On Better-Quasi-Ordering Countable Series-Parallel Orders

We prove that any infinite sequence of countable series-parallel orders contains an increasing (with respect to embedding) infinite subsequence. This result generalizes Laver’s and Corominas’ theorems concerning better-quasi-order of the classes of countable chains and trees. Let C be a class of structures and ≤ an order on C. This class is well-quasi-ordered with respect to ≤ if for any infinite sequence C1, C2, . . . , Ci, . . . in C, there exist i < j such that Ci ≤ Cj . An equivalent definition is: any subset of C has finitely many minimal elements with respect to ≤. We are mainly concerned here with binary relations, the order ≤ being the embedding order. Thus, R ≤ R′ when the relation R is embedded in the relation R′ (in other words, R is an induced relation of R′). One of the very first results concerning well-quasi-order is Higman’s theorem ([5]): the class of finite linear orderings (or chains) labeled by a finite set is well-quasi-ordered with respect to embedding. More precisely, the objects of the class are finite chains whose elements are labeled by a finite set, and embedding means order-preserving and label-preserving injection. For example aabcc embeds into abcabcabc but not into cbacbacba. This result was extended by Kruskal to the class of finite trees [8]. Then Nash-Williams introduced two fundamental tools of the theory: the ‘minimal bad sequence’ which shortened greatly the proofs concerning well-quasi-order (wqo), and the ‘better-quasi-order’ (bqo), a strengthening of wqo, which provides a tool to deal with countable structures [11]. Indeed, wqo is no longer an appropriate tool in the infinite case, since one can construct, from Rado’s counterexample [13], an infinite set of pairwise incomparable countable subsets of a wqo. Laver proved in [9] that the class of countable chains is better-quasi-ordered (and thus well-quasi-ordered) with respect to embedding. Later, Corominas [1] extended the result of Kruskal to countable trees: the class of countable trees labeled by a better-quasiorder is better-quasi-ordered with respect to embedding. The proof is in two parts, one devoted to the construction of countable trees, the other concerned with the preservation of better-quasi-order under certain operations. This latter aspect can be found in Milner [10] and Pouzet [12]. In this paper, Pouzet poses the problem of another class of orders, the N -free or series-parallel orders. This class contains the class of trees, and Pouzet was able to prove the wqo character of the class of finite series-parallel orders. He conjectured also that the class of countable series-parallel orders is better-quasi-ordered. We settle Subject Classification. Primary 05C20; Secondary 05C05,08A65,05C75.