An infinite natural sum

As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessemberg sum is a better alternative, since it shares most of the usual properties of the addition on the naturals. A countably infinite version of the natural sum has been used in a recent paper by V\"a\"an\"anen and Wang, with applications to infinitary logics. We provide an order theoretical characterization of this operation. We show that this countable natural sum differs from the more usual infinite ordinal sum only for an initial finite "head" and agrees on the remaining infinite "tail". We show how to evaluate the countable natural sum just by computing a finite natural sum. Various kinds of infinite mixed sums of ordinals are discussed.Comment: v3 added a remark connected with surreal number

The Rank of Tree-Automatic Linear Orderings

We generalise Delhomm\'e's result that each tree-automatic ordinal is strictly below \omega^\omega^\omega{} by showing that any tree-automatic linear ordering has FC-rank strictly below \omega^\omega. We further investigate a restricted form of tree-automaticity and prove that every linear ordering which admits a tree-automatic presentation of branching complexity at most k has FC-rank strictly below \omega^k.Comment: 20 pages, 3 figure

Some natural zero one laws for ordinals below ε0

We are going to prove that every ordinal α with ε_0 > α ≥ ω^ω satisfies a natural zero one law in the following sense. For α < ε_0 let Nα be the number of occurences of ω in the Cantor normal form of α. (Nα is then the number of edges in the unordered tree which can canonically be associated with α.) We prove that for any α with ω ω  ≤ α < ε_0 and any sentence ϕ in the language of linear orders the asymptotic density of ϕ along α is an element of  {0,1}. We further show that for any such sentence ϕ the asymptotic density along ε_0 exists although this limit is in general in between 0 and 1. We also investigate corresponding asymptotic densities for ordinals below ω^ω

On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

We investigate the ordinal invariants height, length, and width of well quasi orders (WQO), with particular emphasis on width, an invariant of interest for the larger class of orders with finite antichain condition (FAC). We show that the width in the class of FAC orders is completely determined by the width in the class of WQOs, in the sense that if we know how to calculate the width of any WQO then we have a procedure to calculate the width of any given FAC order. We show how the width of WQO orders obtained via some classical constructions can sometimes be computed in a compositional way. In particular, this allows proving that every ordinal can be obtained as the width of some WQO poset. One of the difficult questions is to give a complete formula for the width of Cartesian products of WQOs. Even the width of the product of two ordinals is only known through a complex recursive formula. Although we have not given a complete answer to this question we have advanced the state of knowledge by considering some more complex special cases and in particular by calculating the width of certain products containing three factors. In the course of writing the paper we have discovered that some of the relevant literature was written on cross-purposes and some of the notions re-discovered several times. Therefore we also use the occasion to give a unified presentation of the known results

Direct sums and the Szlenk index

For $\alpha$ an ordinal and $1<p<\infty$, we determine a necessary and sufficient condition for an $\ell_p$-direct sum of operators to have Szlenk index not exceeding $\omega^\alpha$. It follows from our results that the Szlenk index of an $\ell_p$-direct sum of operators is determined in a natural way by the behaviour of the $\epsilon$-Szlenk indices of its summands. Our methods give similar results for $c_0$-direct sums.Comment: The proof of Proposition~2.4 has changed, with some of the arguments transferred to the proof of an added-in lemma, Lemma~2.8. Changes have been made to the Applications sectio