, as a poset, sastisfies the Riesz interpolation property;
if and , then with for some , .
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.
. Given and . Set . Then we have four inequalities, which can be abbreviated as , 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: . From this it is clear that . Set . Since , we have . Also, since , , so that .
. Suppose . Set , and . Then . Since , we have . By the Riesz decomposition property, for some with and . The decomposition equality can be rewritten as , and the last two inequalities can be rewritten as and . Set , so we have . Furthermore, since , we get . Finally from , we have . Gather all the inequalities, we have finally . ∎
Definitions. Let be a po-group.
is a Riesz group if is a directed interpolation group. By directed we mean that , as a poset, is a directed set.
Any lattice-ordered group is an antilattice. Here is an interpolation group that is not an l-group. Let . Define iff for some non-negative integer . This order is a partial order. But is not a lattice, since does not exist. However, if any two elements in have either an upper bound or a lower bound, then the elements are in fact comparable. Therefore, means that form a chain. So any element in the interval “interpolates” and . Note that is not a Riesz group, for otherwise it would be a chain.
|Date of creation||2013-03-22 17:09:18|
|Last modified on||2013-03-22 17:09:18|
|Last modified by||CWoo (3771)|
|Defines||Riesz decomposition property|