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''partially ordered group''
 
''partially ordered group''
  
A [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p1101502.png" /> endowed with a [[Partial order|partial order]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p1101503.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p1101504.png" />,
+
A [[Group|group]] $  \{ G; \cdot, \cle \} $
 +
endowed with a [[Partial order|partial order]] $  \cle $
 +
such that for all $  x,y,z,t \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/p/p110/p110150/p1101505.png" /></td> </tr></table>
+
$$
 +
x \cle y \Rightarrow zxt \cle xyt.
 +
$$
  
(Cf. also [[Partially ordered group|Partially ordered group]].) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p1101506.png" /> is the identity of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p1101507.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p1101508.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p1101509.png" /> is the positive cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015010.png" /> (cf. [[L-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015011.png" />-group]]), then the following relations hold:
+
(Cf. also [[Partially ordered group|Partially ordered group]].) If $  e $
 +
is the identity of a $  po $-
 +
group $  G $
 +
and $  P = P ( G ) = \{ {x \in G } : {x \cge e } \} $
 +
is the positive cone of $  G $(
 +
cf. [[L-group| $  l $-
 +
group]]), then the following relations hold:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015012.png" />;
+
1) $  P \cdot P \subseteq P $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015013.png" />;
+
2) $  P \cap P = \{ e \} $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015014.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015015.png" />.
+
3) $  x ^ {- 1 } Px \subseteq P $
 +
for all $  x $.
  
If, in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015016.png" />, one can find a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015017.png" /> with the properties 1)–3), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015018.png" /> can be made into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015019.png" />-group by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015020.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015021.png" />. It is correct to identify the order of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015022.png" />-group with its positive cone. One often writes a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015023.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015024.png" /> with positive cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015025.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015026.png" />.
+
If, in a group $  G $,  
 +
one can find a set $  P $
 +
with the properties 1)–3), then $  G $
 +
can be made into a $  po $-
 +
group by setting $  x \cle y $
 +
if and only if $  yx ^ {- 1 } \in P $.  
 +
It is correct to identify the order of a $  po $-
 +
group with its positive cone. One often writes a $  po $-
 +
group $  G $
 +
with positive cone $  P $
 +
as $  ( G,P ) $.
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015027.png" /> from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015028.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015029.png" /> into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015030.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015031.png" /> is an order homomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015032.png" /> is a [[Homomorphism|homomorphism]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015033.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015034.png" />,
+
A mapping $  \varphi : G \rightarrow H $
 +
from a $  po $-
 +
group $  G $
 +
into a $  po $-
 +
group $  H $
 +
is an order homomorphism if $  \varphi $
 +
is a [[Homomorphism|homomorphism]] of the group $  G $
 +
and for all $  x,y \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/p/p110/p110150/p11015035.png" /></td> </tr></table>
+
$$
 +
x \cle y \Rightarrow \varphi ( x ) \cle \varphi ( y ) .
 +
$$
  
A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015036.png" /> from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015037.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015038.png" /> into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015039.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015040.png" /> is an order homomorphism if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015041.png" />.
+
A homomorphism $  \varphi $
 +
from a $  po $-
 +
group $  ( G,P ) $
 +
into a $  po $-
 +
group $  ( H,Q ) $
 +
is an order homomorphism if and only if $  \varphi ( P ) \subseteq Q $.
  
A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015042.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015043.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015044.png" /> is called convex (cf. [[Convex subgroup|Convex subgroup]]) if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015045.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015046.png" />,
+
A subgroup $  H $
 +
of a $  po $-
 +
group $  G $
 +
is called convex (cf. [[Convex subgroup|Convex subgroup]]) if for all $  x,y,z $
 +
with $  x,z \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/p/p110/p110150/p11015047.png" /></td> </tr></table>
+
$$
 +
x \cle y \cle z \Rightarrow y \in H.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015048.png" /> is a convex subgroup of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015049.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015050.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015051.png" /> of right cosets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015052.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015053.png" /> is a [[Partially ordered set|partially ordered set]] with the induced order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015054.png" /> if there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015056.png" />. The quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015057.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015058.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015059.png" /> by a convex normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015060.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015061.png" />-group respect with the induced partial order, and the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015062.png" /> is an order homomorphism. The homomorphism theorem holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015063.png" />-groups: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015064.png" /> is an order homomorphism from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015065.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015066.png" /> into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015067.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015068.png" />, then the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015069.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015070.png" /> is a convex [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015071.png" /> and there exists an order isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015072.png" /> from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015073.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015074.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015075.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015076.png" />.
+
If $  H $
 +
is a convex subgroup of a $  po $-
 +
group $  G $,  
 +
then the set $  G/H $
 +
of right cosets of $  G $
 +
by $  H $
 +
is a [[Partially ordered set|partially ordered set]] with the induced order $  Hx \cle Hy $
 +
if there exists an $  h \in H $
 +
such that $  x \cle hy $.  
 +
The quotient group $  G/H $
 +
of a $  po $-
 +
group $  G $
 +
by a convex normal subgroup $  H $
 +
is a $  po $-
 +
group respect with the induced partial order, and the natural homomorphism $  \tau : G \rightarrow {G/H } $
 +
is an order homomorphism. The homomorphism theorem holds for $  po $-
 +
groups: if $  \varphi $
 +
is an order homomorphism from a $  po $-
 +
group $  G $
 +
into a $  po $-
 +
group $  H $,  
 +
then the kernel $  N = \{ {x \in G } : {\varphi ( x ) = e } \} $
 +
of $  \varphi $
 +
is a convex [[Normal subgroup|normal subgroup]] of $  G $
 +
and there exists an order isomorphism $  \psi $
 +
from the $  po $-
 +
group $  G/N $
 +
into $  H $
 +
such that $  \varphi = \tau \psi $.
  
The most important classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015077.png" />-groups are the class of lattice-ordered groups (cf. [[L-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015078.png" />-group]]) and the class of totally ordered groups (cf. [[O-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110150/p11015079.png" />-group]]).
+
The most important classes of $  po $-
 +
groups are the class of lattice-ordered groups (cf. [[L-group| $  l $-
 +
group]]) and the class of totally ordered groups (cf. [[O-group| $  o $-
 +
group]]).
  
 
This article extends and updates the article [[Partially ordered group|Partially ordered group]] (Volume 7).
 
This article extends and updates the article [[Partially ordered group|Partially ordered group]] (Volume 7).

Latest revision as of 08:06, 6 June 2020


partially ordered group

A group $ \{ G; \cdot, \cle \} $ endowed with a partial order $ \cle $ such that for all $ x,y,z,t \in G $,

$$ x \cle y \Rightarrow zxt \cle xyt. $$

(Cf. also Partially ordered group.) If $ e $ is the identity of a $ po $- group $ G $ and $ P = P ( G ) = \{ {x \in G } : {x \cge e } \} $ is the positive cone of $ G $( cf. $ l $- group), then the following relations hold:

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

2) $ P \cap P = \{ e \} $;

3) $ x ^ {- 1 } Px \subseteq P $ for all $ x $.

If, in a group $ G $, one can find a set $ P $ with the properties 1)–3), then $ G $ can be made into a $ po $- group by setting $ x \cle y $ if and only if $ yx ^ {- 1 } \in P $. It is correct to identify the order of a $ po $- group with its positive cone. One often writes a $ po $- group $ G $ with positive cone $ P $ as $ ( G,P ) $.

A mapping $ \varphi : G \rightarrow H $ from a $ po $- group $ G $ into a $ po $- group $ H $ is an order homomorphism if $ \varphi $ is a homomorphism of the group $ G $ and for all $ x,y \in G $,

$$ x \cle y \Rightarrow \varphi ( x ) \cle \varphi ( y ) . $$

A homomorphism $ \varphi $ from a $ po $- group $ ( G,P ) $ into a $ po $- group $ ( H,Q ) $ is an order homomorphism if and only if $ \varphi ( P ) \subseteq Q $.

A subgroup $ H $ of a $ po $- group $ G $ is called convex (cf. Convex subgroup) if for all $ x,y,z $ with $ x,z \in H $,

$$ x \cle y \cle z \Rightarrow y \in H. $$

If $ H $ is a convex subgroup of a $ po $- group $ G $, then the set $ G/H $ of right cosets of $ G $ by $ H $ is a partially ordered set with the induced order $ Hx \cle Hy $ if there exists an $ h \in H $ such that $ x \cle hy $. The quotient group $ G/H $ of a $ po $- group $ G $ by a convex normal subgroup $ H $ is a $ po $- group respect with the induced partial order, and the natural homomorphism $ \tau : G \rightarrow {G/H } $ is an order homomorphism. The homomorphism theorem holds for $ po $- groups: if $ \varphi $ is an order homomorphism from a $ po $- group $ G $ into a $ po $- group $ H $, then the kernel $ N = \{ {x \in G } : {\varphi ( x ) = e } \} $ of $ \varphi $ is a convex normal subgroup of $ G $ and there exists an order isomorphism $ \psi $ from the $ po $- group $ G/N $ into $ H $ such that $ \varphi = \tau \psi $.

The most important classes of $ po $- groups are the class of lattice-ordered groups (cf. $ l $- group) and the class of totally ordered groups (cf. $ o $- group).

This article extends and updates the article Partially ordered group (Volume 7).

References

[a1] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
How to Cite This Entry:
Po-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Po-group&oldid=11826
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article