A
group
with an
order relation
such that for any
in
the inequality
entails
.
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
into an ordered group
is a
homomorphism
of
into
such that
,
,
implies
in
.
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
with respect to a convex subgroup
is totally ordered by putting
if and only if
.
If
is a convex normal subgroup of a totally ordered
group, then this order relation turns the quotient group
into a totally ordered group.
The system
of convex subgroups of a totally ordered group possesses the following properties: a)
is totally ordered by inclusion and closed under intersections and unions; b)
is
infra-invariant,
i.e. for any
and any
one has
;
c) if
is a
jump
in
,
i.e.
,
,
and there is no convex subgroup between them, then
is normal in
,
the quotient group
is an
Archimedean group
and
where
is the normalizer of
in
(cf.
Normalizer of a subset);
and d) all subgroups of
are
strongly isolated,
i.e. for any finite set
in
and any subgroup
the relation
entails
.
An extension
of an ordered group
by an ordered group (cf.
Extension of a group)
is an ordered group if the order in
is stable under all inner automorphisms of
.
An extension
of an ordered group
by a finite group is an ordered group if
is torsion-free and if the order in
is stable under all inner automorphisms of
.
The order type of a countable ordered group has the form
,
where
are the order types of the set of integers and of rational numbers, respectively, and
is an arbitrary countable ordinal. Every ordered group
is a
topological group
relative to the interval topology, in which a base
of open sets consists of the open intervals
A
convex subgroup
of an ordered group is open in this topology.