partially ordered group

partially ordered group

A

partially ordered group

is a group

G

that is a poset at the same time, such that if

a

,

b

∈

G

and

a

≤

b

, then

1.

a

⁢

c

≤

b

⁢

c

, and

2.

c

⁢

a

≤

c

⁢

b

,

for any

c

∈

G

. The two conditions are

equivalent

to the one condition

c

⁢

a

⁢

d

≤

c

⁢

b

⁢

d

for all

c

,

d

∈

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

≤

b

, then

b

-

1

≤

a

-

1

. To see this, left multiply by the first

inequality

by

a

-

1

on both sides to obtain

e

≤

a

-

1

⁢

b

. Then right multiply the resulting inequality on both sides by

b

-

1

to obtain the desired inequality:

b

-

1

≤

a

-

1

.

•

If can be seen that for every

a

∈

G

, the

automorphisms

L

a

,

R

a

:

G

→

G

also preserve order, and hence are order automorphisms as well. For

instance

, if

b

≤

c

, then

L

a

⁢

(

b

)

=

a

⁢

b

≤

a

⁢

c

=

L

a

⁢

(

c

)

.

•

A

element

a

in a po-group

G

is said to be

positive

if

e

≤

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)

(a)

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

∈

G

, then there is a

c

∈

G

such that

a

≤

c

and

b

≤

c

, so that

a

⁢

c

-

1

⁢

b

≤

a

and

a

⁢

c

-

1

⁢

b

≤

b

as well.

*

Also, if

G

is directed, then

G

=

⟨

G

+

⟩

: for any

x

∈

G

, let

a

be the

upper bound

of

{

x

,

e

}

and let

b

=

a

⁢

x

-

1

. Then

e

≤

b

and

x

=

a

-

1

⁢

b

∈

⟨

G

+

⟩

.

(b)

A po-group whose underlying poset is a

lattice

is called a

lattice ordered group

, or an

l-group

.

(c)

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

.

(d)

A po-group is said to be

Archimedean

if

a

n

≤

b

for all

n

∈

ℤ

, then

a

=

e

. Equivalently, if

a

≠

e

, then for any

b

∈

G

, there is some

n

∈

ℤ

such that

b

<

a

n

. This is a

generalization

of the

Archimedean property

on the reals: if

r

∈

ℝ

, then there is some

n

∈

ℕ

such that

r

<

n

. To see this, pick

b

=

r

, and

a

=

1

.

(e)

A po-group is said to be

integrally closed

if

a

n

≤

b

for all

n

≥

1

, then

a

≤

e

. An integrally closed group is Archimedean: if

a

n

≤

b

for all

n

∈

ℤ

, then

a

≤

e

and

e

≤

b

. Since we also have

(

a

-

1

)

-

n

≤

b

for all

n

<

0

, this implies

a

-

1

≤

e

, or

e

≤

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.

Title

partially ordered group

Canonical name

PartiallyOrderedGroup

Date of creation

2013-03-22 16:42:25

Last modified on

2013-03-22 16:42:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

14

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06F05

Classification

msc 06F20

Classification

msc 06F15

Classification

msc 20F60

Synonym

po-group

Synonym

l-group

Synonym

Archimedean po-group

Synonym

integrally closed po-group

Synonym

po-semigroup

Synonym

lattice-ordered group

Synonym

l-semigroup

Related topic

OrderedGroup

Defines

directed group

Defines

positive element

Defines

positive cone

Defines

lattice ordered group

Defines

Archimedean partially ordered group

Defines

integrally closed group

Defines

integrally closed partially ordered group

Defines

partially ordered semigroup

Defines

lattice ordered semigroup

Defines

Archimedean

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