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''right-ordered group''
 
''right-ordered group''
  
A [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r1101102.png" /> endowed with a total order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r1101103.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r1101104.png" />,
+
A [[Group|group]] $  G $
 +
endowed with a total order $  \le $
 +
such that 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/r/r110/r110110/r1101105.png" /></td> </tr></table>
+
$$
 +
x \le y \Rightarrow xz \le yz.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r1101106.png" /> is the positive cone of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r1101107.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r1101108.png" /> (cf. also [[L-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r1101109.png" />-group]]), then:
+
If $  P = P ( G ) = \{ {x \in G } : {x \ge e } \} $
 +
is the positive cone of the $  ro $-
 +
group $  G $(
 +
cf. also [[L-group| $  l $-
 +
group]]), then:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011010.png" />;
+
1) $  P \cdot P \subseteq P $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011011.png" />;
+
2) $  P \cap P ^ {- 1 } = \{ e \} $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011012.png" />. If, in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011013.png" />, there is a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011014.png" /> satisfying 1)–3), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011015.png" /> can given the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011016.png" />-group with positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011017.png" /> by a setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011018.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011019.png" />. The positive cone of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011020.png" />-group is isolated, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011021.png" />.
+
3) $  P \cup P ^ {- 1 } = G $.  
 +
If, in a group $  G $,  
 +
there is a subset $  P $
 +
satisfying 1)–3), then $  G $
 +
can given the structure of a $  ro $-
 +
group with positive cone $  P $
 +
by a setting $  x \le y $
 +
if and only if $  yx ^ {- 1 } \in P $.  
 +
The positive cone of a $  ro $-
 +
group is isolated, i.e., $  x  ^ {n} \in P \Rightarrow x \in P $.
  
The group of order automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011022.png" /> of a totally ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011023.png" /> can be turned into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011024.png" />-group by defining the following relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011025.png" /> on it. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011026.png" /> be any well ordering on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011027.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011028.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011029.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011030.png" /> be the first (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011031.png" />) element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011032.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011033.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011034.png" />-group with respect to the order with positive cone
+
The group of order automorphisms $  { \mathop{\rm Aut} } ( X ) $
 +
of a totally ordered set $  \{ X; \le \} $
 +
can be turned into a $  ro $-
 +
group by defining the following relation $  \le $
 +
on it. Let $  \prec $
 +
be any well ordering on $  X $:  
 +
$  x _ {1} \prec \dots \prec x _  \alpha  \prec \dots $.  
 +
Let $  \varphi \in { \mathop{\rm Aut} } ( X ) $
 +
and let $  x _  \alpha  $
 +
be the first (with respect to $  \prec $)  
 +
element in $  \{ {x \in X } : {x \varphi \neq x } \} $.  
 +
Then $  A ( X ) $
 +
is a $  ro $-
 +
group with respect to the order with positive cone
  
<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/r/r110/r110110/r11011035.png" /></td> </tr></table>
+
$$
 +
P \subset  A ( X ) = \left \{ {\varphi \in { \mathop{\rm Aut} } ( X ) } : {x _  \alpha  \varphi \ge x _  \alpha  } \right \} .
 +
$$
  
Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011036.png" />-group is isomorphic to a subgroup of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011037.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011038.png" /> for some totally ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011039.png" />. There exist simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011040.png" />-groups whose finitely generated subgroups coincide with the commutator subgroup. The class of all groups that can be turned into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011041.png" />-group is a [[Quasi-variety|quasi-variety]], i.e., it is defined by a system of formulas of the form:
+
Any $  ro $-
 +
group is isomorphic to a subgroup of the $  ro $-
 +
group $  { \mathop{\rm Aut} } ( X ) $
 +
for some totally ordered set $  X $.  
 +
There exist simple $  ro $-
 +
groups whose finitely generated subgroups coincide with the commutator subgroup. The class of all groups that can be turned into a $  ro $-
 +
group is a [[Quasi-variety|quasi-variety]], i.e., it is defined by a system of formulas of the form:
  
<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/r/r110/r110110/r11011042.png" /></td> </tr></table>
+
$$
 +
\forall x _ {1} \dots x _ {n} :
 +
$$
  
<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/r/r110/r110110/r11011043.png" /></td> </tr></table>
+
$$
 +
( w _ {1} ( x _ {1} \dots x _ {n} ) = e \& \dots \& w _ {m} ( x _ {1} \dots x _ {n} ) = e )  \Rightarrow
 +
$$
  
<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/r/r110/r110110/r11011044.png" /></td> </tr></table>
+
$$
 +
\Rightarrow
 +
w ( x _ {1} \dots x _ {n} ) = e,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011046.png" /> are the group-theoretical words. This class is closed under formation of subgroups, Cartesian and free products, and extension, and is locally closed.
+
where $  w $,  
 +
$  w _ {i} $
 +
are the group-theoretical words. This class is closed under formation of subgroups, Cartesian and free products, and extension, and is locally closed.
  
The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011047.png" /> of convex subgroups of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011048.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011049.png" /> is a complete chain. It can be non-solvable, non-infra-invariant and non-normal. There exist non-Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011050.png" />-groups without proper convex subgroups.
+
The system $  {\mathcal C} ( G ) $
 +
of convex subgroups of a $  ro $-
 +
group $  G $
 +
is a complete chain. It can be non-solvable, non-infra-invariant and non-normal. There exist non-Abelian $  ro $-
 +
groups without proper convex subgroups.
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011051.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011052.png" /> is Archimedean if for any positive elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011054.png" /> there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011056.png" />. An Archimedean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011057.png" />-group is order-isomorphic to some subgroup of the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011058.png" /> of real numbers with the natural order. The class of Conradian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011060.png" />-groups, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011061.png" />-groups for which the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011062.png" /> is subnormal and the quotient groups of the jumps of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r110/r110110/r11011063.png" /> are Archimedean, is well investigated.
+
A $  ro $-
 +
group $  G $
 +
is Archimedean if for any positive elements $  x,y \in G $
 +
there exists a positive integer $  n $
 +
such that $  x  ^ {n} > y $.  
 +
An Archimedean $  ro $-
 +
group is order-isomorphic to some subgroup of the additive group $  \mathbf R $
 +
of real numbers with the natural order. The class of Conradian $  ro $-
 +
groups, i.e., $  ro $-
 +
groups for which the system $  {\mathcal C} ( G ) $
 +
is subnormal and the quotient groups of the jumps of $  {\mathcal C} ( G ) $
 +
are Archimedean, is well investigated.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. Kopytov,  N.Ya. Medvedev,  "The theory of lattice-ordered groups" , Kluwer Acad. Publ.  (1994)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.T.B. Mura,  A.H. Rhemtulla,  "Orderable groups" , M. Dekker  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.M. Kopytov,  N.Ya. Medvedev,  "The theory of lattice-ordered groups" , Kluwer Acad. Publ.  (1994)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.T.B. Mura,  A.H. Rhemtulla,  "Orderable groups" , M. Dekker  (1977)</TD></TR></table>

Latest revision as of 06:18, 28 March 2023


right-ordered group

A group $ G $ endowed with a total order $ \le $ such that for all $ x,y,z \in G $,

$$ x \le y \Rightarrow xz \le yz. $$

If $ P = P ( G ) = \{ {x \in G } : {x \ge e } \} $ is the positive cone of the $ ro $- group $ G $( cf. also $ l $- group), then:

1) $ P \cdot P \subseteq P $;

2) $ P \cap P ^ {- 1 } = \{ e \} $;

3) $ P \cup P ^ {- 1 } = G $. If, in a group $ G $, there is a subset $ P $ satisfying 1)–3), then $ G $ can given the structure of a $ ro $- group with positive cone $ P $ by a setting $ x \le y $ if and only if $ yx ^ {- 1 } \in P $. The positive cone of a $ ro $- group is isolated, i.e., $ x ^ {n} \in P \Rightarrow x \in P $.

The group of order automorphisms $ { \mathop{\rm Aut} } ( X ) $ of a totally ordered set $ \{ X; \le \} $ can be turned into a $ ro $- group by defining the following relation $ \le $ on it. Let $ \prec $ be any well ordering on $ X $: $ x _ {1} \prec \dots \prec x _ \alpha \prec \dots $. Let $ \varphi \in { \mathop{\rm Aut} } ( X ) $ and let $ x _ \alpha $ be the first (with respect to $ \prec $) element in $ \{ {x \in X } : {x \varphi \neq x } \} $. Then $ A ( X ) $ is a $ ro $- group with respect to the order with positive cone

$$ P \subset A ( X ) = \left \{ {\varphi \in { \mathop{\rm Aut} } ( X ) } : {x _ \alpha \varphi \ge x _ \alpha } \right \} . $$

Any $ ro $- group is isomorphic to a subgroup of the $ ro $- group $ { \mathop{\rm Aut} } ( X ) $ for some totally ordered set $ X $. There exist simple $ ro $- groups whose finitely generated subgroups coincide with the commutator subgroup. The class of all groups that can be turned into a $ ro $- group is a quasi-variety, i.e., it is defined by a system of formulas of the form:

$$ \forall x _ {1} \dots x _ {n} : $$

$$ ( w _ {1} ( x _ {1} \dots x _ {n} ) = e \& \dots \& w _ {m} ( x _ {1} \dots x _ {n} ) = e ) \Rightarrow $$

$$ \Rightarrow w ( x _ {1} \dots x _ {n} ) = e, $$

where $ w $, $ w _ {i} $ are the group-theoretical words. This class is closed under formation of subgroups, Cartesian and free products, and extension, and is locally closed.

The system $ {\mathcal C} ( G ) $ of convex subgroups of a $ ro $- group $ G $ is a complete chain. It can be non-solvable, non-infra-invariant and non-normal. There exist non-Abelian $ ro $- groups without proper convex subgroups.

A $ ro $- group $ G $ is Archimedean if for any positive elements $ x,y \in G $ there exists a positive integer $ n $ such that $ x ^ {n} > y $. An Archimedean $ ro $- group is order-isomorphic to some subgroup of the additive group $ \mathbf R $ of real numbers with the natural order. The class of Conradian $ ro $- groups, i.e., $ ro $- groups for which the system $ {\mathcal C} ( G ) $ is subnormal and the quotient groups of the jumps of $ {\mathcal C} ( G ) $ are Archimedean, is well investigated.

References

[a1] V.M. Kopytov, N.Ya. Medvedev, "The theory of lattice-ordered groups" , Kluwer Acad. Publ. (1994) (In Russian)
[a2] R.T.B. Mura, A.H. Rhemtulla, "Orderable groups" , M. Dekker (1977)
How to Cite This Entry:
Ro-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ro-group&oldid=11213
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article