New construction methods for copulas and the multivariate case

1 Chapter 1 Introduction This chapter introduces general concepts and relevant results for the present perspective on aggregation functions based on copulas. In particular, we discuss supermodular functions on a lattice and we explore some of their basic properties. Various common functional transfor- mations maintain or generate supermodularity and, above all, there is an equivalence between supermodularity and a standard notion of complementarity, known also as “increasing dif- ferences”. The concept of complementarity is well established in economics at least since Edgeworth and the basic idea of complementarity is that the marginal value of an action is increasing in the level of other actions avalaible. The mathematical concept of supermodular- ity formalizes the idea of complementarity and in the literature it is defined for functions on a general lattice, but our aim is to define supermodularity for aggregation functions. 1.1 Partially Ordered Sets and Lattices This section introduces and develops concepts and properties involving order and lattices, by giving also characterizations of sublattice structure. A lattice is a system of elements with two basic operations: formation of meet and formation of join, which are respectively denoted by a^ b and a_ b; this notation has been favoured by Birkhoff and MacLane. To introduce lattices we define first the relation of partial order and then partially ordered sets, including chains. Whenever discussing a general partially ordered set, the associated ordering relation is denoted . Sometimes, the same symbol may be used to denote different ordering relations on different partially ordered sets, where the particular context precludes any ambiguities. Any subset of n is taken to have as the associated ordering relation. Let L be a set of elements; then a relation of partial order over L is any dyadic relation over L which is: (i) reflexive: for every a2 L, a a; (ii) anti-symmetric: if a b and b a, then a= b; (iii) transitive: if a b and b c, then a c.