On operations and linear extensions of well partially ordered sets (Q1765955)

The aim of the present paper is to investigate some operations and linear extensions of well partially ordered sets. In Section 1 the authors introduce some basic concepts. If \((P,\leq _{P})\) is a poset, then they denote by \(P^{*}\) the set of all finite sequences with terms in \(P\) (for an element \(t=(x_{0},x_{1},\dots,x_{n-1})\in \) \(P^{*}\), its length lh\((t)\) is \(n\) and, for all \(i\prec \text{lh}(t)\), \((t)_{i}=x_{i})\). \( P^{*}\) becomes a poset under the following two orders: \(t\subseteq s\) iff \(\text{lh}(t)\prec \text{lh}(s)\) and for all \(i\prec \text{lh}(t)\), \( (t)_{i}\leq _{P} (s)_{i}\), and \(t\subseteq ^{\prime}s\) iff there is a subsequence \(s^{\prime}\) of \(s \) with \(t\subseteq s^{\prime}\). They also construct the poset \(P^{\bigcirc }\) consisting of all weakly decreasing sequences from \(P^{*}\) with the order \(\subseteq \) inherited from \(P^{*}\). Moreover, they introduce the concepts of well-founded poset and partially well-ordered poset (in short p.w.o. set), and they establish some general results concerning these ones. In Section 2 it is shown that if \(P\) is a p.w.o. set, then \(P^{\bigcirc }\) is also a p.w.o. set. If \(P\) is a p.w.o. set, then each linear extension of \(\leq _{P}\) is a well-ordering of \(P\). Let WO\((P)\) be the family of all these extensions. Section 3 is dedicated to presenting some information on ordinals that can be achieved by elements of WO\((P)\). The main goal of Section 4 is to prove that a tree \(T\) with no antichains of size \(\mid T\mid \) has reach not much greater than its rank.