4 1. Cones the vector orderings and cones are in one-to-one correspondence. That is, every cone K of a vector space X induces a vector ordering on X whose positive vectors are precisely the vectors in JC. If we wish to emphasize that the vector order on X is generated by the cone /C, then we shall denote it by /c- Similarly, if is a vector pre-ordering on a vector space X, then the set of positive vectors X+ = {x G X: x 0} is a wedge. Conversely, every wedge W of X gives rise to a vector pre-ordering o n I (by letting x y if x y G W) that satisfies X+ W. Again, if there is a need to emphasize that the vector pre-ordering on X is generated by the wedge W, then we shall denote it by yy. In summary, cones give rise to vector orderings and wedges to vector pre-orderings. It turns out that wedges are simply sums of vector subspaces with cones. The proof of the next result is straightforward and is omitted. Lemma 1.4. A nonempty convex subset W of a vector space is a wedge if and only if it is the sum of a vector subspace and a cone, that is, if and only if there exists a vector subspace V and a cone JC such that W = V + JC. Moreover, whenever W is a wedge, we have the following. (1) IfweletV = Wn(-W) and JC = {0}U(W\V), thenV is a vector subspace and JC is a cone satisfying W V + JC. (2) If W = V + JC, where V is a vector subspace and K is a cone such that V n K = {0}, then V = Wn (-W). (3) The wedge W = V + K is a cone, i.e., V = {0}, if and only if W contains no straight line passing through the origin. Now let W be a wedge of a vector space X, and let x be any vector in the vector subspace generated by W. Choose vectors xi,... , xn G W and real numbers a i , . . . , an such that x = Y17=i a ixi- ^ w e ^ ^i = {^: a i ^ 0} and I2 = {i: a^ 0}, then we have n X = ^^ a iXi = ^2 aiXi ~ ^2/{~ai)Xi ^W ~W i=l ieh ieh Prom this, it easily follows that the vector subspace generated by W is precisely W W. Definition 1.5. A wedge W of a vector space X is said to be generating if X W W, i.e., if the vector subspace generated by W coincides with X. It should be clear that a wedge is generating if and only if it majorizes the space. That is, we have the following simple result.
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