= Convex subgroups of a finite-rank ordered Abelian group
{title2=$\#\{\text{convex subgroups}\}\le r+1$}
For a <torsion-free abelian group> of finite rational rank $r$, each <convex subgroup of an ordered group> is pure: if $ng\in C$ for a positive integer $n$, then $g$ lies between zero and $ng$ or their negatives and hence belongs to $C$. A proper inclusion of convex subgroups therefore increases rational rank. They form a chain, so there are at most $r+1$, including zero and the whole group. A lexicographically ordered copy of $\mathbb Z^r$ attains the bound.
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