Hi there, I have a totally ordered group $(\mathbb{R};\leq,\oplus,0,-)$ with the reals as base set satisfying monotonicity, i.e. for all $x,y,z$ we have that if $y\leq z$ then $x\oplus y \leq x\oplus z$, and I am wondering if this already suffices to show that the group is archimedean. (i.e., that for all $a,b$ with $a \leq b$ there exists an $n$ such that $\underbrace{a\oplus\ldots\oplus a}_n \geq b$).

I strongly suppose it is, as I couldn't find any counterexample, but neither can I find an convincing proof. (My further goal is to show that the group is isomorphic to $(\mathbb{R};\leq,+,0,-)$ which should be immediate once I can proof archimedeanness)

thanks in advance, Chris

orderedgroup? etc.) and define terms which you are not sure are standard. $\endgroup$ – Noah Stein May 28 '12 at 21:39