
Section 1 .4 Arithmetic ope rations a n d roundings 35
Finally we show by simple examples that in general Lemma 1.29 and Theorem 1.30
are not valid in the case of a complete but not linearly ordered lattice. Let fM , g be
the complete lattice that appears in Figure 1.11 (a).
i.M/
(a) (b)
o.M/
e
c
a
b
d
f
c
ab
d
z
Figure 1.11. Roundings in nonlinearly ordered sets.
The subset fS, g consisting of the solid points in Figure 1.11 (a) obviously is a
screen of fM , g. We define a mapping
: M ! S by the following properties:
1. All screenpoints are fixed points of the mapping.
2.
a D b, c D d , e D f . See Figure 1.11 (a).
Then
is a monotone rounding. However, neither Lemma ...