# Associativity equation for topological rings and logarithms

Let $$R$$ be a topological ring of characteristic zero. Assume that $$R$$ is commutative. Let $$G :R \times R \to R$$ be a continuous function satisfying $$G(G(x,y),z)=G(x,G(y,z))$$ and $$G(0,x)=x$$, for all $$x,y,z \in R$$.

If we assume that $$G$$ is a convergent power series then $$G$$ is also a formal group law and the Honda's argument(cf. Hazewinkel, "Formal Groups and Applications", Chapter 1, 5.8, p.37) proves that G admits a logarithm, i.e. $$G(x,y)=h^{-1}(h(x)+h(y))$$.

Question : Does $$G$$ admit a logarithm if it is only continuous?

EDIT Assuming that $$G$$ is continuous, what are the additional assumptions that should be made on $$R$$ and $$G$$ to guarantee the existence of logarithms?

• But if $G$ is only continuous then the ring structure plays no rôle at all – მამუკა ჯიბლაძე Aug 9 '20 at 18:18
• As @მამუკაჯიბლაძე says: If $R$ has the discrete topology, then continuity means nothing, and we are just requiring that $G$ is a binary operation that makes $R$ a semigroup with left identity $0$. Such operations exist in abundance; not all of them are groups, so not all of them have logarithms. – darij grinberg Aug 9 '20 at 18:22
• What about $R=Q_p$? – A413 Aug 9 '20 at 18:40
• For $Q_p$ you get arbitrary topological semigroup structure on the Cantor set with a point removed. – მამუკა ჯიბლაძე Aug 9 '20 at 19:35