104 L
.A. Sholomov Mathematical Social Sciences 39 2000 81 –107
2
all factors having types a or g, the class Q is formed by all factors of types b or
ˆ h, and all factors of types d 2 f , j produce the class Q. The classes do not contain
21
9
the factors r r of type c only. The result of operation q
+ q on elementary factors q
i i i
i i
ˆ ˆ
9 9
9
and q is determined in the following way. If q [ Q or q [ Q then q + q 5 1. Now, let
i i
i i
i 1
2
ˆ
9 9
q , q [
⁄ Q. In this case, if q and q are contained in the same class Q
or Q then
i i
i i
1 2
9 9
9
q + q 5 q
∧ q . If they belong to different classes Q
and Q then q
+ q 5 1. If some
i i
i i
i i
21
ˆ
9 9
9
of the factors q ,q coincides with r r and q , q [
⁄ Q then the result of q + q is
i i
i i i
i i
i
equal to the other of the factors.
9 9
For conjunctions K 5 q , . . . , q , K9 5 q , . . . , q and DNF’s
c 5 K ∨
? ? ? ∨
K ,
1 n
1 n
1 s
9 9
c9 5 K ∨
? ? ? ∨
K , the result of operation K + K9 and c + c9 are defined by 14 and
1 t
15. For the operators F 5 c and G 5 c9, we set F + G 5 c + c9 it can be proved that
the result of composition operation F + G does not depend on chosen DNF’s c and c9.
Any operator F satisfying the condition F + F 5 F is referred to as closed with respect
to composition. The role of the composition operation is clarified by the following assertion:
F [ F 7, 7 iff F is closed . In this way, the explicit description of all operators
n
7 →
7, mentioned in the statement a, can be found.
12. Summary table
Recall previously introduced notations:
J J
ˆ ˆ
ˆ LL , L9L 9, L L are a linear hierarchy generalized, a partial LH general-
ized, a LH on the set J generalized, respectively—Section 8; ˆ
TT is a partial transitive LH generalized—Section 11; ˆ
QQ is a weighted majority operator generalized with right of veto—Section 10;
1
GG is any monotone operator;
a 21
r a [
h1, 2 1j is r at a 5 1 and is r at
a 5 2 1.
n
Now, we shall present the obtained results on explicit forms of operators 5 →
5
1 2
for all classes 5 , 5 of 3 provided 5 5 in Table 3. Rows and columns of this
1 2
1 2
table are associated with classes 5 and 5 , respectively. If condition 5 5 is false
1 2
1 2
the corresponding entry the square contains the line. In the case of monotone operators F, it is necessary to all entries of the table except
Table 3 Domain
Range +
6 3
7 ]
a a
a a
a a
a a
i i
i
+ r
r r
r r
r , r r G
s d
i i
i i
i i
i i
i i
i J
ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ –
L 9 L 9
L 9 L
L , L L G
s d
i i
i i
i i
a a
a a
i i
ˆ 6
– –
r r
r r
Q G
i i
i i
i i
a a
a a
i i
– –
– r
r r
r G
i i
i i
i i
a a
a i
i
3 –
– –
– r
r r G
i i
i i
i
ˆ 7
– –
– –
– T
–
i i
a
– –
– –
– –
r G
i
L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107
105
a a
i
for the ones mentioned later the following replacements: r and r
are to be replaced
i i
J
ˆ ˆ
ˆ ˆ
ˆ ˆ
by r ; instead of generalized operators L, L , L 9, L , T , Q one has to use the suitable
i i
i i
J
operators L, L , L9, L , T , Q; an arbitrary operator G must be replaced by a monotone
i i
i 1
n n
operator G . Exceptions are operators + →
7 of the second type, operators 0 →
3, ]
n 21
operators 0 →
7 of the first type. First of them is equivalent to the operator r ,
i i
9
and other two are equivalent to L .
i i
13. Aggregation operators with given relation of voters power
The idea to reduce problems for aggregation operators to similar problems for ordinal
n
relations on R turns out to be useful also for design of specific operators in practical
social choice problems. In this section we consider an example of such a problem. Let N 5
h1, . . . , nj be the set of voters, and R be an irreflexive binary relation on N. The notation iRj is interpreted as ‘the voter i is stronger than j’, and R is called the
relation of voters power . Subsets M N will be referred to as coalitions. A coalition M
1
is stronger than M , denoted by M RM , if there exists an injection j : M
→ M such
2 1
2 2
1
that
jiRi for all i [ M . With a profile r 5 r , . . . , r and two alternatives x, y [ A
2 1
n
we associate the coalition V r 5 hiuxr yj. An operator Fr is said to be compatible
xy i
with the relation R of voters power if V rRV r
⇒
xFry for all x, y, r. We will
xy yx
suppose that the operators also satisfy the normative constraints of binariness B, neutrality to alternatives N, non-imposition nI, and some rationality condition
n
5 →
5 .
1 2
We will consider the case of 5 5 0, 5 5 3, R [ 3. The operator, satisfying
1 2
chosen conditions and compatible with R, is nonunique. The intersection of all such operators will be denoted by F . It is easy to see that F is the least with respect to
R R
inclusion operator which is compatible with R and satisfies the constraints B, N, nI, and
n
condition 0 →
3. Moreover, F automatically satisfies the condition M of mono-
