1.1. Wedges and cones 3 Recall that a binary relation on a set X is said to be an order relation on the set X (and X equipped with is called a partially ordered set) if it satisfies the following three properties: (i) reflexivity: x x for all x G l , (ii) antisymmetry: x y and y x imply x y, (iii) transitivity: x y and y z imply x z. If a binary relation on a set X satisfies properties (i) and (iii), i.e., if is reflexive and transitive but not necessarily antisymmetric, then is called a pre-order relation on the set X (and X equipped with is called a pre-partially ordered set). Definition 1.3. In a pre-ordered set, the notation x y means x y and x y^ y. We shall also express x y by saying that x dominates y, and when x y is true, we shall say that x strictly dominates y. Alternatively, the symbol x y will be denoted by y x, and x y will be denoted by y x. A subset A of a pre-ordered set is said to majorize another set B if for each b e B there exists some a G A such that a b. Wedges are associated with vector pre-orderings and cones with vector orderings. An order relation (resp. a pre-order relation) on a vector space L is said to be a vector ordering (resp. a vector pre-ordering) if, in addition to being reflexive, antisymmetric and transitive (resp. reflexive and transitive), is also compatible with the algebraic structure of L in the sense that if x y, then (a) x + z y + z for each z G L and (b) ax ay for all a 0. A vector space L equipped with a vector ordering is called an ordered vector space or a partially ordered vector space, denoted (L, ) or simply L if is well understood. In an ordered vector space (L, ) any vector satisfying x 0 is known as a positive vector and the collection of all positive vectors L+ = {x G L: x 0} is referred to as the positive cone or simply as the cone of L. Notice that L + is indeed a cone in L. Similar definitions can be given for a vector pre-ordering. The cone L+ also will be denoted by L + . An arbitrary cone K of a vector space X defines a vector ordering on X by letting x y whenever x y G /C. Equivalently, x y whenever x G y + /C or y G x /C see the companion figure. It is easy to check that is indeed a vector ordering whose positive cone coincides with /C, that is, X + /C. Consequently,
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