tree (set theory)

In set theory, a tree is a partially ordered set (''T'', <) such that for each ''t'' ∈ ''T'', the set is well-ordered by the relation <. Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.

Definition

A tree is a partially ordered set (poset) (''T'', <) such that for each ''t'' ∈ ''T'', the set is well-ordered by the relation <. In particular, each well-ordered set (''T'', <) is a tree. For each ''t'' ∈ ''T'', the order type of is called the ''height'' of ''t'', denoted ht(''t'', ''T''). The ''height'' of ''T'' itself is the least ordinal greater than the height of each element of ''T''. A ''root'' of a tree ''T'' is an element of height 0. Frequently trees are assumed to have only one root. Trees in set theory are often defined to grow downward making the root the greatest node.

Trees with a single root may be viewed as rooted trees in the sense of graph theory in one of two ways: either as a tree (graph theory) or as a trivially perfect graph. In the first case, the graph is the undirected Hasse diagram of the partially ordered set, and in the second case, the graph is simply the underlying (undirected) graph of the partially ordered set. However, if ''T'' is a tree of height > ω, then the Hasse diagram definition does not work. For example, the partially ordered set $\omega + 1 = \left\$ does not have a Hasse Diagram, as there is no predecessor to ω. Hence a height of at most ω is required in this case.

A ''branch'' of a tree is a maximal chain in the tree (that is, any two elements of the branch are comparable, and any element of the tree ''not'' in the branch is incomparable with at least one element of the branch). The ''length'' of a branch is the ordinal that is order isomorphic to the branch. For each ordinal α, the ''α-th level'' of ''T'' is the set of all elements of ''T'' of height α. A tree is a κ-tree, for an ordinal number κ, if and only if it has height κ and every level has cardinality less than the cardinality of κ. The ''width'' of a tree is the supremum of the cardinalities of its levels.

Any single-rooted tree of height $\leq \omega$ forms a meet-semilattice, where meet (common ancestor) is given by maximal element of intersection of ancestors, which exists as the set of ancestors is non-empty and finite well-ordered, hence has a maximal element. Without a single root, the intersection of parents can be empty (two elements need not have common ancestors), for example $\left\$ where the elements are not comparable; while if there are an infinite number of ancestors there need not be a maximal element – for example, $\left\$ where $\omega_0, \omega_0'$ are not comparable.

A ''subtree'' of a tree $(T,<)$ is a tree $(T',<)$ where $T' \subseteq T$ and $T'$ is downward closed under $<$, i.e., if $s, t \in T$ and $s < t$ then $t \in T' \implies s \in T'$.

Set-theoretic properties

There are some fairly simply stated yet hard problems in infinite tree theory. Examples of this are the Kurepa conjecture and the Suslin conjecture. Both of these problems are known to be independent of Zermelo–Fraenkel set theory. By Kőnig's lemma, every ω-tree has an infinite branch. On the other hand, it is a theorem of ZFC that there are uncountable trees with no uncountable branches and no uncountable levels; such trees are known as Aronszajn trees. Given a cardinal number κ, a ''κ-Suslin tree'' is a tree of height κ which has no chains or antichains of size κ. In particular, if κ is singular then there exists a κ-Aronszajn tree and a κ-Suslin tree. In fact, for any infinite cardinal κ, every κ-Suslin tree is a κ-Aronszajn tree (the converse does not hold).

The Suslin conjecture was originally stated as a question about certain total orderings but it is equivalent to the statement: Every tree of height ω₁ has an antichain of cardinality ω₁ or a branch of length ω₁.

See also

* Cantor tree
* Kurepa tree
* Laver tree
* Tree (descriptive set theory)
* Continuous graph
* Prefix order

References

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* Chapter 2, Section 5.
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* {{Cite book, last = Kechris , first = Alexander S. , authorlink = Alexander S. Kechris , title = Classical Descriptive Set Theory , others = Graduate Texts in Mathematics 156 , publisher = Springer , year = 1995 , id = {{isbn, 0-387-94374-9 {{isbn, 3-540-94374-9

External links

Sets, Models and Proofs
by Ieke Moerdijk an
Jaap van Oosten
see Definition 3.1 and Exercise 56 on pp. 68–69.


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Set theory
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