Riesz group

Let G be a po-group and G+ the positive conePlanetmathPlanetmathPlanetmathPlanetmath of G. The following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    G, as a poset, sastisfies the Riesz interpolation property;

  2. 2.

    if x,y1,y2G+ and xy1y2, then x=z1z2 with ziyi for some ziG+, i=1,2.

The second property above, put it plainly, says that any positive elementMathworldPlanetmath that is bounded from above by a productPlanetmathPlanetmath of positive elements, can be “decomposed” as a product of positive elements. This property is known as the Riesz decomposition property.


(12). Given xy1y2 and ex,y1,y2. Set r=y1-1x. Then we have four inequalities, which can be abbreviated as {r,e}{x,y2}, 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: {r,e}z2{x,y2}. From this it is clear that ez2y1. Set z1=xz2-1. Since z2x, we have exz2-1=z1. Also, since y1-1x=rz2, z2-1x-1y1, so that z1x(x-1y1)=y1.

(21). Suppose {a,b}{c,d}. Set x=a-1c, y1=a-1d and y2=b-1c. Then x,y1,y2G+. Since edb-1, we have x=a-1c=a-1eca-1(db-1)c=(a-1d)(b-1c)=y1y2. By the Riesz decomposition property, a-1c=x=z1z2 for some z1,z2G with ez1y1=a-1d and ez2y2=b-1c. The decomposition equality can be rewritten as c=az1z2, and the last two inequalities can be rewritten as az1d and bz2c. Set s=az1, so we have aaz1=saz1z2=c. Furthermore, since bz2c=az1z2, we get baz1=s. Finally from z1a-1d, we have s=az1d. Gather all the inequalities, we have finally {a,b}s{c,d}. ∎

Definitions. Let G be a po-group.

  • G is called an interpolation group if G satisfies one of the two equivalent conditions in the theoremMathworldPlanetmath above.

  • G is a Riesz group if G is a directed interpolation group. By directed we mean that G, as a poset, is a directed setMathworldPlanetmath.

  • G is an antilattice if G is a Riesz group with the property that if a,bG have a greatest lower boundMathworldPlanetmath, then a and b are comparablePlanetmathPlanetmath.

Any lattice-ordered group is an antilattice. Here is an interpolation group that is not an l-group. Let G=×. Define (a,b)(c,d) iff (c,d)-(a,b)=(0,n) for some non-negative integer n. This order is a partial orderMathworldPlanetmath. But G is not a latticeMathworldPlanetmath, since (1,0)(0,0) does not exist. However, if any two elements in G have either an upper bound or a lower bound, then the elements are in fact comparable. Therefore, {a,b}{c,d} means that a,b,c,d form a chain. So any element in the interval [ab,cd] “interpolates” {a,b} and {c,d}. Note that G is not a Riesz group, for otherwise it would be a chain.

Title Riesz group
Canonical name RieszGroup
Date of creation 2013-03-22 17:09:18
Last modified on 2013-03-22 17:09:18
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 06F20
Classification msc 20F60
Defines Riesz decomposition property
Defines interpolation group
Defines antilattice