Exercise 2, § 2

Let $G$ be a non-commutative linearly ordered group (for example the multiplicative group of a non-commutative ordered division ring). In the set $\mathbf{R}_+ \times G$ consider the set $E$ consisting of $(0, e)$ ($e$ being the identity element of $G$) and all pairs $(x, y)$ where $x$ is any real number $> 0$ and $y$ is any element of $G$. Define a law of composition on $E$ by the rule $(x, y)(x', y') = (x + x', yy')$, and order $E$ lexicographically [i.e. $(x, y) < (x', y')$ if $x < x'$ or if $x = x'$ and $y < y'$]. If we take $I = E$, show that axioms (GR_I), (GR_{II}), (GR_{IIIb}) (Exercise 1) and (GR_{IV}) are satisfied; but that if $f$ is an increasing mapping of $E$ into $\mathbf{R}_+$ such that $f(zz') = f(z) + f(z')$ for all $a$ and $z'$ in $E$, then $f$ is not strictly increasing.