example of tree (set theoretic)

The set + is a tree with <T=<. This isn’t a very interesting tree, since it simply consists of a line of nodes. However note that the height is ω even though no particular node has that height.

A more interesting tree using + defines m<Tn if ia=m and ib=n for some i,a,b+{0}. Then 1 is the root, and all numbers which are not powers of another number are in T1. Then all squares (which are not also fourth powers) for T2, and so on.

To illustrate the concept of a cofinal branch, observe that for any limit ordinalMathworldPlanetmath κ we can construct a κ-tree which has no cofinal branches. We let T={(α,β)|α<β<κ} and (α1,β1)<T(α2,β2)α1<α2β1=β2. The tree then has κ disjoint branches, each consisting of the set {(α,β)|α<β} for some β<κ. No branch is cofinal, since each branch is capped at β elements, but for any γ<κ, there is a branch of height γ+1. Hence the supremum of the heights is κ.

Title example of tree (set theoretic)
Canonical name ExampleOfTreesetTheoretic
Date of creation 2013-03-22 12:52:27
Last modified on 2013-03-22 12:52:27
Owner uzeromay (4983)
Last modified by uzeromay (4983)
Numerical id 5
Author uzeromay (4983)
Entry type Example
Classification msc 05C05
Classification msc 03E05
Related topic CofinalBranch