L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107
101
s
into account that the reduction of a GLH gives r , we obtain the following results about
i n
operators + →
5 perhaps, most of them are known.
a
a F [ F +, 5 , 5 [ h+, 0, 6, j iff F is equivalent to an operator r , a [ h1,
i 1
2 1 j to an operator r if F [ F +, 5 . For F [ F+, + the result is in Monjardet
i
1978, for instance.
a
s
b F [ F +, 3 iff F is equivalent to an operator r ,
a [ h1, 2 1j to an
1s k i s
s
1
operator r
if F [ F +, 3 .
1s k i
s
a
s
c F [ F +, 7 iff F is equivalent to an operator in the form r
or
1s k i
]
s
]
a 21
1
s
r , a [
h1, 2 1j r
or r
if F [ F +, 7 .
1s k i
s 1s k i
1s k i
s s
s
a
d F [ F +, iff F is equivalent to an operator r G, a [
h1, 2 1j, where G is
i 1
1
any operator in form 27 to an operator r G where G
is any monotone operator
i 1
1
in form 27, if F [ F +, . For F [ F +, the result is in Blau and Deb 1977.
n
10. Explicit form of operators 4
→ 5, 4 [
h6, , 3, j
We were not able to formulate a general theorem for these cases. However, the
n
approach to reduce the synthesis of operators 4 →
5 to an analysis of describing functions g of ordinal relations
r [ 5 can be preserved. Now we will formulate results
r
obtained by means of the approach Sholomov, 1990b, 1994.
a
s
a F [ F 4, 3 , 4 [ h6, , 3 j, iff F is equivalent to an operator
r ,
1s k i
s
1
a [ h1, 2 1j to an operator
r if F [ F 4, 3 .
s 1s k i
s
n
This fact for monotone mappings 3 →
3 only is contained in Brown 1975, and for
n
arbitrary operators 3 →
3 the result is in Vladimirov 1987. The form of operators
n
4 →
3, 4 [ h6, , 3 j, in monotone case under the additional restriction of neutrality
to voters, follows from Barthelemy 1983. The next statement is a simple corollary of this a and of a in Section 8.
a
b F [ F 4, 5 4, 5 [ h6, j, 4 5, iff F is equivalent to an operator r ,
i 1
a [ h1, 2 1j to an operator r if F [ F 4, 5 .
i n
The following proposition gives a general presentation of operators 4 →
, 4 .
a
c F [ F 4, , 4 [ h, 3, j, iff F is equivalent to an operator r G where
i 1
1
a [ h1, 2 1j, G [ F a 5 1, G [ F , if F [ F 4, .
n
Mappings 6 →
turn out to be more complicated, and they lead to a new kind of operators. At first, let us introduce a number of notions. A Boolean function fx , . . . ,
1
x is called a threshold function if there exist real numbers w , . . . , w weights and t
n 1
n
a threshold such that fx , . . . , x 5 1
⇔ w x 1 ? ? ? 1 w x t.
1 n
1 1
n n
The collection w , . . . , w , t is referred to as the realization of f. Henceforth, we
1 n
assume that the function f is monotone. Then the weights and the threshold can be taken to be nonnegative.
We denote by Th the set of all threshold functions f which have a realization with
1 2
102 L
.A. Sholomov Mathematical Social Sciences 39 2000 81 –107
t 5 w 1 ? ? ? 1 w 2. After the normalization of w , . . . , w , t, the realization is
1 n
1 n
reduced to w , . . . , w , 1 2 where w 1 . . . 1 w 5 1.
1 n
1 n
To each monotone Boolean function f 5
k x . . . x ,
i i
1 s
hi , . . . ,i j
1 s
put in correspondence the function
9
u 5 k
n p
n p
f i
j hi , . . . , i j i [hi , . . . , i j
j [ ⁄
hi , . . . , i j
1 s
1 s
1 s
of P , and denote by
Q the aggregation operator associated with u . Also denote by
3,2 f
f o
7 the set of operators
Q for all f [ Th , and by 7
the set of operators similar
1 2 f
1 2 1 2
o
to operators of 7 a similar operator comes out the initial one by replacements of
1 2 21
o
some r by r . The sense of operators
Q [ 7 can be explained in the following
i i
f 1 2
manner. Each individual i, 1 i n, has w votes, and the alternative x is preferred over
i
y in the collective decision iff none of voters prefers y and at least one half of votes
o
support x. According to that, operators of the class 7 of 7
will be called
1 2 1 2
generalized weighted majority operators with right of veto. The following statement is valid.
d F [ F 6, iff F is equivalent to an operator Q G where Q [ 7
, G [ F
1 2 o
1 1
Q [ 7 , G [ F , if F [ F 6, .
1 2
n
11. Explicit form of operators 7
