Finite dimensional scattered posets

We discuss a possible characterization, bymeans of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains. © 2013 Elsevier Ltd. All rights reserved. 0. Introduction and presentation of the results A fundamental result, due to Szpilrajn [33], states that every order on a set is the intersection of a family of linear orders on this set. The dimension of the order, also called the dimension of the ordered set, is then defined as the minimum cardinality of such a family (Dushnik, Miller [13]). Specialization of Szpilrajn’s result to several types of orders has been studied [3]. An ordered set (in short poset), or its order, is scattered if it does not contain a subset which is ordered as the chain η of rational numbers. Bonnet andPouzet [2] proved that a poset is scattered if and only if the order is the intersection of scattered linear orders. It turns out that there are scattered posetswhose order is the intersection of finitelymany linear orders but which cannot be the intersection of finitely many scattered linear orders. We give nine examples in Theorem 1. This naturally leads to the following question: Question 1. If an order is the intersection of finitely many linear scattered orders, is this order the intersection of n many scattered linear orders, where n is the dimension of this order? We do not have the answer even for dimension two orders. We cannot even answer this: Question 2. If an order of dimension two is the intersection of three scattered linear orders, is this order the intersection of two scattered linear orders? ✩ Research done under the auspices of the CMCU Franco-Tunisien program ‘‘Outils mathématiques pour l’informatique’’. This research was completed while the authors visited each other. The support provided by the University of Ottawa is gratefully acknowledged. E-mail addresses: [email protected] (M. Pouzet), [email protected] (H. Si Kaddour), [email protected] (N. Zaguia). 0195-6698/$ – see front matter© 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ejc.2013.07.008 80 M. Pouzet et al. / European Journal of Combinatorics 37 (2014) 79–99 Question 1 is a special instance of the following general question: Given a positive integer n, which orders are intersection of at most n scattered linear orders? We propose an approach based on the notion of an obstruction. For a positive integer n denote by L(n), resp. LS(n), the class of posets P whose order is the intersection of at most n linear orders, resp. at most n scattered linear orders. Set L(< ω) :=  n<ω L(n) and LS(< ω) :=  n<ω LS(n). All the four classes are closed under embeddability, that is if C is one of the above classes, then a poset Q belongs to C whenever it is embeddable in some P ∈ C (that is Q is isomorphic to an induced subposet of P). An obstruction to a class C is any poset not belonging to C. Then such a class C can be characterized by obstructions as the class of posets embedding no obstruction to C. But, it can be also characterized by means of smaller collections of obstructions. If B is a class of posets, denote by Forb(B) the class of posets in which no member of B is embeddable. With this terminology, we may ask: Find B as simple as possible such that LS(n) = Forb(B). If n = 1, then since LS(1) is the class of scattered chains, we may take B = {η}. If n ≥ 2, the following question emerges immediately: Question 3. Is there a cardinal λ such that every obstruction to LS(n) contains an obstruction of size at most λ? As it can be easily seen, the existence of such a cardinal for an arbitrary class closed under embeddability follows readily from the Vopěnka principle, a strong set theoretical principle possibly inconsistent with usual set-theoretical axioms. It implies the existence of large cardinal numbers (e.g. supercompact cardinals) and its consistency is implied by the existence of huge cardinals (see [19] pp. 413–415). In the case of LS(n), we conjecture that λ is countable. After the submission of this paper, it has been proved true for n = 2 [23]. The same question for L(n) has a simple answer: each obstruction contains a finite one. Indeed, as it is well known, a poset P belongs to L(n) whenever for every finite subset A of P the poset induced by P on A is also in L(n) (this striking fact is a consequence of the compactness theorem of first order logic—for a proof, see the survey [20]). LetCrit(L(n)) be the collection ofminimal obstructions (that is the collection of finite posets Q whose dimension is larger than n, whereas every proper subposet has dimension at most n), then L(n) = Forb(Crit(L(n))). Members of Crit(L(n)) have dimension n+ 1; these posets are the so-called (n + 1)-irreducible posets [35]. For n = 1, there is just one: the two element antichain. For n = 2, a complete description has been given by D. Kelly in 1972 (see [20]). For n > 2 a description seems to be hopeless; in fact, the problem to decide whether or not a finite poset belongs toL(n) is NP-complete (see [37]). IfC = L(< ω), then every obstruction contains a countable one (this easily follows from the finitary result mentioned above), hence L(< ω) = Forb(B) where B is a set of countable posets, each with a countable dimension. In terms of obstructions, Question 1 amounts to: Question 4. Does Crit(L(n)) determine LS(n) within LS(< ω)? We rather consider the following: Question 5. Can LS(< ω) be determined within L(< ω) by a finite set BS of obstructions? We provide ten examples of obstructions. All are countable and have dimension at most 3. In order to present these examples, for an ordered set P denote by P the dual of P , denote by P̌ the set P equipped with the strict order <. We denote by B(P̌) the poset defined as follows: the underlying set is P × {0, 1}, the ordering defined by (x, i) < (y, j) if i < j and x < y. This poset is called the open split of P . It is clearly bipartite, moreover B(P̌) is order-isomorphic to B(P̌). Let T2 be the infinite binary tree (that is the set of finite 0,1 sequences equipped with the prefix order) and letΩ(η) be the infinite binary tree in which each level is totally ordered by an increasing way from the left to the right (see Fig. 1 for an equivalent representation). M. Pouzet et al. / European Journal of Combinatorics 37 (2014) 79–99 81