Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 4 / 3 / C

For a faithful transitive ordered action, primitivity here means that the only invariant equivalence relations with convex classes are equality and the universal relation. The trichotomy is: a regular Archimedean action, an order two-transitive action, or a periodic action. In the regular case the chain is an ordered additive subgroup of $\mathbb{R}$, acting by translations. In the periodic case there is a fixed-point-free period $z$ on the Dedekind completion, commuting with the action and having cofinal and coinitial integer iterates; inside a period interval the relevant stabilizer acts order two-transitively. These are the cases of the McCleary trichotomy theorem.

Only the regular Archimedean case satisfies the inequality. It is a linearly ordered group. For positive $a,b$, if $a\le b$, then $ab\le b^2\le b^2{a}^2$; if $b\le a$, then $ab\le a^2\le b^2{a}^2$. Taking $a=|f|$, $b=|g|$ proves the displayed inequality.

To exclude an order two-transitive action, choose $x<y<z<t$. Choose an automorphism taking $(x,y)$ to $(y,z)$ and replace it by its pointwise join with the identity, giving positive $a$ with $xa=y$, $ya=z$. Similarly obtain positive $b$ with $xb=x$, $yb=t$. ThenThus $ab\nleqq b^2{a}^2$, while $|a|=a$ and $|b|=b$. The same construction inside a period interval excludes the periodic case, using the order two-transitive stabilizer supplied by the classification.