Dimensions of chains of relations

Abstract

Abstract Any partial order is an intersection of linear orders, generally in many ways. The smallest number of linear orders required in such an intersection is called the dimension of the given partial order. This concept of dimension can be trivially extended to quasi orders (i.e. reflexive and transitive relations), by using weak orders instead of linear orders. A further extension to general relations is obtained by replacing weak orders with biorders (also called Ferrers relations). These notions are now classical, see e.g. Trotter, Combinatorics and Partially Ordered Sets: Dimension Theory (1992). In this talk, we introduce several concepts of dimension for chains of relations. The motivation comes from the analysis of valued relations in preference modelling. A forerunner was Ovchinnikov (1984) who however worked in another theoretical setting than ours. Valverde (1985) investigation of the transitive closure of a valued relation paved the way to a fairly general concept of dimension. After having made a choice of definitions among many possible ones, we show that any valued quasi order is an intersection of valued weak orders. Our goal is to study the combinatorics of the resulting dimension concept, which is defined thus for valued quasi-orders, or equivalently for chains of quasi-orders. A similar dimension can also be defined for any valued relation (that is, for any chain of relations) if valued biorders take the place of valued weak orders as the one-dimensional relations. This remark directly follows from Fodor and Roubens (1995). Interesting, combinatorial questions arise about the new concept of dimension for a chain of quasi orders. We stress that the dimension of the chain is not just a plain function of the dimensions of the quasi orders. For instance, if the chain has dimension at most two, all the quasi orders have also dimension at most two; however, the converse does not necessarily hold. Characterizing two-dimensional chains of quasi orders thus requires additional conditions that force the (two-dimensional) quasi orders to be coherent in some way. We describe several such conditions that are inferred from the study of transitive orientations of comparability graphs. Although progress towards a solution is reported, we leave open the problem of characterizing two-dimensional chains. Moreover, the computational complexity of the corresponding decision problem is presently unknown. The talk is based on joint work with Jutta Mitas.