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,