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A [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p0717101.png" /> on which a [[Partial order|partial order]] relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p0717102.png" /> is given such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p0717103.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p0717104.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p0717105.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p0717106.png" />.
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$#C+1 = 31 : ~/encyclopedia/old_files/data/P071/P.0701710 Partially ordered group
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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p0717107.png" /> in a partially ordered group is called the positive cone, or the integral part, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p0717108.png" /> and satisfies the properties: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p0717109.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171010.png" />; and 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171011.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171012.png" />. Any subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171014.png" /> that satisfies the conditions 1)–3) induces a partial order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171016.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171017.png" />) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171018.png" /> is the positive cone.
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Examples of partially ordered groups. The additive group of real numbers with the usual order relation; the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171019.png" /> of functions from an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171020.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171021.png" />, with the operation
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A [[Group|group]]  $  G $
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on which a [[Partial order|partial order]] relation $  \leq  $
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is given such that for all  $  a , b , x , y $
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in  $  G $
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the inequality  $  a \leq  b $
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implies  $  x a y \leq  x b y $.
  
<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/p071/p071710/p07171022.png" /></td> </tr></table>
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The set  $  P = \{ {x \in G } : {x \geq  1 } \} $
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in a partially ordered group is called the positive cone, or the integral part, of  $  G $
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and satisfies the properties: 1)  $  P P \subseteq P $;  
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2)  $  P \cap P  ^ {-} 1 = \{ 1 \} $;  
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and 3)  $  x  ^ {-} 1 P x \subseteq P $
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for all  $  x \in G $.  
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Any subset  $  P $
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of  $  G $
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that satisfies the conditions 1)–3) induces a partial order on  $  G $(
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$  x \leq  y $
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if and only if  $  x  ^ {-} 1 y \in P $)
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for which  $  P $
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is the positive cone.
  
and order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171023.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171025.png" />; the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171026.png" /> of all automorphisms of a totally ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171027.png" /> with respect to composition of functions, and with order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171029.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071710/p07171031.png" />.
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Examples of partially ordered groups. The additive group of real numbers with the usual order relation; the group  $  F ( X , \mathbf R ) $
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of functions from an arbitrary set  $  X $
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into  $  \mathbf R $,
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with the operation
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$$
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( f + g ) ( x)  =  f ( x) + g ( x)
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$$
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and order relation $  f \leq  g $
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if $  f ( x) \leq  g( x) $
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for all $  x \in X $;  
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the group $  A ( M) $
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of all automorphisms of a totally ordered set $  M $
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with respect to composition of functions, and with order relation $  \phi \leq  \psi $
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if $  \phi ( m) \leq  \psi ( m) $
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for all $  m \in M $,  
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where $  \phi , \psi \in A ( M) $.
  
 
The basic concepts of the theory of partially ordered groups are those of an order homomorphism (cf. [[Ordered group|Ordered group]]), a [[Convex subgroup|convex subgroup]], and Cartesian and lexicographic products.
 
The basic concepts of the theory of partially ordered groups are those of an order homomorphism (cf. [[Ordered group|Ordered group]]), a [[Convex subgroup|convex subgroup]], and Cartesian and lexicographic products.

Latest revision as of 08:05, 6 June 2020


A group $ G $ on which a partial order relation $ \leq $ is given such that for all $ a , b , x , y $ in $ G $ the inequality $ a \leq b $ implies $ x a y \leq x b y $.

The set $ P = \{ {x \in G } : {x \geq 1 } \} $ in a partially ordered group is called the positive cone, or the integral part, of $ G $ and satisfies the properties: 1) $ P P \subseteq P $; 2) $ P \cap P ^ {-} 1 = \{ 1 \} $; and 3) $ x ^ {-} 1 P x \subseteq P $ for all $ x \in G $. Any subset $ P $ of $ G $ that satisfies the conditions 1)–3) induces a partial order on $ G $( $ x \leq y $ if and only if $ x ^ {-} 1 y \in P $) for which $ P $ is the positive cone.

Examples of partially ordered groups. The additive group of real numbers with the usual order relation; the group $ F ( X , \mathbf R ) $ of functions from an arbitrary set $ X $ into $ \mathbf R $, with the operation

$$ ( f + g ) ( x) = f ( x) + g ( x) $$

and order relation $ f \leq g $ if $ f ( x) \leq g( x) $ for all $ x \in X $; the group $ A ( M) $ of all automorphisms of a totally ordered set $ M $ with respect to composition of functions, and with order relation $ \phi \leq \psi $ if $ \phi ( m) \leq \psi ( m) $ for all $ m \in M $, where $ \phi , \psi \in A ( M) $.

The basic concepts of the theory of partially ordered groups are those of an order homomorphism (cf. Ordered group), a convex subgroup, and Cartesian and lexicographic products.

Important classes of partially ordered groups are totally ordered groups and lattice-ordered groups (cf. Totally ordered group; Lattice-ordered group).

References

[1] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[2] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
How to Cite This Entry:
Partially ordered group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partially_ordered_group&oldid=13204
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article