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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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| + | $#A+1 = 31 n = 0 |
| + | $#C+1 = 31 : ~/encyclopedia/old_files/data/P071/P.0701710 Partially ordered group |
| + | Automatically converted into TeX, above some diagnostics. |
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| + | if TeX found to be correct. |
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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
| + | A [[Group|group]] $ G $ |
| + | on which a [[Partial order|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 $. |
| | | |
− | <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>
| + | 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. |
| | | |
− | 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" />. | + | 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|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=48137
This article was adapted from an original article by V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article