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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