Riesz interpolation property

Riesz interpolation property

The

interpolation property

, in its most general form, may be interpreted as follows: given a set

S

and a

transitive relation

⪯

defined on

S

, we say that

(

S

,

⪯

)

, or

S

for short, has the

interpolation property

if for any

a

,

b

∈

S

with

a

⪯

b

, there is a

c

∈

S

such that

a

⪯

c

⪯

b

.

Let

P

be a poset. Let

𝒜

be the set of all finite subsets of

P

. Define

⪯

on

𝒜

as follows: for any

A

,

B

∈

𝒜

,

A

⪯

B

iff

a

≤

b

for every

a

∈

A

and every

b

∈

B

. It is not hard to see that

⪯

is a transitive relation on

𝒜

. The following are

equivalent

:

1.

(

𝒜

,

⪯

)

has the interpolation property

2.

for every pair of doubletons

{

a

1

,

a

2

}

and

{

b

1

,

b

2

}

with

a

i

≤

b

j

for

i

,

j

∈

𝟐

, there is a

c

∈

P

such that

a

i

≤

c

≤

b

j

for

i

,

j

∈

𝟐

.

3.

for every pair of

finite sets

{

a

1

,

…

,

a

n

}

and

{

b

1

,

…

,

b

m

}

with

a

i

≤

b

j

for

i

∈

𝐧

and

j

∈

𝐦

, there is a

c

∈

P

such that

a

i

≤

c

≤

b

j

for

i

∈

𝐧

, and

j

∈

𝐦

.

Here,

𝐧

denotes the set

{

1

,

…

,

n

}

.

Proof.

Clearly

1

⇒

2

and

3

⇒

1

. To see that

2

⇒

3

, we use

induction

twice:

if

𝐧

=

𝟐

=

𝐦

, then we are done. Now, fix

𝐧

=

𝟐

and induct on

𝐦

first. Let

i

∈

𝟐

. If

a

i

≤

b

j

for

j

∈

𝐦

+

𝟏

, then

a

i

≤

b

j

for

j

∈

𝐦

in particular, so there is a

c

∈

P

such that

a

i

≤

c

≤

b

j

for

j

∈

𝐦

(induction step). This means

a

i

≤

c

and

a

i

≤

b

m

+

1

. Apply

2

to get a

d

∈

P

with

a

i

≤

d

and

d

≤

c

and

d

≤

b

m

+

1

. As a result,

a

i

≤

d

≤

b

j

for

j

∈

𝐦

+

𝟏

.

Next, fix

𝐦

and induct on

𝐧

. Let

j

∈

𝐦

. If

a

i

≤

b

j

for

i

∈

𝐧

+

𝟏

, then

a

i

≤

b

j

for

i

∈

𝐧

in particular, so there is an

e

∈

P

such that

a

i

≤

e

≤

b

j

for

i

∈

𝐧

(induction step). This means

a

n

+

1

≤

b

j

and

e

≤

b

j

. Apply the result from the previous induction step, we find an

f

∈

P

such that

a

n

+

1

≤

f

and

e

≤

f

and

f

≤

b

j

. As a result,

a

i

≤

f

≤

b

j

for

i

∈

𝐧

+

𝟏

.
∎

Definition

. A poset is said to have the

Riesz interpolation property

if it

satisfies

any of the three equivalent conditions above.

In other words, if one finite set, say

A

, is bounded above by another finite set

B

, then there is an

element

c

that serves as an

upper bound

for

A

and a lower bound for

B

. One readily sees that any

lattice

has the Riesz interpolation property. In fact, a poset having the Riesz interpolation property can be thought of as an intermediate

concept

between an arbitrary poset and a lattice.

A poset having the Riesz interpolation property can be illustrated by the following

Hasse diagrams

:

\xymatrix

@

!

=

40

p

t

b

1

\ar

@

-

[

r

d

]

|

!

"

2

,

1

"

;

"

1

,

2

"

\hole

\ar

@

-

[

d

]

&

b

2

\ar

@

-

[

l

d

]

\ar

@

-

[

d

]

a

1

&

a

2

\xymatrix

@

!

=

7

p

t

&

&

&

implies

&

&

&

\xymatrix

@

!

=

7

p

t

b

1

\ar

@

-

[

r

d

]

&

&

b

2

\ar

@

-

[

l

d

]

&

c

\ar

@

-

[

r

d

]

\ar

@

-

[

l

d

]

&

a

1

&

&

a

2

Remark

. One can generalize the Riesz interpolation property on a poset

P

to the

countable

interpolation property

, if

𝒜

is to be the set of countable subsets of

P

, or a

universal

interpolation property

, if

𝒜

=

2

P

, the

powerset

of

P

.

Title

Riesz interpolation property

Canonical name

RieszInterpolationProperty

Date of creation

2013-03-22 17:04:22

Last modified on

2013-03-22 17:04:22

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06F15

Classification

msc 06A99

Classification

msc 06F20

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