No abstract

Given a class of linear order types C, we identify and study several different classes of trees, naturally associated with C in terms of how the paths in those trees are related to the order types belonging to C. We investigate and completely determine the set-theoretic relationships between these classes of trees and between their corresponding first-order theories. We then obtain some general results about the axiomatization of the first-order theories of some of these classes of trees in terms of the first-order theory of the generating class C, and indicate the problems obstructing such general results for the other classes. These problems arise from the possible existence of nondefinable paths in trees, that need not satisfy the first-order theory of C, so we have started analysing first-order definable and undefinable paths in trees.

Tree paths are investigated using first-order logic. The following results are obtained: (i) every definable path can be defined by a firstorder formula using at most one parameter chosen from the path itself; (ii) a canonical representation of the formulas that define definable paths is obtained; and (iii) every tree that has only finitely many paths that are not definable is n-equivalent to a tree of which all paths are definable. Moreover, a certain property that might be expected to hold, involving the transfer of n-equivalence between trees, is shown not to be true.

Trees are partial orderings where every element has a linearly ordered set of smaller elements. We define and study several natural notions of completeness of trees, extending Dedekind completeness of linear orders and Dedekind-MacNeille completions of partial orders. We then define constructions of tree completions that extend any tree to a minimal one satisfying the respective completeness property.