Riesz group

Riesz group

Let

G

be a

po-group

and

G

+

the

positive cone

of

G

. The following are

equivalent

:

1.

G

, as a poset, sastisfies the

Riesz interpolation property

;

2.

if

x

,

y

1

,

y

2

∈

G

+

and

x

≤

y

1

⁢

y

2

, then

x

=

z

1

⁢

z

2

with

z

i

≤

y

i

for some

z

i

∈

G

+

,

i

=

1

,

2

.

The second

property

above, put it plainly, says that any

positive element

that is

bounded from above

by a

product

of positive elements, can be “decomposed” as a product of positive elements. This property is known as the

Riesz decomposition property

.

Proof.

(

1

⇒

2

)

. Given

x

≤

y

1

⁢

y

2

and

e

≤

x

,

y

1

,

y

2

. Set

r

=

y

1

-

1

⁢

x

. Then we have four

inequalities

, which can be abbreviated as

{

r

,

e

}

≤

{

x

,

y

2

}

, where each of the

elements

in the first set is less than or equal to each of the elements in the second set. By the Riesz interpolation property, we can insert an element between the sets:

{

r

,

e

}

≤

z

2

≤

{

x

,

y

2

}

. From this it is clear that

e

≤

z

2

≤

y

1

. Set

z

1

=

x

⁢

z

2

-

1

. Since

z

2

≤

x

, we have

e

≤

x

⁢

z

2

-

1

=

z

1

. Also, since

y

1

-

1

⁢

x

=

r

≤

z

2

,

z

2

-

1

≤

x

-

1

⁢

y

1

, so that

z

1

≤

x

⁢

(

x

-

1

⁢

y

1

)

=

y

1

.

(

2

⇒

1

)

. Suppose

{

a

,

b

}

≤

{

c

,

d

}

. Set

x

=

a

-

1

⁢

c

,

y

1

=

a

-

1

⁢

d

and

y

2

=

b

-

1

⁢

c

. Then

x

,

y

1

,

y

2

∈

G

+

. Since

e

≤

d

⁢

b

-

1

, we have

x

=

a

-

1

⁢

c

=

a

-

1

⁢

e

⁢

c

≤

a

-

1

⁢

(

d

⁢

b

-

1

)

⁢

c

=

(

a

-

1

⁢

d

)

⁢

(

b

-

1

⁢

c

)

=

y

1

⁢

y

2

. By the Riesz decomposition property,

a

-

1

⁢

c

=

x

=

z

1

⁢

z

2

for some

z

1

,

z

2

∈

G

with

e

≤

z

1

≤

y

1

=

a

-

1

⁢

d

and

e

≤

z

2

≤

y

2

=

b

-

1

⁢

c

. The decomposition

equality

can be rewritten as

c

=

a

⁢

z

1

⁢

z

2

, and the last two inequalities can be rewritten as

a

⁢

z

1

≤

d

and

b

⁢

z

2

≤

c

. Set

s

=

a

⁢

z

1

, so we have

a

≤

a

⁢

z

1

=

s

≤

a

⁢

z

1

⁢

z

2

=

c

. Furthermore, since

b

⁢

z

2

≤

c

=

a

⁢

z

1

⁢

z

2

, we get

b

≤

a

⁢

z

1

=

s

. Finally from

z

1

≤

a

-

1

⁢

d

, we have

s

=

a

⁢

z

1

≤

d

. Gather all the inequalities, we have finally

{

a

,

b

}

≤

s

≤

{

c

,

d

}

.
∎

Definitions

. Let

G

be a po-group.

•

G

is called an

interpolation group

if

G

satisfies

one of the two equivalent conditions in the

theorem

above.

•

G

is a

Riesz group

if

G

is a directed interpolation group. By directed we mean that

G

, as a poset, is a

directed set

.

•

G

is an

antilattice

if

G

is a Riesz group with the property that if

a

,

b

∈

G

have a

greatest lower bound

, then

a

and

b

are

comparable

.

Any lattice-ordered group is an antilattice. Here is an interpolation group that is not an l-group. Let

G

=

ℤ

×

ℤ

. Define

(

a

,

b

)

≤

(

c

,

d

)

iff

(

c

,

d

)

-

(

a

,

b

)

=

(

0

,

n

)

for some non-negative integer

n

. This order is a

partial order

. But

G

is not a

lattice

, since

(

1

,

0

)

∨

(

0

,

0

)

does not exist. However, if any two elements in

G

have either an upper bound or a lower bound, then the elements are in fact comparable. Therefore,

{

a

,

b

}

≤

{

c

,

d

}

means that

a

,

b

,

c

,

d

form a chain. So any element in the

interval

[

a

∨

b

,

c

∧

d

]

“interpolates”

{

a

,

b

}

and

{

c

,

d

}

. Note that

G

is not a Riesz group, for otherwise it would be a chain.

Title

Riesz group

Canonical

name

RieszGroup

Date of creation

2013-03-22 17:09:18

Last modified on

2013-03-22 17:09:18

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06F20

Classification

msc 20F60

Defines

Riesz decomposition property

Defines

interpolation group

Defines

antilattice

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