# Convex subgroup

A subgroup $H$ of a (partially) ordered group $G$ which is a convex subset of $G$ with respect to the given order relation. Normal convex subgroups are exactly the kernels of homomorphisms of the partially ordered group which preserve the order. A subgroup of an orderable group which is convex for any total order is called an absolutely convex subgroup; if it is convex only for a certain total order, it is called a relatively convex subgroup. The intersection of all non-trivial relatively convex subgroups of an orderable group is an absolutely convex subgroup; the union of all proper relatively convex subgroups is also an absolutely convex subgroup. Torsion-free Abelian groups have no non-trivial absolutely convex subgroups. A subgroup $H$ of a completely ordered group $G$ is absolutely convex if and only if for any elements $g\not\in H$, $a\in H$ the intersection $S(g)\cap S(ga)$ is non-empty, where $S(x)$ is the minimal invariant sub-semi-group of $G$ containing $x$. A convex $l$-subgroup $H$ of a lattice-ordered group is isolated, i.e. for any natural number $n$, it follows from $x^n\in H$ that $x\in H$.