solid set

solid set

Let

V

be a

vector lattice

and

|

⋅

|

be the

absolute value

defined on

V

. A subset

A

⊆

V

is said to be

solid

, or

absolutely convex

, if,

|

v

|

≤

|

u

|

implies that

v

∈

A

, whenever

u

∈

A

in the first place.

From this

definition

, one deduces immediately that

0

belongs to every non-empty solid set. Also, if

a

is in a solid set, so is

a

+

, since

|

a

+

|

=

a

+

≤

a

+

+

a

-

=

|

a

|

. Similarly

a

-

∈

S

, and

|

a

|

∈

S

, as

||

a

||

=

|

a

|

. Furthermore, we have

Proposition 1

.

If

S

is a solid subspace of

V

, then

S

is a vector sublattice.

Proof.

Suppose

a

,

b

∈

S

. We want to show that

a

∧

b

∈

S

, from which we see that

a

∨

b

=

a

+

b

-

(

a

∧

b

)

∈

S

also since

S

is a vector subspace. Since both

a

∧

b

,

a

∨

b

∈

S

, we have that

S

is a

sublattice

.

To show that

a

∧

b

∈

S

, we need to find

c

∈

S

with

|

a

∧

b

|

≤

|

c

|

. Let

c

=

|

a

|

+

|

b

|

. Since

a

,

b

∈

S

,

|

a

|

,

|

b

|

∈

S

, and so

c

∈

S

as well. We also have that

|

c

|

=

c

. So to show

a

∧

b

∈

S

, it is enough to show that

|

a

∧

b

|

≤

c

. To this end, note first that

a

≤

|

a

|

and

b

≤

|

b

|

, so

a

∧

b

≤

|

a

|

∧

|

b

|

≤

|

a

|

∨

|

b

|

. Also, since

-

a

≤

|

a

|

and

-

b

≤

|

b

|

,

-

(

a

∧

b

)

=

(

-

a

)

∨

(

-

b

)

≤

|

a

|

∨

|

b

|

. As a result,

|

a

∧

b

|

=

-

(

a

∧

b

)

∨

(

a

∧

b

)

≤

|

a

|

∨

|

b

|

. But

|

a

|

∨

|

b

|

≤

|

a

|

∨

|

b

|

+

|

a

|

∧

|

b

|

=

|

a

|

+

|

b

|

=

c

, we have that

|

a

∧

b

|

≤

|

a

|

∨

|

b

|

≤

c

.
∎

Examples

Let

V

be a vector lattice.

•

0

and

V

itself are solid subspaces.

•

If

V

is finite dimensional, the only solid subspaces are the improper ones.

•

An example of a proper solid subspace of a vector lattice is found, when we take

V

to be the

countably infinite

direct product

of

ℝ

, and

S

to be the countably infinite

direct sum

of

ℝ

.

•

An example of a solid set that is not a subspace is the

unit disk

in

ℝ

2

, where the

ordering

is defined componentwise.

•

Given any set

A

, the smallest solid set containing

A

is called the

solid closure

of

A

. For example, if

A

=

{

a

}

, then its solid closure is

{

v

∈

V

∣

|

v

|

≤

|

a

|

}

. In

ℝ

2

, the solid closure of any point

p

is the disk centered at

O

whose radius is

|

p

|

.

•

The solid closure of

V

+

, the

positive cone

, is

V

.

Proposition 2

.

If

V

is a vector lattice and

S

is a solid subspace of

V

, then

V

/

S

is a vector lattice.

Proof.

Since

S

is a subspace

V

/

S

has the

structure

of a

vector space

, whose vector space

operations

are inherited from the operations on

V

. Since

S

is solid, it is a sublattice, so that

V

/

S

has the structure of a

lattice

, whose lattice operations are inherited from those on

V

. It remains to show that the partial ordering is “

compatible

” with the vector operatons. We break this down into two steps:

•

for any

u

+

S

,

v

+

S

,

w

+

S

∈

V

/

S

, if

(

u

+

S

)

≤

(

v

+

S

)

, then

(

u

+

S

)

+

(

w

+

S

)

≤

(

v

+

S

)

+

(

w

+

S

)

. This is a disguised form of the following: if

u

-

v

≤

a

∈

S

, then

(

u

+

w

)

-

(

v

+

w

)

≤

b

∈

S

for some

b

. This is

obvious

: just pick

b

=

a

.

•

if

0

+

S

≤

u

+

S

∈

V

/

S

, then for any

0

<

λ

∈

k

(

k

an

ordered field

),

0

+

S

≤

λ

⁢

(

u

+

S

)

. This is the same as saying: if

c

≤

u

for some

b

∈

S

, then

d

≤

λ

⁢

u

for some

d

∈

S

. This is also obvious: pick

d

=

λ

⁢

c

.

The proof is now

complete

.
∎

Title

solid set

Canonical

name

SolidSet

Date of creation

2013-03-22 17:03:19

Last modified on

2013-03-22 17:03:19

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06F20

Classification

msc 46A40

Synonym

absolutely convex

Defines

vector lattice homomorphism

Defines

solid closure

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