# Ordered group

A group $G$ with an order relation $\leq$ such that for any $a , b , x , y$ in $G$ the inequality $a \leq b$ entails $x a y \leq x b y$. If the order is total (respectively, partial), one speaks of a totally ordered group (respectively, a partially ordered group).

An order homomorphism of a (partially) ordered group $G$ into an ordered group $H$ is a homomorphism $\phi$ of $G$ into $H$ such that $x \leq y$, $x , y \in G$, implies $x \phi \leq y \phi$ in $H$. The kernels of order homomorphisms are the convex normal subgroups (cf. Convex subgroup; Normal subgroup). The set of right cosets of a totally ordered group $G$ with respect to a convex subgroup $H$ is totally ordered by putting $H x \leq H y$ if and only if $x \leq y$. If $H$ is a convex normal subgroup of a totally ordered group, then this order relation turns the quotient group $G / H$ into a totally ordered group.

The system $\Sigma (G)$ of convex subgroups of a totally ordered group possesses the following properties: a) $\Sigma (G)$ is totally ordered by inclusion and closed under intersections and unions; b) $\Sigma (G)$ is infra-invariant, i.e. for any $H \in \Sigma (G)$ and any $x \in G$ one has $x ^ {-1} H x \in \Sigma (G)$; c) if $A < B$ is a jump in $\Sigma (G)$, i.e. $A , B \in \Sigma (G)$, $A \subset B$, and there is no convex subgroup between them, then $A$ is normal in $B$, the quotient group $B / A$ is an Archimedean group and

$$[ [ N _ {G} (B) , N _ {G} (B) ] , B ] \subseteq A ,$$

where $N _ {G} (B)$ is the normalizer of $B$ in $G$( cf. Normalizer of a subset); and d) all subgroups of $\Sigma (G)$ are strongly isolated, i.e. for any finite set $x, g _ {1} \dots g _ {n}$ in $G$ and any subgroup $H \in \Sigma (G)$ the relation

$$x \cdot g _ {1} ^ {-1} x g _ {1} \dots g _ {n} ^ {-1} x g _ {n} \in H$$

entails $x \in H$.

An extension $G$ of an ordered group $H$ by an ordered group (cf. Extension of a group) is an ordered group if the order in $H$ is stable under all inner automorphisms of $G$. An extension $G$ of an ordered group $H$ by a finite group is an ordered group if $G$ is torsion-free and if the order in $H$ is stable under all inner automorphisms of $G$.

The order type of a countable ordered group has the form $\eta ^ \alpha \xi$, where $\eta , \xi$ are the order types of the set of integers and of rational numbers, respectively, and $\alpha$ is an arbitrary countable ordinal. Every ordered group $G$ is a topological group relative to the interval topology, in which a base of open sets consists of the open intervals

$$( a , b ) = \{ {x \in G } : {a < x < b } \} .$$

A convex subgroup of an ordered group is open in this topology.

#### References

 [1] A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)

If the order relation on the partially ordered group defines a lattice (i.e. for all $a,b \in G$ there exists a greatest lower bound $a \wedge b$ and a least upper bound $a \lor b$), then one speaks of a lattice-ordered group or $l$- group; cf. also Ordered semi-group. These turn up naturally in many branches of mathematics. For a survey of the current state-of-the-art in this field see [a1][a3].