6 1. Cones Full sets that are not convex Clearly, the intersection of a family of full sets is a full set. From this, we easily see that every subset B of L is included in a smallest (with respect to inclusion) full subset—called the full hull of B and denoted [B]. A moment's thought reveals that [B] = (B + L+)n(B-L+) = | J [x,y\. x,yeB From the first formula we conclude that (1) the full hull of a convex set is convex, and (2) the full hull of a circled set is circled.5 A subset A of an ordered vector space L is said to be bounded above if there exists some vector x (called an upper bound of A) such that a x holds true for all a G A. Similarly, A is bounded below if there exists some vector x (called a lower bound of A) satisfying x a for all a G A. A subset A of L is order bounded if A is bounded from above and below or, equivalently, if it is included in an order interval. A vector u in an ordered vector space L is called the least upper bound (or the supremum) of a nonempty subset A of L, written u = sup A, if (a) u is an upper bound of A1 i.e., a u holds for all a E i , and (b) u is the least upper bound of A in the sense that for any upper bound v of A we have u v. It should be clear from the above definition that a nonempty subset of an ordered vector space can have at most one supremum. The definition of the greatest lower bound (or the infimum) of a nonempty set A, denoted inf A, is introduced analogously. The classical lattice notations for the supremum and infimum of a finite set {xi,... ,x n } are sup{xi,... ,x n } = \J Xi and inf{xi,. ..,x n } = f\ xi. 2 = 1 1 = 1 We also write sup{x, y} = xV y and inf {x, y} = x A y. Recall that a subset S of a vector space is called circled (or balanced) whenever Ax G S holds for all x 6 S and all A G R with |A| 1.
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