Let $A$ be an $R$-algebra.


Suppose $A$ has a $R$-coalgebra structure compatible with the algebra structure.
(I.e. there is a comultiplication map $\Delta$ and counit map $\epsilon$ compatible with the multiplication map and unit map of $A$.)


Then I am wondering whether $A$ can have another $R$-coalgebra structure other than $(\Delta, \epsilon)$?



Is a comultiplication structure unique?
Yes, $A$ can have multiple compatible $R$-coalgebra structures. Let $G$ be a finite set and let $A=R(G)$ be the commutative algebra of functions on $G$ with pointwise multiplication. Then any group structure on $G$ gives an Hopf structure on $A$ and non-isomorphic group structures on $G$ leads to non-isomorphic Hopf structures on $A$.


