Riesz interpolation property
The interpolation property, in its most general form, may be interpreted as follows: given a set $S$ and a transitive relation $\u2aaf$ defined on $S$, we say that $(S,\u2aaf)$, or $S$ for short, has the interpolation property if for any $a,b\in S$ with $a\u2aafb$, there is a $c\in S$ such that $a\u2aafc\u2aafb$.
Let $P$ be a poset. Let $\mathcal{A}$ be the set of all finite subsets of $P$. Define $\u2aaf$ on $\mathcal{A}$ as follows: for any $A,B\in \mathcal{A}$, $A\u2aafB$ iff $a\le b$ for every $a\in A$ and every $b\in B$. It is not hard to see that $\u2aaf$ is a transitive relation on $\mathcal{A}$. The following are equivalent^{}:

1.
$(\mathcal{A},\u2aaf)$ has the interpolation property

2.
for every pair of doubletons $\{{a}_{1},{a}_{2}\}$ and $\{{b}_{1},{b}_{2}\}$ with ${a}_{i}\le {b}_{j}$ for $i,j\in \mathrm{\U0001d7d0}$, there is a $c\in P$ such that ${a}_{i}\le c\le {b}_{j}$ for $i,j\in \mathrm{\U0001d7d0}$.

3.
for every pair of finite sets^{} $\{{a}_{1},\mathrm{\dots},{a}_{n}\}$ and $\{{b}_{1},\mathrm{\dots},{b}_{m}\}$ with ${a}_{i}\le {b}_{j}$ for $i\in \mathbf{n}$ and $j\in \mathbf{m}$, there is a $c\in P$ such that ${a}_{i}\le c\le {b}_{j}$ for $i\in \mathbf{n}$, and $j\in \mathbf{m}$.
Here, $\mathbf{n}$ denotes the set $\{1,\mathrm{\dots},n\}$.
Proof.
Clearly $1\Rightarrow 2$ and $3\Rightarrow 1$. To see that $2\Rightarrow 3$, we use induction^{} twice:
if $\mathbf{n}=\mathrm{\U0001d7d0}=\mathbf{m}$, then we are done. Now, fix $\mathbf{n}=\mathrm{\U0001d7d0}$ and induct on $\mathbf{m}$ first. Let $i\in \mathrm{\U0001d7d0}$. If ${a}_{i}\le {b}_{j}$ for $j\in \mathbf{m}+\mathrm{\U0001d7cf}$, then ${a}_{i}\le {b}_{j}$ for $j\in \mathbf{m}$ in particular, so there is a $c\in P$ such that ${a}_{i}\le c\le {b}_{j}$ for $j\in \mathbf{m}$ (induction step). This means ${a}_{i}\le c$ and ${a}_{i}\le {b}_{m+1}$. Apply $2$ to get a $d\in P$ with ${a}_{i}\le d$ and $d\le c$ and $d\le {b}_{m+1}$. As a result, ${a}_{i}\le d\le {b}_{j}$ for $j\in \mathbf{m}+\mathrm{\U0001d7cf}$.
Next, fix $\mathbf{m}$ and induct on $\mathbf{n}$. Let $j\in \mathbf{m}$. If ${a}_{i}\le {b}_{j}$ for $i\in \mathbf{n}+\mathrm{\U0001d7cf}$, then ${a}_{i}\le {b}_{j}$ for $i\in \mathbf{n}$ in particular, so there is an $e\in P$ such that ${a}_{i}\le e\le {b}_{j}$ for $i\in \mathbf{n}$ (induction step). This means ${a}_{n+1}\le {b}_{j}$ and $e\le {b}_{j}$. Apply the result from the previous induction step, we find an $f\in P$ such that ${a}_{n+1}\le f$ and $e\le f$ and $f\le {b}_{j}$. As a result, ${a}_{i}\le f\le {b}_{j}$ for $i\in \mathbf{n}+\mathrm{\U0001d7cf}$. ∎
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^{}:
$$\text{xymatrix}\mathrm{@}!=40pt{b}_{1}\text{ar}\mathrm{@}[rd]!\mathrm{"}2,1\mathrm{"};\mathrm{"}1,2\mathrm{"}\text{hole}\text{ar}\mathrm{@}[d]\mathrm{\&}{b}_{2}\text{ar}\mathrm{@}[ld]\text{ar}\mathrm{@}[d]{a}_{1}\mathrm{\&}{a}_{2}\text{xymatrix}\mathrm{@}!=7pt\mathrm{\&}\mathrm{\&}\mathrm{\&}\text{implies}\mathrm{}\mathrm{}\mathrm{}\text{xymatrix}\mathrm{@}!=7pt{b}_{1}\text{ar}\mathrm{@}[rd]\mathrm{}\mathrm{}{b}_{2}\text{ar}\mathrm{@}[ld]\mathrm{}c\text{ar}\mathrm{@}[rd]\text{ar}\mathrm{@}[ld]\mathrm{}{a}_{1}\mathrm{}\mathrm{}{a}_{2}$$ 
Remark. One can generalize the Riesz interpolation property on a poset $P$ to the countable^{} interpolation property, if $\mathcal{A}$ is to be the set of countable subsets of $P$, or a universal^{} interpolation property, if $\mathcal{A}={2}^{P}$, the powerset of $P$.
Title  Riesz interpolation property 

Canonical name  RieszInterpolationProperty 
Date of creation  20130322 17:04:22 
Last modified on  20130322 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 