On Scattered Posets with Finite Dimension

We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains. Introduction and presentation of the results A fundamental result, due to Szpilrajn [26], 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 [11]). Specialization of Szpilrajn’s result to several types of orders have 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 and Pouzet [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 posets whose order is the intersection of finitely many 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, does 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, does this order the intersection of two scattered linear orders? Question 1 is a special instance of the following general qestion: 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 obstruction. Let n be a non negative integer; let L(n), resp. LS(n) be the class of posets P whose order is the 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). Date: December 9, 2008. 2000 Mathematics Subject Classification. 06 A06, 06 A15, 54G12.