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A [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701201.png" /> with an [[Order relation|order relation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701202.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701203.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701204.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701205.png" /> entails <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701206.png" />. If the order is total (respectively, partial), one speaks of a [[Totally ordered group|totally ordered group]] (respectively, a [[Partially ordered group|partially ordered group]]).
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An order homomorphism of a (partially) ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701207.png" /> into an ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701208.png" /> is a [[Homomorphism|homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o0701209.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012010.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012013.png" />, implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012015.png" />. The kernels of order homomorphisms are the convex normal subgroups (cf. [[Convex subgroup|Convex subgroup]]; [[Normal subgroup|Normal subgroup]]). The set of right cosets of a totally ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012016.png" /> with respect to a convex subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012017.png" /> is totally ordered by putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012018.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012020.png" /> is a convex normal subgroup of a totally ordered group, then this order relation turns the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012021.png" /> into a totally ordered group.
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{{TEX|done}}
  
The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012022.png" /> of convex subgroups of a totally ordered group possesses the following properties: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012023.png" /> is totally ordered by inclusion and closed under intersections and unions; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012024.png" /> is infra-invariant, i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012025.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012026.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012027.png" />; c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012028.png" /> is a jump in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012029.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012031.png" />, and there is no convex subgroup between them, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012032.png" /> is normal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012033.png" />, the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012034.png" /> is an [[Archimedean group|Archimedean group]] and
+
A [[Group|group]]  $  G $
 +
with an [[Order relation|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|totally ordered group]] (respectively, a [[Partially ordered group|partially ordered group]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012035.png" /></td> </tr></table>
+
An order homomorphism of a (partially) ordered group  $  G $
 +
into an ordered group  $  H $
 +
is a [[Homomorphism|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|Convex subgroup]]; [[Normal 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.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012036.png" /> is the normalizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012038.png" /> (cf. [[Normalizer of a subset|Normalizer of a subset]]); and d) all subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012039.png" /> are strongly isolated, i.e. for any finite set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012041.png" /> and any subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012042.png" /> the relation
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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|Archimedean group]] and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012043.png" /></td> </tr></table>
+
$$
 +
[ [ N _ {G} (B) , N _ {G} (B) ] , B ]  \subseteq  A ,
 +
$$
  
entails <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012044.png" />.
+
where  $  N _ {G} (B) $
 +
is the normalizer of  $  B $
 +
in  $  G $(
 +
cf. [[Normalizer of a subset|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
  
An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012045.png" /> of an ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012046.png" /> by an ordered group (cf. [[Extension of a group|Extension of a group]]) is an ordered group if the order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012047.png" /> is stable under all inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012048.png" />. An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012049.png" /> of an ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012050.png" /> by a finite group is an ordered group if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012051.png" /> is torsion-free and if the order in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012052.png" /> is stable under all inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012053.png" />.
+
$$
 +
x \cdot g _ {1}  ^ {-1} x g _ {1} \dots g _ {n}  ^ {-1}
 +
x g _ {n}  \in H
 +
$$
  
The order type of a countable ordered group has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012055.png" /> are the order types of the set of integers and of rational numbers, respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012056.png" /> is an arbitrary countable ordinal. Every ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012057.png" /> is a [[Topological group|topological group]] relative to the interval topology, in which a base of open sets consists of the open intervals
+
entails  $  x \in H $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012058.png" /></td> </tr></table>
+
An extension  $  G $
 +
of an ordered group  $  H $
 +
by an ordered group (cf. [[Extension of a group|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|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|convex subgroup]] of an ordered group is open in this topology.
 
A [[Convex subgroup|convex subgroup]] of an ordered group is open in this topology.
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====References====
 
====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></table>
 
<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></table>
 
 
  
 
====Comments====
 
====Comments====
If the order relation on the partially ordered group defines a [[Lattice|lattice]] (i.e. for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012059.png" /> there exists a greatest lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012060.png" /> and a least upper bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012061.png" />), then one speaks of a [[Lattice-ordered group|lattice-ordered group]] or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070120/o07012063.png" />-group; cf. also [[Ordered semi-group|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 [[#References|[a1]]]–[[#References|[a3]]].
+
If the order relation on the partially ordered group defines a [[Lattice|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|lattice-ordered group]] or $  l $-
 +
group; cf. also [[Ordered semi-group|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 [[#References|[a1]]]–[[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Anderson,  T. Feil,  "Lattice-ordered groups. An introduction" , Reidel  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M.W. Glass (ed.)  W.Ch. Holland (ed.) , ''Lattice-ordered groups. Advances and techniques'' , Kluwer  (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Martinez (ed.) , ''Ordered algebraic structures'' , Kluwer  (1989)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Anderson,  T. Feil,  "Lattice-ordered groups. An introduction" , Reidel  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M.W. Glass (ed.)  W.Ch. Holland (ed.) , ''Lattice-ordered groups. Advances and techniques'' , Kluwer  (1989)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Martinez (ed.) , ''Ordered algebraic structures'' , Kluwer  (1989)</TD></TR></table>

Latest revision as of 16:40, 31 March 2020


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)

Comments

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].

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)
How to Cite This Entry:
Ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ordered_group&oldid=44948
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article