positive cone

positive cone

Let

R

be a

commutative ring

with 1. A subset

P

of

R

is called a

pre-positive cone

of

R

provided that

1.

P

+

P

⊆

P

(

P

is additively closed)

2.

P

⋅

P

⊆

P

(

P

is

multiplicatively closed

)

3.

-

1

∉

P

4.

sqr

⁡

(

R

)

:=

{

r

2

∣

r

∈

R

}

⊆

P

.

As it turns out, a field endowed with a pre-positive cone has an order

structure

. The field is called a formally real (

http://planetmath.org/FormallyRealField

), orderable, or

ordered field

. Before defining what this “order” is, let’s do some preliminary work. Let

P

0

be a pre-positive cone of a field

F

. By Zorn’s Lemma, the set of pre-positive cones extending

P

0

has a

maximal element

P

. It can be shown that

P

has two additional

properties

:

5.

P

∪

(

-

P

)

=

F

6.

P

∩

(

-

P

)

=

(

0

)

.

Proof.

First, suppose there is

a

∈

F

-

(

P

∪

(

-

P

)

)

. Let

P

¯

=

P

+

P

⁢

a

. Then

a

∈

P

¯

and so

P

is

strictly

contained

in

P

¯

. Clearly,

sqr

⁡

(

F

)

⊆

P

¯

and

P

¯

is easily seen to be additively closed. Also,

P

¯

is multiplicatively closed as the equation

(

p

1

+

q

1

⁢

a

)

⁢

(

p

2

+

q

2

⁢

a

)

=

(

p

1

⁢

p

2

+

q

1

⁢

q

2

⁢

a

2

)

+

(

p

1

⁢

q

2

+

q

1

⁢

p

2

)

⁢

a

demonstrates. Since

P

is a maximal and

P

¯

properly contains

P

,

P

¯

is not a pre-positive cone, which means

-

1

∈

P

¯

. Write

-

1

=

p

+

q

⁢

a

. Then

q

⁢

(

-

a

)

=

p

+

1

∈

P

. Since

q

∈

P

,

1

/

q

=

q

⁢

(

1

/

q

)

2

∈

P

,

-

a

=

(

1

/

q

)

⁢

(

p

+

1

)

∈

P

, contradicting the

assumption

that

a

∉

-

P

. Therefore,

P

∪

(

-

P

)

=

F

.

For the second part, suppose

a

∈

P

∩

(

-

P

)

. Since

a

∈

-

P

,

-

a

∈

P

. If

a

≠

0

, then

-

1

=

a

⁢

(

-

a

)

⁢

(

1

/

a

)

2

∈

P

, a

contradiction

.
∎

A subset

P

of a field

F

satisfying conditions 1, 2, 5 and 6 is called a

positive cone

of

F

.
A positive cone is a pre-positive cone. If

a

∈

F

, then either

a

∈

P

or

-

a

∈

P

. In either case,

a

2

∈

P

.
Next, if

-

1

∈

P

, then

1

∈

-

P

. But

1

=

1

2

∈

P

, we have

1

∈

P

∩

(

-

P

)

, contradicting Condition 6 of

P

.

Now, define a

binary relation

≤

, on

F

by:

a

≤

b

⟺

b

-

a

∈

P

It is not hard to see that

≤

is a

total order

on

F

. In

addition

, with the

additive

and multiplicative structures on

F

, we also have the
following two rules:

1.

a

≤

b

⇒

a

+

c

≤

b

+

c

2.

0

≤

a

and

0

≤

b

⇒

0

≤

a

⁢

b

.

Thus,

F

is a field ordered by

≤

.

Remark

. Positive cones may be defined for more general ordered

algebraic structures

, such as partially ordered groups, or partially ordered rings.

References

1

A. Prestel,

Lectures on Formally Real Fields

, Springer, 1984

Title

positive cone

Canonical

name

PositiveCone

Date of creation

2013-03-22 14:46:54

Last modified on

2013-03-22 14:46:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 13J25

Classification

msc 12D15

Related topic

PositivityInOrderedRing

Related topic

FormallyRealField

Defines

pre-positive cone

Generated on Fri Feb 9 17:34:28 2018 by

LaTeXML