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,bS with ab, there is a cS such that acb.

Let P be a poset. Let 𝒜 be the set of all finite subsets of P. Define on 𝒜 as follows: for any A,B𝒜, AB iff ab for every aA and every bB. It is not hard to see that is a transitive relation on 𝒜. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    (𝒜,) has the interpolation property

  2. 2.

    for every pair of doubletons {a1,a2} and {b1,b2} with aibj for i,j𝟐, there is a cP such that aicbj for i,j𝟐.

  3. 3.

    for every pair of finite setsMathworldPlanetmath {a1,,an} and {b1,,bm} with aibj for i𝐧 and j𝐦, there is a cP such that aicbj for i𝐧, and j𝐦.

Here, 𝐧 denotes the set {1,,n}.


Clearly 12 and 31. To see that 23, we use inductionMathworldPlanetmath twice:

if 𝐧=𝟐=𝐦, then we are done. Now, fix 𝐧=𝟐 and induct on 𝐦 first. Let i𝟐. If aibj for j𝐦+𝟏, then aibj for j𝐦 in particular, so there is a cP such that aicbj for j𝐦 (induction step). This means aic and aibm+1. Apply 2 to get a dP with aid and dc and dbm+1. As a result, aidbj for j𝐦+𝟏.

Next, fix 𝐦 and induct on 𝐧. Let j𝐦. If aibj for i𝐧+𝟏, then aibj for i𝐧 in particular, so there is an eP such that aiebj for i𝐧 (induction step). This means an+1bj and ebj. Apply the result from the previous induction step, we find an fP such that an+1f and ef and fbj. As a result, aifbj for i𝐧+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 latticeMathworldPlanetmath 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 diagramsMathworldPlanetmath:

\xymatrix@!=40ptb1\ar@-[rd]|!"2,1";"1,2"\hole\ar@-[d]&b2\ar@-[ld]\ar@-[d]a1&a2\xymatrix@!=7pt&&& implies &&&\xymatrix@!=7ptb1\ar@-[rd]&&b2\ar@-[ld]&c\ar@-[rd]\ar@-[ld]&a1&&a2

Remark. One can generalize the Riesz interpolation property on a poset P to the countableMathworldPlanetmath interpolation property, if 𝒜 is to be the set of countable subsets of P, or a universalPlanetmathPlanetmath interpolation property, if 𝒜=2P, 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