Definition (Formal group). Let $R$ be a ring. A formal group over $R$ is a power series $F(X,Y)\in R[[X,Y]]$ satisfying

• (i)
  $F(X,Y)=F(Y,X)$
• (ii)
  $F(X,0)=X$ and $F(0,Y)=Y$
• (iii)
  $F(X,F(Y,Z))=F(F(X,Y),Z)$

This looks like it would only define a monoid, but in fact inverses are guaranteed to exist in this context.