Riesz interpolation property
The interpolation property, in its most general form, may be interpreted as follows: given a set and a transitive relation defined on , we say that , or for short, has the interpolation property if for any with , there is a such that .
Let be a poset. Let be the set of all finite subsets of . Define on as follows: for any , iff for every and every . It is not hard to see that is a transitive relation on . The following are equivalent:
has the interpolation property
for every pair of doubletons and with for , there is a such that for .
for every pair of finite sets and with for and , there is a such that for , and .
Here, denotes the set .
Clearly and . To see that , we use induction twice:
if , then we are done. Now, fix and induct on first. Let . If for , then for in particular, so there is a such that for (induction step). This means and . Apply to get a with and and . As a result, for .
Next, fix and induct on . Let . If for , then for in particular, so there is an such that for (induction step). This means and . Apply the result from the previous induction step, we find an such that and and . As a result, for . ∎
In other words, if one finite set, say , is bounded above by another finite set , then there is an element that serves as an upper bound for and a lower bound for . 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:
Remark. One can generalize the Riesz interpolation property on a poset to the countable interpolation property, if is to be the set of countable subsets of , or a universal interpolation property, if , the powerset of .
|Title||Riesz interpolation property|
|Date of creation||2013-03-22 17:04:22|
|Last modified on||2013-03-22 17:04:22|
|Last modified by||CWoo (3771)|