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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