1.1. Wedges and cones 7
Of course, in any ordered vector space x/\x x\Jx x. In very
special ordered vector spaces any finite set of points has a supremum and
an infimum. Other ordered vector spaces that are equally rare have the
property that some pairs of vectors do not have a supremum or an infimum
while other pairs do have least upper and greatest lower bounds. In most
ordered vector spaces for each pair of points x, y that are not comparable
(i.e., x ^ y ^ x) the supremum x\f y and infimum xAt/do not exist.
There is a useful geometric interpreta-
tion of the supremum and the infimum of
a set of two points. Suppose that K is a
cone of a vector space L. Then for any two
vectors x,y G L there exists some z G L sat-
isfying (x + /C) D (y + /C) = z + K, if and only
if z x V y. Similarly, there exists some
w G L such that (y /C) fl (x K) = w K
if and only if w = x A y. Notice that in the
companion figure xVy exists but x\f u does not exist. We shall return to
the lattice properties in Section 1.3.
We continue by recalling the notions of monotone nets. A net {xa}
in a partially ordered set is said to be increasing, written xa t (resp.
decreasing, written xa j) if a (3 implies xa x@ (resp. xa xp). The
symbol xa] x means that xa | and x = sup{xa}. Likewise the notation
x
a
| x (resp. x xa I) means that the net {xa} is increasing (resp.
decreasing) and xa x (resp. x xa) holds for each index a. The meaning
xa | x is analogous. The decreasing and increasing nets are referred to as
monotone nets.
A nonempty subset D of a partially ordered set is said to be directed
upward, in symbols D], if for every pair x,y G D there exists some z G D
satisfying x z and y z. The directed downward sets are defined
similarly.
Next, we shall indicate how one can obtain new ordered vector spaces
from old ones. First, notice that every subspace of an ordered vector space
can be viewed as an ordered vector space in its own right. Indeed, if Y is a
vector subspace of an ordered vector space X, then it should be clear that Y
equipped with the induced order from X is automatically an ordered vector
space. Its positive cone is simply the cone Y+ = Y D X+. Unless otherwise
stated, the vector subspaces of ordered vector spaces will be considered as
ordered vector spaces with the induced vector ordering.
Another way of constructing ordered vector spaces is by means of Carte-
sian products. If {(Li,L^~)}iei is a family of ordered vector spaces, then it
should be easy to see that the product L
+
= Yliei ^+
ls a c o n e
°f the
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