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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