On trees and tree dimension of ordered sets (Q1318361)

We call an ordered set \((X,\leq)\) a tree if no pair of incomparable elements of \(X\) has an upper bound. We denote by \(\text{St}(X)\) the set of stalagmites of \((X,\leq)\) which have a greatest element. Theorem. Let \((X,\leq)\) be an ordered set, \((Z,\leq)\) a tree and \(\tau: Z\to X\) an order-preserving surjection such that for every tree \((Y,\leq)\) and every order-preserving surjection \(f: Y\to X\) there exists an order- preserving mapping \(g: Y\to Z\) such that \(f= g\tau\). Then \((Z,\leq)\) contains a subset which, with respect to the induced order, is isomorphic to \((\text{St}(X),\leq)\). We define the tree dimension \(\text{td}(X,\leq)\) of an ordered set as the minimal number of extensions of \((X,\leq)\) which are trees such that the given order is the intersection of those tree orders. The author characterizes the tree dimension, describes relations between dimension and tree dimension and gives removal theorems.