View source for Convex subgroup

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A subgroup $H$ of a (partially) [[Ordered group|ordered group]] $G$ which is a [[Convex subset|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|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|lattice-ordered group]] is isolated, i.e. for any natural number $n$, it follows from $x^n\in H$ that $x\in H$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Kokorin,   V.M. Kopytov,   "Fully ordered groups" , Israel Program Sci. Transl.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Fuchs,   "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>