88 L
.A. Sholomov Mathematical Social Sciences 39 2000 81 –107
The synthesis problem for operators turns out to parallel to the problem of description of so-called ordinal relations. These relations are considered in Section 4.
4. Ordinal relations
n
Let R be the n-dimensional real space. To every pair of points x5x , . . . , x and
1 n
n
y5 y , . . . , y in R we associate the n-tuple
1 n
n
Dx, y 5 sgnx 2 y , . . . , sgnx 2 x [
h 2 1, 0, 1 1j ,
1 1
n n
n
where sgnz is 11, 21, 0 if z . 0, z , 0, z 5 0, respectively. A relation r on R is
called ordinal if for all x, y, z, v Dx, y 5 Dz, v
⇒
x ry
⇔
z rv.
An ordinal relation
r is referred to as regular if xry
∧
z x
⇒
z ry where z x
⇔ z
1
x ∧
? ? ? ∧
z x .
1 n
n
Ordinal relations are often used in multicriteria choice models in which alternatives
are described by tuples x 5 x , . . . , x of estimates according to n criteria. The
1 n
following examples of regular relations are well known:
the lexicography: x ly
⇔ ix . y , x 5 y for j , i,
i i
j j
the Pareto relation: x py
⇔
x y
∧
x ± y.
˜ Let us denote by P
the set of all two-valued functions gu , . . . , u 5 gu in
3,2 1
n n
three-valued variables, g: h 2 1,0, 1 1j
→ h0,1j. Any ordinal relation r can be uniquely
˜ presented by its describing function g u [ P
: g Dx, y 5 1
⇔
x ry. And conversely,
r 3,2
r
any function g [ P describes the single ordinal relation
r for which g 5 g. It is easy
3,2 r
˜ ˜
to see that r is a regular relation iff the function g is monotone i.e. u v implies
r
˜ ˜
g u g v . We denote by M the set of all monotone functions in P
.
r r
3,2 3,2
We will use some special presentations of functions g [ P . Let us introduce
3,2
functions pu, p9u [ P in a single three-valued variable by
3,2
1 if
u 5 1, 1
if u 5 0 or u 5 1,
pu 5 p9u 5
H H
if u 5 1 or u 5 0,
if u 5 2 1.
¯ ¯
It is obvious that pu ∨
p9u 5 p9u, pu ∧
p9u 5 pu, p2u 5 p 9u, p92u 5 pu, ¯
where ∨
, ∧
and are Boolean operations .
Any function g [ P can be presented in the form
3,2
gu , . . . ,u 5 w pu , . . . , pu , p9u , . . . , p9u ,
11
1 n
1 n
1 n
˜ where
w is a Boolean function. To this end, associate to each n-tuple s 5 s , . . . ,
1 n
˜ s [
h 2 1, 0, 1 1j the conjunction K u 5 q u ∧
? ? ? ∧
q u where q u is
˜ n
s s
1 s
n s
1 n
¯ ¯
pu, pu ∧
p9u, p 9u, if s 5 1, s 5 0, s 5 2 1, respectively. It is easy to see that
˜ ˜
˜ q u 5 1
⇔ u 5
s , hence K u 5 1 ⇔
u 5 s. The presentation 11 of g can be formed
˜ s
i i
i s
i
L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107
89
˜ ˜
˜ as the disjunction of K u for all
s such that gs 5 1 if g ; 0 put w ; 0. It is easy to
˜ s
prove for monotone g that w can be taken to be monotone. Denoting pu 5 p ,
i i
˜ ˜
9 9
9
p9u 5 p , p 5 p , . . . , p , p 9 5 p , . . . , p , we will write 11 as
i i
1 n
1 n
˜ ˜ ˜
gu 5 wp, p 9.
12 As examples, let us present the describing functions of both the lexicography and the
Pareto relation:
9 9 9
9
g 5 p ∨
p p ∨
? ? ? ∨
p p ? ? ? p p ,
13
l 1
1 2
1 2
n 21 n
9 9 9
g 5 p p ? ? ? p p ∨
p ∨
? ? ? ∨
p
p 1
2 n
1 2
n
the symbols ∧
of conjunction are omitted. ¯
¯ ¯
9 9
9
A conjunction K 5 q . . . q is called elementary if q [ h p , p , p , p , p p , 1j,
1 n
i i
i i
i i
i
1 i n q 5 1 indicates the absence of ith factor. A disjunction of elementary
i
conjunctions is called a disjunctive normal form DNF. The formula 13 is an example of DNF. Any function g [ P
can be presented by some DNF for instance, as described
3,2
above.
5. Techniques for study of ordinal relations
