(Hopefully, this question falls under analysis. I was unable to match it well with any of the forums.) The proof that the identity element of a binary operation, f: X x X [itex]\rightarrow[/itex] X, is unique is simple and quite convincing: for any e and e' belonging to X, e=f(e,e')=f(e',e)=e'. However, if we take f(m,n)=max(m,n), it appears that any m will have multiple identity elements--the elements of the set of numbers n less than m. There must be something that I'm missing here. Any help would be appreciated. Thanks!