example of tree (set theoretic)

The set $\mathbb{Z}^{+}$ 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 $\omega$ even though no particular node has that height.

A more interesting tree using $\mathbb{Z}^{+}$ defines $m<_{T}n$ if $i^{a}=m$ and $i^{b}=n$ for some $i,a,b\in\mathbb{Z}^{+}\cup\{0\}$ . Then $1$ is the root, and all numbers which are not powers of another number are in $T_{1}$ . Then all squares (which are not also fourth powers) for $T_{2}$ , and so on.

To illustrate the concept of a cofinal branch, observe that for any limit ordinal $\kappa$ we can construct a $\kappa$ -tree which has no cofinal branches. We let $T=\{(\alpha,\beta)|\alpha<\beta<\kappa\}$ and $(\alpha_{1},\beta_{1})<_{T}(\alpha_{2},\beta_{2})\leftrightarrow\alpha_{1}<% \alpha_{2}\wedge\beta_{1}=\beta_{2}$ . The tree then has $\kappa$ disjoint branches, each consisting of the set $\{(\alpha,\beta)|\alpha<\beta\}$ for some $\beta<\kappa$ . No branch is cofinal, since each branch is capped at $\beta$ elements, but for any $\gamma<\kappa$ , there is a branch of height $\gamma+1$ . Hence the supremum of the heights is $\kappa$ .

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