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Difference between revisions of "Ro-group"

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A [[Group|group]]  $  G $
 
A [[Group|group]]  $  G $
endowed with a total order  $  \cle $
+
endowed with a total order  $  \le $
 
such that for all  $  x,y,z \in G $,
 
such that for all  $  x,y,z \in G $,
  
 
$$  
 
$$  
x \cle y \Rightarrow xz \cle yz.
+
x \le y \Rightarrow xz \le yz.
 
$$
 
$$
  
If  $  P = P ( G ) = \{ {x \in G } : {x \cge e } \} $
+
If  $  P = P ( G ) = \{ {x \in G } : {x \ge e } \} $
 
is the positive cone of the  $  ro $-
 
is the positive cone of the  $  ro $-
 
group  $  G $(
 
group  $  G $(
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can given the structure of a  $  ro $-
 
can given the structure of a  $  ro $-
 
group with positive cone  $  P $
 
group with positive cone  $  P $
by a setting  $  x \cle y $
+
by a setting  $  x \le y $
 
if and only if  $  yx ^ {- 1 } \in P $.  
 
if and only if  $  yx ^ {- 1 } \in P $.  
 
The positive cone of a  $  ro $-
 
The positive cone of a  $  ro $-
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The group of order automorphisms  $  { \mathop{\rm Aut} } ( X ) $
 
The group of order automorphisms  $  { \mathop{\rm Aut} } ( X ) $
of a totally ordered set  $  \{ X; \cle \} $
+
of a totally ordered set  $  \{ X; \le \} $
 
can be turned into a  $  ro $-
 
can be turned into a  $  ro $-
group by defining the following relation  $  \cle $
+
group by defining the following relation  $  \le $
 
on it. Let  $  \prec $
 
on it. Let  $  \prec $
 
be any well ordering on  $  X $:  
 
be any well ordering on  $  X $:  
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$$  
 
$$  
P \subset  A ( X ) = \left \{ {\varphi \in { \mathop{\rm Aut} } ( X ) } : {x _  \alpha  \varphi \cge x _  \alpha  } \right \} .
+
P \subset  A ( X ) = \left \{ {\varphi \in { \mathop{\rm Aut} } ( X ) } : {x _  \alpha  \varphi \ge x _  \alpha  } \right \} .
 
$$
 
$$
  

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=53509
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article