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''totally ordered group''
 
''totally ordered group''
  
A [[Po-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o1100102.png" />-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o1100103.png" /> whose [[Partial order|partial order]] is total (cf. also [[Totally ordered group|Totally ordered group]]). A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o1100104.png" />-group is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o1100105.png" />-group if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o1100106.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o1100107.png" /> is the positive cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o1100108.png" />. Every Abelian torsion-free group, every locally nilpotent-torsion free group and every free group can be turned into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o1100109.png" />-group. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001010.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001011.png" /> is a quotient group of a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001012.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001013.png" /> with suitable order by some convex normal subgroup. Direct, Cartesian, free, and direct wreath products of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001014.png" />-groups can be turned in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001015.png" />-groups by extending the orders of the factors. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001016.png" />-group is a [[Topological group|topological group]] respect with to the interval topology. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001017.png" />-group can be imbedded into simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001018.png" />-groups. There exist non-Hopfian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001019.png" />-groups.
+
A [[Po-group| $  po $-
 +
group]] $  G $
 +
whose [[Partial order|partial order]] is total (cf. also [[Totally ordered group|Totally ordered group]]). A $  po $-
 +
group is an o $-
 +
group if and only if $  P ( G ) \cup P ( G ) ^ {- 1 } = G $,  
 +
where $  P ( G ) $
 +
is the positive cone of $  G $.  
 +
Every Abelian torsion-free group, every locally nilpotent-torsion free group and every free group can be turned into an o $-
 +
group. Any o $-
 +
group $  G $
 +
is a quotient group of a free o $-
 +
group $  F $
 +
with suitable order by some convex normal subgroup. Direct, Cartesian, free, and direct wreath products of o $-
 +
groups can be turned in o $-
 +
groups by extending the orders of the factors. Any o $-
 +
group is a [[Topological group|topological group]] respect with to the interval topology. Any o $-
 +
group can be imbedded into simple o $-
 +
groups. There exist non-Hopfian o $-
 +
groups.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001020.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001021.png" />-group, then the group-theoretical structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001022.png" /> is very nice. In particular, any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001023.png" />-group has a subnormal solvable system of subgroups, it is a torsion-free group and a group with unique roots. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001024.png" />-groups are suitable examples in the study of many classes of groups. The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001025.png" /> of convex subgroups of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001026.png" />-group is well studied (cf. [[Convex subgroup|Convex subgroup]]). In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001027.png" /> is the central series of isolated normal subgroups for any locally nilpotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001028.png" />-group.
+
If $  G $
 +
is an o $-
 +
group, then the group-theoretical structure of $  G $
 +
is very nice. In particular, any o $-
 +
group has a subnormal solvable system of subgroups, it is a torsion-free group and a group with unique roots. Thus, o $-
 +
groups are suitable examples in the study of many classes of groups. The system $  {\mathcal C} ( G ) $
 +
of convex subgroups of an o $-
 +
group is well studied (cf. [[Convex subgroup|Convex subgroup]]). In particular, $  {\mathcal C} ( G ) $
 +
is the central series of isolated normal subgroups for any locally nilpotent o $-
 +
group.
  
It is useful to apply methods from the theory of semi-groups to questions about orderability of a group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001029.png" /> be the normal sub-semi-group of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001030.png" /> generated by a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001031.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001032.png" /> is orderable (i.e., it is possible to turn <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001033.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001034.png" />-group) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001035.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001036.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001037.png" />). Every partial order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001038.png" /> can be extended to a total order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001039.png" /> if and only if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001041.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001042.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o110/o110010/o11001043.png" />,
+
It is useful to apply methods from the theory of semi-groups to questions about orderability of a group. Let $  S ( X ) $
 +
be the normal sub-semi-group of a group $  G $
 +
generated by a set $  X \subset  G $.  
 +
Then $  G $
 +
is orderable (i.e., it is possible to turn $  G $
 +
in an o $-
 +
group) if and only if $  e \notin S ( a _ {1} \dots a _ {n} ) $
 +
for all $  a _ {1} \dots a _ {n} \in G $(
 +
$  a _ {i} \neq e $).  
 +
Every partial order $  Q $
 +
can be extended to a total order on $  G $
 +
if and only if for all $  x \in G $,  
 +
$  x \neq e $
 +
implies $  e \notin S ( x ) $
 +
and for all $  x,y,z \in G $,
  
<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/o110/o110010/o11001044.png" /></td> </tr></table>
+
$$
 +
x,y \in S ( z ) \Rightarrow S ( x ) \cap S ( y ) \neq \emptyset.
 +
$$
  
 
Groups with this property are called fully orderable. If a group is torsion-free and Abelian, or locally nilpotent or orderable metabelian, then it is fully orderable. The class of fully orderable groups is closed under formation of direct products and is locally closed. It is not closed under formation of subgroups, Cartesian products, free products. It is non-axiomatizable.
 
Groups with this property are called fully orderable. If a group is torsion-free and Abelian, or locally nilpotent or orderable metabelian, then it is fully orderable. The class of fully orderable groups is closed under formation of direct products and is locally closed. It is not closed under formation of subgroups, Cartesian products, free products. It is non-axiomatizable.

Latest revision as of 08:03, 6 June 2020


totally ordered group

A $ po $- group $ G $ whose partial order is total (cf. also Totally ordered group). A $ po $- group is an $ o $- group if and only if $ P ( G ) \cup P ( G ) ^ {- 1 } = G $, where $ P ( G ) $ is the positive cone of $ G $. Every Abelian torsion-free group, every locally nilpotent-torsion free group and every free group can be turned into an $ o $- group. Any $ o $- group $ G $ is a quotient group of a free $ o $- group $ F $ with suitable order by some convex normal subgroup. Direct, Cartesian, free, and direct wreath products of $ o $- groups can be turned in $ o $- groups by extending the orders of the factors. Any $ o $- group is a topological group respect with to the interval topology. Any $ o $- group can be imbedded into simple $ o $- groups. There exist non-Hopfian $ o $- groups.

If $ G $ is an $ o $- group, then the group-theoretical structure of $ G $ is very nice. In particular, any $ o $- group has a subnormal solvable system of subgroups, it is a torsion-free group and a group with unique roots. Thus, $ o $- groups are suitable examples in the study of many classes of groups. The system $ {\mathcal C} ( G ) $ of convex subgroups of an $ o $- group is well studied (cf. Convex subgroup). In particular, $ {\mathcal C} ( G ) $ is the central series of isolated normal subgroups for any locally nilpotent $ o $- group.

It is useful to apply methods from the theory of semi-groups to questions about orderability of a group. Let $ S ( X ) $ be the normal sub-semi-group of a group $ G $ generated by a set $ X \subset G $. Then $ G $ is orderable (i.e., it is possible to turn $ G $ in an $ o $- group) if and only if $ e \notin S ( a _ {1} \dots a _ {n} ) $ for all $ a _ {1} \dots a _ {n} \in G $( $ a _ {i} \neq e $). Every partial order $ Q $ can be extended to a total order on $ G $ if and only if for all $ x \in G $, $ x \neq e $ implies $ e \notin S ( x ) $ and for all $ x,y,z \in G $,

$$ x,y \in S ( z ) \Rightarrow S ( x ) \cap S ( y ) \neq \emptyset. $$

Groups with this property are called fully orderable. If a group is torsion-free and Abelian, or locally nilpotent or orderable metabelian, then it is fully orderable. The class of fully orderable groups is closed under formation of direct products and is locally closed. It is not closed under formation of subgroups, Cartesian products, free products. It is non-axiomatizable.

This article complements the article Totally ordered group (Vol. 9).

References

[a1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
[a2] A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Halstad (1974) (In Russian)
[a3] R.T.B. Mura, A.H. Rhemtulla, "Orderable groups" , M. Dekker (1977)
[a4] V.M. Kopytov, N.Ya. Medvedev, "The theory of lattice-ordered groups" , Kluwer Acad. Publ. (1994) (In Russian)
How to Cite This Entry:
O-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=O-group&oldid=48034
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article