Oops Email is great. I cited Peter before he could correct himself. A possible remedy would be to consider not just monoids with zero, but those equipped with a homomorphism from R, so that points are retractions of that. This cuts down on the automorphisms, at least the naturally available ones. Bill On Wed Mar 5 6:22 , "Prof. Peter Johnstone" sent:
As George Janelidze pointed out to me, there was an error in what I wrote yesterday: the multiplicative monoid structure of C(X) determines X, and hence the ring structure of C(X), up to isomorphism (this is a 1949 result of A.N. Milgram), but it doesn't determine the additive structure uniquely, since one can take the standard addition and "conjugate" it by a multiplicative automorphism of R, in the same way that Steve points out for finite fields (e.g. one could define a new addition by f +' g = (f^3 + g^3)^{1/3}).
Peter Johnstone