PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (230)
Orphanage
Unclass'd
Unproven (519)
Corrections (17)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
partially ordered group (Definition)

A partially ordered group is a group $ G$ that is a poset at the same time, such that if $ a,b\in G$ and $ a\le b$, then

  1. $ ac\le bc$, and
  2. $ ca\le cb$,
for any $ c\in G$. The two conditions are equivalent to the one condition $ cad\le cbd$ for all $ c,d\in G$. A partially ordered group is also called a po-group for short.

Remarks.

  • One of the immediate properties of a po-group is this: if $ a\le b$, then $ b^{-1}\le a^{-1}$. To see this, left multiply by the first inequality by $ a^{-1}$ on both sides to obtain $ e\le a^{-1}b$. Then right multiply the resulting inequality on both sides by $ b^{-1}$ to obtain the desired inequality: $ b^{-1}\le a^{-1}$.
  • If can be seen that for every $ a\in G$, the automorphisms $ L_a,R_a:G\to G$ also preserve order, and hence are order automorphisms as well. For instance, if $ b\le c$, then $ L_a(b)=ab\le ac = L_a(c)$.
  • A element $ a$ in a po-group $ G$ is said to be positive if $ e\le a$, where $ e$ is the identity element of $ G$. The set of positive elements in $ G$ is called the positive cone of $ G$.
  • (special po-groups)
    1. A po-group whose underlying poset is a directed set is called a directed group.
      • If $ G$ is a directed group, then $ G$ is also a filtered set: if $ a,b\in G$, then there is a $ c\in G$ such that $ a\le c$ and $ b\le c$, so that $ ac^{-1}b\le a$ and $ ac^{-1}b\le b$ as well.
      • Also, if $ G$ is directed, then $ G=\langle G^+\rangle$: for any $ x\in G$, let $ a$ be the upper bound of $ \lbrace x,e\rbrace$ and let $ b=ax^{-1}$. Then $ e\le b$ and $ x=a^{-1}b\in \langle G^+\rangle$.
    2. A po-group whose underlying poset is a lattice is called a lattice ordered group, or an l-group.
    3. If the partial order on a po-group $ G$ is a linear order, then $ G$ is called a totally ordered group, or simply an ordered group.
    4. A po-group is said to be Archimedean if $ a^n\le b$ for all $ n\in \mathbb{Z}$, then $ a=e$. Equivalently, if $ a\ne e$, then for any $ b\in G$, there is some $ n\in \mathbb{Z}$ such that $ b<a^n$. This is a generalization of the Archimedean property on the reals: if $ r\in \mathbb{R}$, then there is some $ n\in \mathbb{N}$ such that $ r<n$. To see this, pick $ b=r$, and $ a=1$.
    5. A po-group is said to be integrally closed if $ a^n\le b$ for all $ n\ge 1$, then $ a\le e$. An integrally closed group is Archimedean: if $ a^n\le b$ for all $ n\in\mathbb{Z}$, then $ a\le e$ and $ e\le b$. Since we also have $ (a^{-1})^{-n}\le b$ for all $ n<0$, this implies $ a^{-1}\le e$, or $ e\le a$. Hence $ a=e$. In fact, an directed integrally closed group is an Abelian po-group.
  • Since the definition above does not involve any specific group axioms, one can more generally introduce partial ordering on a semigroup in the same fashion. The result is called a partially ordered semigroup, or a po-semigroup for short. A lattice ordered semigroup is defined similarly.



"partially ordered group" is owned by CWoo.
(view preamble)

View style:

See Also: ordered group

Other names:  po-group, l-group, Archimedean po-group, integrally closed po-group, po-semigroup, lattice-ordered group, l-semigroup
Also defines:  directed group, positive element, positive cone, lattice ordered group, Archimedean partially ordered group, integrally closed group, integrally closed partially ordered group, partially ordered semigroup, lattice ordered semigroup, Archimedean

Attachments:
distributivity in po-groups (Definition) by CWoo
Log in to rate this entry.
(view current ratings)

Cross-references: semigroup, axioms, abelian, implies, integrally closed, reals, Archimedean property, archimedean, ordered group, totally ordered, linear order, lattice, upper bound, filtered set, directed set, identity element, positive, order, preserve, automorphisms, right, sides, inequality, properties, equivalent, poset, group
There are 10 references to this entry.

This is version 11 of partially ordered group, born on 2007-02-17, modified 2007-05-27.
Object id is 8922, canonical name is PartiallyOrderedGroup.
Accessed 5891 times total.

Classification:
AMS MSC: 20F60 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Ordered groups)
 06F15 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered groups)
 06F20 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered abelian groups, Riesz groups, ordered linear spaces)
 06F05 (Order, lattices, ordered algebraic structures :: Ordered structures :: Ordered semigroups and monoids)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)