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.