ lead to different Hopf algebra structure on $M(G)$.

There shouldn't be any relations in general. Given an algebra $A$, there can be multiple different co-multiplications on $A$ to make it a bialgebra, or even a Hopf algebra. For instance, let $M(G)$ be the set of functions $G\rightarrow k$ where $G$ is a finite group and $k$ is a field. This is a $k$-algebra using pointwise multiplication. Every different group structures on $G$ lead to different Hopf algebra structure on $M(G)$.