Ordered group
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
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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
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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) |
Comments
If the order relation on the partially ordered group defines a lattice (i.e. for all there exists a greatest lower bound
and a least upper bound
), then one speaks of a lattice-ordered group or
-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].
References
[a1] | M. Anderson, T. Feil, "Lattice-ordered groups. An introduction" , Reidel (1988) |
[a2] | A.M.W. Glass (ed.) W.Ch. Holland (ed.) , Lattice-ordered groups. Advances and techniques , Kluwer (1989) |
[a3] | J. Martinez (ed.) , Ordered algebraic structures , Kluwer (1989) |
Ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordered_group&oldid=16970