Tree (set theory)

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In set theory, a tree is a partially ordered set (poset) (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation <. For each t ∈ T, the order type of {s ∈ T : s < t} is called the idxheight, or 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 (as the typical questions that are investigated in this field are easily reduced to questions about single-rooted trees).

Trees with a single root in which each element has finite height can be naturally viewed as rooted trees in the graph-theoretical sense: there is an edge from x to y if and only y is a direct successor of x (i.e., x<y, but there is no element between x and y). However, if T is a tree of height > ω, then there is no natural edge relation that will make T a tree in the sense of graph theory.

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. A tree is a κ-tree if and only if it has height κ and every level has size less than κ. For each ordinal α, the α-th level of T is the set of all elements of T of height α. The width of a tree is the supremum of the cardinalities of its levels.

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. König proved that 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. 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 (i.e. not regular) 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).

Note: 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.