1.1. Wedges and cones
5
Lemma 1.6. A wedge W of a vector space X is generating if and only if
for each x G X there exists some y EW satisfying y yy x.
Let W be a wedge of a vector space X. We shall say that a vector e EW
is an order unit (or more precisely a W-order unit in case the wedge W
must be indicated) if for each x G X there exists some A 0 such that
x Xe. Clearly, if W has an order unit, then the wedge W is automatically
generating. Also, if e G W is an order unit, then so are ae for all a 0
and e + x for each x G W.
The order units of a wedge coincide with its internal
points.3
Lemma 1.7. Let W be a wedge of some vector space X. A vector e G W
is an order unit if and only if it is an internal point of W.
Proof. Assume first that e is an internal point of W and let x G X. Then
there exists some a 0 such that e + a(—x) 0. This implies x ^e, so
that e is an order unit.
For the converse, suppose that e is an order unit and let x G X. Fix
some Ao 0 such that Ao(—x) e. Now notice that for each 0 A Ao we
have
H-x) = (j-0)M-x)Toee-
Consequently, e + Xx 0 for all 0 A Ao, which shows that e is an
internal point of W.
There are some useful sets associated with an ordered vector space that
will play an important role in our study in this book. They are introduced
next. If x and y are two vectors in an ordered vector
space L satisfying x y, then the order interval [x, y]
is the set defined by [x,y] = {z G L: x z y} . If
x y is not true, then we let [x,y] = 0, so that the
order interval is defined for all vectors x and y.4 Clearly,
for all x and y we have
[x,y] = (x + L+)
n(y-L+)=x
+ [0,y-x}.
In particular, each order interval is a convex set.
Definition 1.8. A subset A of an ordered vector space L is said to be full
(or order-convex) if for every x,y G A we have [x,y] C A.
o
Recall that a point a £ A, where A is a subset of a vector space X, is called an internal
point of A if for each x G X there exists some Ao 0 such that a + Xx A for all 0 A Ao-
Notice that a point a G A is an internal point if and only if for each x G X there exists some
0 ao 1 such that aa + (1 a)x G A for all ao a 1.
If A is a convex set, then a vector a G A is an internal point of A if and only if for each x £ X
there exists some a 0 such that a + ax G A.
This "order interval" definition is applicable to any pre-ordered set as well.
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