On Better-Quasi-Ordering Countable Series-Parallel Orders

Abstract

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 betterquasi-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-quasiordered with respect to < if for any infinite sequence C1,, C2... , Ck, ... 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, 1? < 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 einbedding. Later, Corominas [1] extended the result of Kruskal to countable trees: the class of countable trees labeled by a better-quasi-order 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 Received by the editors October 4, 1995 and, in revised form, January 6, 1998. 1991 Mathematics Subject Classification. Primary 05C20; Secondary 05C05,08A65,05C75.